==== TABLE OF CONTENTS ====
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//descriptors.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//introduction.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//lines.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//main.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//moments.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//ordered-slit.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//preamble.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//ratios.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//regularization.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//resolutions.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//stopping.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//transmission.tex
Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//references.bib
===========================
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//descriptors.tex ====
\section{Finite stopping-band descriptors}\label{de:section}
Local erosion assigns a barrier to a neighborhood of each admissible
datum. We now choose finitely many of these neighborhoods covering all
but a prescribed amount of the reference probability. This is a finite
approximation in probability; the whole space of slit data need not be
compact, and no deterministic bound on every possible slit count is
required.
\begin{theorem}[Stopping-band transfer]\label{if:band-transfer}
Fix an inward or outward band, target, and ambient square chart as in
Section~\ref{if:section}, with all the stated positive clearances.
Use the critical plus-boundary law in the fixed larger reference square.
For every exceptional probability
$\epsilon>0$ and target error $\eta>0$, generic contour levels can be
chosen in any prescribed separated subwindows so that the following
hold for all sufficiently fine meshes.
\begin{enumerate}
\item The stopping observation of Lemma~\ref{if:stopping} has a finite
descriptor $Z^\delta\in\{0,1,\ldots,m\}$, computed from band
spins. State $0$ is exceptional and has probability at most
$\epsilon$ in the fixed reference chart.
\item Every good state $i$ has a deterministic regular signed barrier
$\Gamma_i$. For every realizable transcript with $Z^\delta=i$,
adding its plus pins and then its minus pins changes the target
law by a sum of at most $\eta$ in total variation. The resulting
target-side Gibbs law, denoted $Q_{i,\delta}$, is the same for
every such transcript. This assertion is uniform over all
bounded observables on the target.
\item In the inward case one may also move the full barrier to a
slightly farther signed contour, retaining the entire enclosed
free component as the kept volume. Counting the two old-pin
removals as well, the sum of target errors can still be made
at most $\eta$. The final kept volume has a positive
macroscopic buffer from its fixed boundary. All geometric
clearances have positive minima over the good states.
\item For a fixed finite list of bands the descriptors have convergent
joint laws. The bins can be chosen to be continuity sets of
the limiting data law; hence this finite-space convergence is
in total variation.
\end{enumerate}
The number of states, their barriers, and their positive clearances
depend on the fixed geometry and on $\epsilon,\eta$, not on the mesh.
The threshold for the mesh is chosen afterward. If a different law,
possibly conditional on remote pins, has band density at most $K$
times that of the reference chart by
Corollary~\ref{fd:buffered-ratio}, its exceptional probability is at most
$K\epsilon$. The same applies to mixtures of those conditional laws.
\end{theorem}
\begin{proof}
Choose generic levels by Lemma~\ref{if:visible-data}. At each admissible
limiting datum apply Lemma~\ref{if:erosion}, and, for the inward version,
Lemma~\ref{if:barrier-buffer}. Allocate the requested total-variation
budget among the finitely many ordered steps. This produces an open
neighborhood and a barrier construction valid uniformly for every
realizable datum in that neighborhood on all sufficiently fine meshes.
Refine this cover to balls whose closures lie in such neighborhoods
and whose boundaries have limiting probability zero. For a fixed
center, only countably many radii can give positive boundary mass.
Separability from Remark~\ref{if:metric-separable} supplies a countable
such subcover. By continuity of probability along its increasing finite
unions, take finitely many balls whose union has limiting
probability larger than $1-\epsilon/2$. Order them and form disjoint
bins by assigning a datum to the first ball containing it. Their
boundaries are contained in the finite union of the selected sphere
boundaries and therefore have probability zero. The remaining bin
is exceptional. Convergence of the data makes its lattice probability
at most $\epsilon$ for all sufficiently fine meshes.
Associate to a good bin the barrier of the neighborhood containing
its ball. The local lemmas hold for every transcript in that bin,
with no dependence on an observable. Take the minimum of the
finitely many positive clearances and mesh thresholds. This proves
the first three assertions. For a finite list use the same argument
in each coordinate and Lemma~\ref{if:visible-data} jointly. Every
descriptor string is a continuity set, proving convergence of all
its finitely many probabilities. Finally the bad event depends
only on band spins, so the density comparison gives the last
assertion directly, including after conditioning on the stated
remote information.
\end{proof}
\begin{corollary}[Lookup approximation of conditional averages]
\label{if:descriptor-averages}
In the outward setting of Theorem~\ref{if:band-transfer}, let
$\mathcal F^\delta$ be the stopping information. For any family
of target observables $h_\delta$ satisfying $|h_\delta|\le H$, define
\[
a_i(\delta)=\mathbb E_{Q_{i,\delta}}h_\delta
\quad (i\ge1),
\]
and choose any exceptional-bin value in $[-H,H]$. Then, on every
good transcript,
\begin{equation}\label{if:lookup-pointwise}
\left|\mathbb E[h_\delta\mid\mathcal F^\delta]
-a_{Z^\delta}(\delta)\right|\le 2H\eta.
\end{equation}
In particular, in the reference chart,
\begin{equation}\label{if:lookup-ltwo}
\mathbb E\left|\mathbb E[h_\delta\mid\mathcal F^\delta]
-a_{Z^\delta}(\delta)\right|^2
\le 4H^2(\eta^2+\epsilon).
\end{equation}
The good bins, barriers, and mesh threshold work for all these
observables simultaneously. Neither the observables nor their
lookup values need converge as $\delta\downarrow0$. More explicitly,
fix a reference continuation law $\nu$ and regard
$f_\delta^\nu(T)=\mathbb E_\nu[h_\delta\mid\mathcal F^\delta=T]$
as a function of the stopping transcript $T$. Evaluating this same
function under any buffered band distribution with density bound
$K$ gives the integrated estimate with exceptional term $K\epsilon$.
It is the reference conditional-average function that is transferred.
It also equals the conditional average for the new law when that
law has the same continuation and distant wall given $T$.
If the distant wall is changed to an assignment $b$, the uniform
wall extension in Lemma~\ref{if:erosion} supplies the corresponding
completed laws $Q_{i,\delta}^{b}$. The conditional averages for this
continuation use the separate lookup values
$a_i^{b}(\delta)=\mathbb E_{Q_{i,\delta}^{b}}h_\delta$.
\end{corollary}
\begin{proof}
The conditional target law of a stopping transcript is its Gibbs
continuation by Lemma~\ref{if:stopping}. On a good bin its total
variation distance from the single law $Q_{i,\delta}$ is at most
$\eta$. The defining inequality for total variation gives
Equation~\eqref{if:lookup-pointwise}. On the exceptional bin both
quantities are bounded by $H$ in absolute value. Squaring and
integrating proves Equation~\eqref{if:lookup-ltwo}. All constructions
in Theorem~\ref{if:band-transfer} preceded the choice of $h_\delta$.
The pointwise good-bin estimate and the transferred exceptional
probability prove the last assertion. This is the reason that
mesh-dependent likelihoods can be transferred without assigning
them a continuum limit.
\end{proof}
\begin{corollary}[Random levels and finite ball probes]
\label{if:random-levels}
Theorem~\ref{if:band-transfer} and
Corollary~\ref{if:descriptor-averages} remain valid when the two contour radii are
drawn with a bounded density in a fixed compact rectangle of separated
radii lying strictly inside the permitted windows. Different band
observations may use independent draws. Their joint data laws
converge, including the auxiliary randomness.
Fix a countable dense collection of marked rational-walk prototypes
in the ambient space $\widehat{\mathcal X}$ of
Definition~\ref{if:datum}; probe balls are intersected with the
admissible-data space $\mathcal X_{\mathrm{band}}$. Allow a probe to test
whether the data lie within distance $r$ of one prototype, where
$r$ is drawn uniformly in a rational interval with closure in $(0,1)$.
For any prescribed $\epsilon,\eta$, finitely many such probes suffice
to obtain the lookup approximation in
Corollary~\ref{if:descriptor-averages}, with adjusted exceptional
probability at most $\epsilon$. The finite probe list depends only
on the reference local geometry, the radius rectangle, and the
requested tolerances. It is independent of the bounded observables
and of the size of an ambient buffered array.
\end{corollary}
\begin{proof}
For almost every radius pair the deterministic continuity argument
in Lemma~\ref{if:visible-data} applies. Couple the radius draws
identically on all meshes, independently of the converging loops.
The records take values in the union over the permitted radii specified
in Definition~\ref{if:datum}. The deterministic rounding rule is
measurable in those draws, and its recording maps $R_\delta$ have
uniformly vanishing displacement. Thus the continuous mapping theorem,
or dominated convergence after this coupling, proves joint convergence
of the encoded data and the radius draws. The full stopping information
includes those draws. The compact radius rectangle
provides a uniform positive separation of contours and uniform
clearance from the window edges. Thus the local erosion construction
still has a neighborhood for every admissible resulting datum.
For each datum choose a neighborhood $U_d$ as in
Lemma~\ref{if:erosion}. By density, there are a prototype $p$ and
rational numbers $0<a<b<1$ such that
\[
d\in B(p,a),\qquad B(p,b)\subset U_d.
\]
Choose a rational threshold interval strictly between $a$ and $b$.
A positive test then certifies membership in $U_d$, while every
datum in $B(p,a)$ tests positive for every threshold in that interval.
The inner balls form an open cover of the admissible data. Select
finitely many covering probability at least $1-\epsilon/2$ in the
reference law, and decode by taking the first positive test in
their fixed order. If no test is positive, use the exceptional
state. For a positive test use the single barrier law belonging
to its neighborhood $U_d$; hence
Equation~\eqref{if:lookup-pointwise} holds without dependence on
the random threshold or on $h_\delta$.
Every probe tie has probability zero because its threshold is an
independent continuous random variable, conditional on the data.
The finite string of bits consequently converges in law, and the
exceptional probability is at most $\epsilon$ for fine meshes.
Only finitely many entries in any fixed dense enumeration of the
prototypes and rational intervals have been used. For finitely
many permissible radius windows, take the union of the corresponding
lists. Finally the choices were made in one fixed local reference
chart. Density comparison transfers their error bounds to other
buffered chart laws with the same fixed multiplicative constant,
regardless of the number of tiles in the larger array.
\end{proof}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//descriptors.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//introduction.tex ====
\section{Introduction}\label{in:section}
An exploration of an Ising interface fixes spins along a random boundary.
To use the unexplored region, one needs estimates that remain valid after
this conditioning. Two difficulties arise. The fixed signs can alternate,
and the exposed boundary can contain many narrow excursions. We give
comparison estimates that retain arbitrary common signs, then replace
the boundary visible from a distant target by one of finitely many
regular signed barriers. The replacement controls the whole target
spin law, including observables that vary with the mesh.
\subsection{Finite graphs and pointwise comparison}
For a finite graph $G=(V,E)$, nonnegative couplings $K_e$, and finite real
fields $h_v$, the Ising law is
\[
\mu(\sigma)=Z^{-1}\exp\!\left(
\sum_{\{u,v\}\in E}K_{uv}\sigma_u\sigma_v
+\sum_{v\in V}h_v\sigma_v\right),
\qquad \sigma\in\{-1,+1\}^{V}.
\]
Prescribing a spin removes that variable and adds its incident
interactions to the fields of its neighbors. By contrast, deleting a
vertex means removing the vertex and all its incident edges.
Let $I,J,F$ be disjoint, with $I,J$ nonempty, and prescribe common spins
on $F$. Form a comparison graph by deleting $F$, contracting $I$ to one
terminal and $J$ to another, and setting all fields to zero. Let $q_0$
be the probability that the two terminals are connected in its
random-cluster model with cluster weight $2$ and edge parameters
$p_e=1-e^{-2K_e}$. Parallel couplings are added on contraction. Write
$\|\nu-\nu'\|_{\mathrm{TV}}=\sup_A|\nu(A)-\nu'(A)|$.
\begin{theorem}[Finite-graph comparison]\label{in:comparison}
For every two configurations $b,b'$ on $J$ and every configuration $a$
on $I$,
\[
e^{-4\operatorname{arctanh}q_0}
\leq
\frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)}
{\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)}
\leq e^{4\operatorname{arctanh}q_0}.
\]
Either conditional law may be replaced by an arbitrary mixture of
these laws. The bound is symmetric in the two terminal sets.
\end{theorem}
Lemma~\ref{fd:ratio} proves the theorem. For the square lattice at
\[
\beta_c=\tfrac12\log(1+\sqrt2),\qquad
B_r=[-r,r]^2\cap\mathbb Z^2,
\]
dual crossings turn it into a uniform buffered comparison. If
$I\subset z+B_R$ and $J\subset(z+B_{(1+c)R})^c$, the density ratio
is bounded above and below by constants depending only on $c>0$.
If $I\subset z+B_w$, $J\subset(z+B_R)^c$, and $R\ge8w$, its
logarithm is at most $C(w/R)^\alpha$ in absolute value, for an
absolute $\alpha>0$. Common pins may occur anywhere off the terminals.
The comparison also passes through a common conditional randomization,
such as internal Edwards--Sokal bond sampling. Precise geometric and
cross-domain forms appear in Corollaries~\ref{fd:buffered-ratio}
and~\ref{fd:finite-cover}.
\subsection{Exploration across a band}
For the exploration results we use the critical nearest-neighbor Ising
model with zero external field and prescribed boundary spins.
Put $Q_r=\{z:\|z\|_\infty<r\}$. Fix $0<u<v$, a larger square
$Q_L$, and a target at positive distance from the band
$\overline Q_v\setminus Q_u$ and the boundary of $Q_L$.
In the inward construction, reveal the outside of $Q_v$ and explore
towards $Q_u$; the target is a compact subset of $Q_u$. In the outward
construction, reveal $Q_u$ and explore towards $Q_v$; the target lies
between $\partial Q_v$ and a more distant fixed wall. All the
clearances are fixed before the lattice mesh $\delta$ tends to zero.
They can be arbitrarily small positive numbers.
When the two contour levels vary, each ranges over a prescribed open
interval. Their closures are disjoint and preserve all these clearances;
these are the separated level windows.
Choose a fixed diagonal in every lattice face and linearly interpolate
the spins on the resulting triangles. The zero set gives noncrossing
polygonal strands. Explore each strand meeting the starting contour
until its first return to that contour or its first hit of the other
contour, revealing the traversed triangles' spins. Include the starting
contour vertices in the initial reveal. The complete transcript is the
record of this adaptive reveal. The Gibbs property holds conditional on
each transcript, as proved in Lemma~\ref{if:stopping}.
A \emph{regular signed barrier} is a deterministic Jordan contour with
finitely many prescribed plus and minus arcs, together with a lattice
boundary layer following those arcs. Its fresh side is the side containing
the target. A finite descriptor is a function of the band spins with values
in a fixed finite set; one value is allowed to be exceptional.
\begin{theorem}[Finite conditional resolution]\label{in:resolution}
Fix the preceding band geometry and use the critical plus-boundary law
in a larger reference square. For every $\epsilon,\eta>0$, one can
choose contour levels in any prescribed separated subwindows,
a finite descriptor $Z^\delta\in\{0,1,\ldots,m\}$, and regular
signed barriers $\Gamma_1,\ldots,\Gamma_m$ with the following properties
for all sufficiently fine meshes.
The exceptional state $0$ has probability at most $\epsilon$. For every
transcript in a good state $i\in\{1,\ldots,m\}$, insert the plus pins of $\Gamma_i$ and
then its minus pins. The sum of the two resulting total-variation
changes on the target is at most $\eta$. The final target-side law
$Q_{i,\delta}$ is the same for every transcript in that state.
Consequently this conclusion holds simultaneously for all bounded target
observables, including mesh-dependent observables. For any fixed finite
list of bands the descriptors have convergent joint laws.
The finite list of barriers and its positive geometric clearances are
chosen before the mesh threshold. If another law has band density at
most $K$ times the reference density, its exceptional probability is
at most $K\epsilon$.
\end{theorem}
The full statement, including a further inward barrier displacement,
is Theorem~\ref{if:band-transfer}. Density comparison transfers the
probability of the exceptional event. To transfer a conditional target
law, the continuation beyond a common source contour must also be the
same; Corollary~\ref{if:descriptor-averages} states this distinction
precisely.
\subsection{Inputs and proof mechanism}
Critical spatial comparison already follows from FK crossing estimates.
Duminil-Copin, Hongler, and Nolin prove polynomial relative decorrelation
for separated FK events and discuss ratio mixing
\cite[Section~5.3, Proposition~5.11, author-hosted version]{DHN2011}.
Here we formulate comparison directly for full spin configurations on
arbitrary terminal sets of a finite ferromagnetic graph, retaining
arbitrary common pins and fields. The positive-association and cluster
representation framework is due to Fortuin--Kasteleyn--Ginibre, Holley,
and Edwards--Sokal \cite{FKG1971,Holley1974,EdwardsSokal1988};
marginal closure of the lattice condition belongs to the multivariate
total positivity theory of Karlin and Rinott \cite{KarlinRinott1980}.
We give the binary marginalization proof and reduce terminal log-odds
rectangles to the extreme rectangle. Balancing the two terminal fields
then permits the arbitrary-field covariance inequality of Ding, Song,
and Sun \cite[Corollary~1.3]{DSS2023}. That inequality is the imported
two-spin result; the pointwise terminal comparison is the finite-graph
deduction proved here. The FK crossing and one-arm estimates of
\cite{DHN2011} supply its critical geometric consequences.
To control a conditional target law, we construct an ordered coupling
in a doubled spin model. A disagreement that reaches a patch carries
magnetic mass of order $w^{7/8}$ on a nearby counting segment of scale
$w$. Pointwise comparison bounds the effect of changing that patch on
a target at distance $s$. Summing over an admissible cut gives ordered
transmission, including its signed-source and total-variation forms
(Section~\ref{tr:section}). As a separate quantitative consequence of
the spin-correlation limits of Chelkak, Hongler, and Izyurov
\cite{CHI2021}, we prove a mesh-first $L^2$ bound of order $(w/s)^{3/8}$
between normalized magnetizations on nearby contours, and a prediction
bound from a nearby contour (Section~\ref{li:section}). The stopping
construction below uses transmission directly.
Benoist and Hongler prove convergence of critical Ising interfaces to
nested $\mathrm{CLE}_3$ and a simultaneous common-edge approximation
for all macroscopic Ising loops \cite{BH2019}. We use that stronger
approximation to preserve coverage, orientation, and multiplicity under
every noncrossing complete resolution of the lattice interfaces
(Section~\ref{re:section}). At generic stopping levels the part of the
boundary visible from the target is a Jordan base with finitely many
slits (Section~\ref{if:section}).
The main geometric step replaces this slit boundary by regular signed
outlines. Finite graph embeddings and planar extension theorems
\cite{Schaefer2021,Moise1977} provide a topological chart. Narrow ribbons
retain the two boundary sides of each slit and the sign switch at its
tip. The spin-correlation limits of \cite{CHI2021} control magnetic
means in fixed regular upper and lower comparison domains. Small
neighborhoods of roots and tips can be overwritten with common pins at
a vanishing cost on a distant test contour. Away from them, the signs
seen by the common free component order every nearby realized wall
between the regular comparisons. Ordered transmission then turns the
small mean gap into target total variation. For an outward band, a
common bounded weight also permits arbitrary prescribed signs on the
distant wall.
This construction gives one barrier for every realizable transcript
in a neighborhood of the limiting slit datum. Its neighborhood and
mesh threshold are chosen before the target observable. A countable
cover, reduced to finitely many continuity sets carrying arbitrarily
high probability, yields Theorem~\ref{in:resolution}
(Sections~\ref{rg:section}--\ref{de:section}).
The positive width of every comparison collar is a fixed geometric
hypothesis; the constants may depend on it. Pointwise ratios allow
arbitrary common pins, whereas the magnetic mass charts and test
contours used in transmission have the free neighborhoods specified in
Definition~\ref{tr:admissible}. These separate hypotheses will be kept
explicit throughout.
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//introduction.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//lines.tex ====
\section{Nearby contours and prediction}\label{li:section}
A second consequence of the correlation estimates is that a normalized
magnetization can be predicted from a nearby contour. This result
supplements ordered transmission when one compares observations on
successive cuts; it is independent of the stopping-band construction
that follows. We first estimate the difference of two line sums and
then use the minimizing property of conditional expectation.
The relative displacement $h=w/s$ is held fixed while $s\to\infty$.
The proof takes the continuum limit of the two-point sums before
estimating their change under displacement. It therefore requires no
discrete derivative estimate for the two-point function.
\begin{proposition}[Differences of nearby line sums]
\label{tr:line-difference}
Let $J_s,J'_s$ be two parallel horizontal or vertical lattice segments of lengths $O(s)$ whose
matching portions differ by a displacement of length at most $Cw$ and
whose unmatched end pieces have total length $O(w)$. The same statement
is allowed for two concentric square contours of radii comparable to
$s$ and radius difference at most $Cw$, by matching their four sides.
Assume the union has a free neighborhood of width $cs$ and is covered
by a fixed number of buffered planar charts. Set
\[
M_{J_s}=s^{-7/8}\sum_{J_s}\sigma_x,
\qquad M_{J'_s}=s^{-7/8}\sum_{J'_s}\sigma_x.
\]
There is a constant $C_0$, depending only on the fixed geometry and the
buffer, such that, for each fixed $h=w/s\in(0,h_0]$,
\begin{equation}
\label{tr:line-difference-eq}
\limsup_{s\to\infty}
\|M_{J_s}-M_{J'_s}\|_2\leq C_0 h^{3/8}.
\end{equation}
The law may have arbitrary common pins beyond the buffer or be a mixture
of such laws. The same conclusion holds for nearby full periodic rows of period $W$,
normalized by $s^{-7/8}$, when $W/s$ is fixed and positive; its constant
may depend on this ratio.
\end{proposition}
\begin{proof}
The pointwise comparison in Corollary~\ref{fd:buffered-ratio}, applied to
the nonnegative square of a line difference, reduces each buffered chart
to the whole-plane law at the cost of a fixed factor. A finite chart
decomposition and the triangle inequality in $L^2$ will therefore suffice.
In the whole-plane law the mean is zero. By
Lemma~\ref{fd:static-correlations}, normalized two-point sums on
rescaled segments converge to the corresponding integrals of
\[
K(x-y)=c_\sigma |x-y|^{-1/4}.
\]
If the endpoint data vary with $s$, first pass to a subsequence realizing
the upper limit and then, after a common translation, to a further
subsequence on which their rescaled lengths and relative endpoints
converge. These data range over a compact set, and all the estimates
below are uniform on that set.
Here is the required control at the diagonal. The uniform bound
$\mathbb E(\sigma_x\sigma_y)\leq C(1\vee|x-y|)^{-1/4}$ shows that
parameter pairs whose rescaled distance is at most $r$ contribute at
most $Cr^{3/4}+o(1)$ to the second moment of a normalized segment sum.
The equal-site terms are $O(s^{-3/4})$. Away from the diagonal the
two-point asymptotic is uniform, so ordinary Riemann sums converge.
These observations justify the integral limit for both self and cross
terms, including shifted segments.
For two matching segments, pair their points by a displacement $v$ with
$|v|\leq Ch$. In the limiting squared difference, split pairs of their
one-dimensional parameters at distance $2Ch$. The near part is bounded
by a constant times
\[
\int_0^{C'h} r^{-1/4}\,dr=O(h^{3/4});
\]
this bound also applies if one of the shifted kernels has its singularity
at a nearby parameter value. For the far part,
$|\nabla K(u)|\leq C|u|^{-5/4}$, and the mean value theorem bounds the
combined kernel difference in absolute value by $Ch|u|^{-5/4}$.
Its integral is at most
\[
Ch\int_{2Ch}^{C'}r^{-5/4}\,dr=O(h^{3/4}).
\]
Thus the limiting squared norm of the matching-segment difference is
$O(h^{3/4})$. An unmatched end segment of length $O(h)$ has second
moment $O(h^{7/4})$ by the same kernel integral, and hence contributes
$O(h^{7/8})$ in norm. There are only a bounded number of such pieces.
Since $h^{7/8}\leq h^{3/8}$ for $h\leq1$, the triangle inequality proves
\eqref{tr:line-difference-eq}.
For concentric squares, trim corresponding sides to equal lengths,
translate their matching portions in the perpendicular direction, and
apply the segment result to each side and the short remainders.
For full periodic rows, subdivide both rows at the same fixed locations
into matching segments of length a sufficiently small fixed fraction of
$W$. Each pair, together with its buffer, lies in a planar chart; if
necessary decrease $h_0$ by a constant depending on $W/s$. Apply the
preceding comparison and sum the finitely many segment norms. Mixtures
are permitted since the density comparison is uniform before mixing.
\end{proof}
\begin{corollary}[Prediction from a nearby contour]
\label{tr:line-regularity}
In the setting of Proposition~\ref{tr:line-difference}, let $F$ be a
sigma-field containing the spins of $J'_s$, and write
$P_FX=\mathbb E[X\mid F]$. Then
\begin{equation}
\label{tr:line-regularity-eq}
\limsup_{s\to\infty}\|M_{J_s}-P_FM_{J_s}\|_2
\leq C_0(w/s)^{3/8}.
\end{equation}
The normalization on the approximating sum remains $s^{-7/8}$ even
when the two contours have different lengths.
\end{corollary}
\begin{proof}
The nearby sum $M_{J'_s}$ is $F$-measurable. The minimizing property of
conditional expectation gives
$\|M_{J_s}-P_FM_{J_s}\|_2\leq\|M_{J_s}-M_{J'_s}\|_2$.
Use Proposition~\ref{tr:line-difference}.
\end{proof}
\begin{remark}[Order of scales]\label{tr:quantifiers}
The transmission exponent, the aspect ratios in the local mass construction,
and the constants for a fixed family of cuts are chosen first. The bounds
are uniform for sufficiently large patch scales. For each of $N$ concentric cuts
at spacings comparable to $s/N$, the local geometry is the same for every
$N$ and $w/s\asymp N^{-1}$. Thus the constant in
\eqref{tr:transmission-lower} is independent of $N$. For the nearby-line
estimate, fix $N$ and then take the mesh sufficiently fine; this threshold
may depend on $N$. A large fixed $N$ may therefore be selected using the
transmission and line constants before taking the mesh limit.
\end{remark}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//lines.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Buffered comparison and stopping-band resolution in critical Ising}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove pointwise comparison and finite stopping-band approximation
theorems for the critical nearest-neighbor Ising model on the square
lattice. Arbitrary common pinned spins are allowed in the comparison:
the likelihood ratio is controlled by a zero-field FK connection
probability on the graph with those vertices deleted. For an interface
exploration across a band of fixed positive width, finitely many regular
signed barriers approximate every good conditional target law in total
variation, uniformly over bounded observables and sufficiently fine
meshes.
\end{abstract}
\input{sections/introduction}
\input{sections/ratios}
\input{sections/moments}
\input{sections/transmission}
\input{sections/lines}
\input{sections/resolutions}
\input{sections/stopping}
\input{sections/regularization}
\input{sections/descriptors}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//main.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//moments.tex ====
\section{Spin correlations and magnetic moments}\label{mo:section}
The magnetic estimates in this section concern the critical nearest-neighbor
Ising model with no external fields except those induced by prescribed spins.
Pins outside the stated free neighborhoods may have arbitrary signs.
\begin{lemma}[Static correlation inputs]\label{fd:static-correlations}
The following statements hold with constants depending only on the indicated
fixed bulk separations and aspect ratios.
\begin{enumerate}
\item In a square of radius \(r\), under any FK boundary wiring,
\[
c r^{-1/8}\le
\phi(x\longleftrightarrow\partial B_r)
\le C r^{-1/8}
\]
for \(x\) in a fixed central subsquare. In the plus Ising law this is
the corresponding magnetization estimate.
\item In a plus rectangle of scale \(r\) and fixed aspect ratio, for two
distinct points in separated fixed compact bulk sets,
\[
c r^{-1/4}\le\operatorname{Cov}_+(\sigma_x,\sigma_y)
\le C r^{-1/4}.
\]
\item The infinite-volume zero-field law on \(\mathbb Z^2\) is unique and
spin-flip symmetric. For a lattice-dependent constant \(c_\sigma>0\),
\[
\mathbb E(\sigma_x\sigma_y)
=c_\sigma |x-y|^{-1/4}(1+o(1))
\quad (|x-y|\longrightarrow\infty),
\qquad
0\le\mathbb E(\sigma_x\sigma_y)
\le C(1+|x-y|)^{-1/4}.
\]
The asymptotic is uniform in direction.
\end{enumerate}
\end{lemma}
\begin{proof}
The centered square assertion is
\cite[arXiv version~1, Lemma~26]{DHN2011}.
For another bulk point, take a square of radius \(c'r\) about that
point contained in the original square. Its free-boundary arm lower
bound remains a lower bound for the arm inside the larger domain.
Require in addition an open circuit in its outer half-annulus and
a rectangle crossing from its inner half to the original boundary.
The arm, circuit, and crossing then join. RSW gives the latter two
events jointly a fixed positive probability, and FK association
multiplies this by the arm lower bound. The upper bound uses the
wired arm in a contained square. This proves the stated uniformity.
The magnetization tends to zero as the boundary recedes. The monotone
coupling of plus and minus boundary laws has disagreement probability
equal to half their mean difference at a vertex. A union bound on any
fixed finite set therefore makes the extremal infinite-volume laws equal,
proving uniqueness and symmetry.
For the second assertion, apply bulk one- and two-spin convergence
\cite[Theorem~1.3]{CHI2021} in the fixed rescaled rectangle. We spell out
strict positivity of the limiting connected function. In the upper
half-plane let
\(q=|z-z'|/|z-\overline{z'}|\in(0,1)\).
The plus one- and two-spin formulae
\cite[Theorem~7.1, Equation~(7.3)]{CHI2021} give
\[
\frac{\langle\sigma_z\sigma_{z'}\rangle_+}
{\langle\sigma_z\rangle_+\langle\sigma_{z'}\rangle_+}
=\left(\frac{q^{1/2}+q^{-1/2}}2\right)^{1/2}>1.
\]
Conformal covariance transfers this strict inequality to a rectangle.
Continuity and compactness give a positive minimum for the connected
function on separated compact bulk sets. Uniform bulk convergence gives
the discrete lower bound for sufficiently large \(r\); finite energy
absorbs the finitely many smaller scales. The upper bound also follows
from convergence, or from two disjoint wired arm tests.
For completeness, put \(R=|x-y|\) and \(u=(y-x)/R\), and bracket the
plane two-point function between free and plus laws in a square centered
at \(x\) of radius \(TR\). Fix \(T>2\) first and take \(R\to\infty\).
Theorem~1.3 of \cite{CHI2021} gives the continuum limits
\(F_T^b(0,u)\), for \(b\in\{\mathrm{free},+\}\), uniformly for
\(|u|=1\). Conformal scaling gives
\(F_T^b(0,u)=T^{-1/4}F_1^b(0,u/T)\).
The normalization and two-spin fusion in
\cite[Proposition~5.3, Theorem~6.3, Equation~(6.12)]{CHI2021} imply
\(F_1^b(0,u/T)=T^{1/4}(1+o_T(1))\) for both boundary conditions.
Remark~6.4 there makes the error uniform in direction: the colliding
pair remains in the bulk and there are no other insertions. Letting
\(T\to\infty\) after the fixed-\(T\) mesh limit proves the plane
asymptotic with the common lattice conversion constant \(c_\sigma\),
uniformly in direction. Finally, connection of \(x\) to \(y\) forces arms in the two
disjoint boxes of radius \(\lfloor|x-y|/4\rfloor\). Conditioning on all
other edges and using the wired upper arm bound proves the uniform upper
bound. Nonnegativity follows from the FK representation.
\end{proof}
\begin{lemma}[Product arm bound]\label{fd:fk-product}
Let \(Q\) be a lattice square with an ordinary FK bond law under
arbitrary boundary wiring. After sampling the bonds, assign each
cluster touching the boundary any sign, and give every other cluster
an independent fair sign. Conditional on the bonds, all nonboundary
cluster signs are mutually independent and independent of the entire
collection of boundary-cluster signs. Boundary-cluster signs may have
any joint law conditional on the bonds; the bonds are not further conditioned on compatibility
with prescribed mixed signs. Suppose
\(x_1,\ldots,x_n\) lie a distance at least \(cs\) from \(\partial Q\)
and in a set of diameter at most \(As\), where \(n\ge2\).
For
\[
r_i=\max\left(1,\left\lfloor
\min\left(\tfrac{cs}{4},
\tfrac14\min_{j\ne i}\|x_i-x_j\|_\infty\right)
\right\rfloor\right),
\]
one has
\[
\left|\mathbb E\prod_{i=1}^n\sigma_{x_i}\right|
\le C_{n,c}\prod_{i=1}^n r_i^{-1/8}.
\]
The same bound holds conditionally on arbitrary bonds exterior to \(Q\),
with the induced wiring, before cluster colors are sampled.
\end{lemma}
\begin{proof}
First cancel insertions that occur an even number of times at one vertex.
At a vertex occurring repeatedly, the original \(r_i\)'s are one, so
discarding those factors costs nothing. All remaining radii are the
original radii; none is recomputed after cancellation. Each remaining singleton whose
ball has radius larger than one must have an open arm to that ball's
boundary if the conditional cluster-sign average is nonzero: otherwise
its free cluster contains no other odd insertion and has mean sign zero.
A cluster reaching a prescribed-sign boundary also requires this arm.
The balls of radius greater than one are disjoint, and are contained in
\(Q\). Their arm probabilities, conditionally on all exterior edges,
are at most their fully wired probabilities. Lemma~\ref{fd:static-correlations}
bounds each by \(Cr_i^{-1/8}\). Sequential conditioning multiplies
the bounds. Absolute cluster-sign averages are at most one. Radius-one
balls and parity cancellations can only remove requirements, proving the
claim. Exterior bond conditioning supplies only boundary wiring,
so the same proof applies conditionally. Prescribed mixed-spin
boundaries are handled by the comparison in Lemma~\ref{fd:moments}.
\end{proof}
\begin{lemma}[Buffered line and area moments]\label{fd:moments}
Fix \(A,c>0\), an integer \(p\ge1\), and \(\nu\in\{1,2\}\).
Let \(S\) be a set of lattice points of diameter at most \(As\) such that
\[
|S\cap B_t(x)|\le A(t+1)^\nu\quad(t\ge0),
\qquad |S|\le A s^\nu.
\]
Assume the Ising law has a free square-lattice neighborhood of width
\(cs\) about \(S\), with arbitrary pins beyond that neighborhood.
Then, for weights \(|a_x|\le1\),
\begin{equation}
\mathbb E\left|\sum_{x\in S}a_x\sigma_x\right|^{2p}
\le C_{p,A,c}\,s^{2p(\nu-1/8)}.
\label{fd:moment-bound}
\end{equation}
This includes a fixed number of line segments or square contours
\((\nu=1)\), and area sums \((\nu=2)\). It applies as well in a
buffered chart of a torus or cylinder. No pin on \(S\) is allowed in
this assertion.
\end{lemma}
\begin{proof}
Use Corollary~\ref{fd:buffered-ratio}, and a fixed finite covering if
necessary, to compare to a plus square with a proportional buffer.
It is enough to prove the bound there. Expand the even power and take
absolute values of the spin-product expectations. Lemma~\ref{fd:fk-product}
and the diameter bound reduce the desired sum to a constant times
\[
\sum_{x_1,\ldots,x_{2p}\in S}
\prod_{i=1}^{2p}
\sum_{j\ne i}(1+\|x_i-x_j\|_\infty)^{-1/8}.
\]
Indeed the capped nearest distance in that lemma is bounded below by
a fixed multiple of the uncapped nearest distance plus one, since all
points lie within diameter \(As\).
Expand the product by choosing one neighbor \(j(i)\ne i\) for each
index. The directed graph on the indices has outdegree one, and each
component consists of an oriented cycle of length at least two with
in-trees attached. Put \(q=1/8\). The counting hypothesis gives, uniformly
in \(y\),
\[
\sum_{x\in S}(1+\|x-y\|_\infty)^{-q}
\le C s^{\nu-q},
\qquad
\sum_{x\in S}(1+\|x-y\|_\infty)^{-2q}
\le C s^{\nu-2q}.
\]
These follow by dyadic distance shells; both \(q\) and \(2q\) are less
than \(\nu\). Summing an in-tree leaf removes one edge and contributes
\(Cs^{\nu-q}\). Once only a cycle remains, sum one vertex incident to
two cycle edges by Cauchy--Schwarz, at cost \(Cs^{\nu-2q}\).
For a two-cycle the two edges have the same endpoints and this is the
second displayed bound. The remaining graph is a path; remove its leaves
at cost \(Cs^{\nu-q}\) each, and its last vertex at cost \(Cs^\nu\).
A component with \(m\) vertices therefore costs at most
\(C_m s^{m(\nu-q)}\). There are at most \((2p-1)^{2p}\) neighbor
choices, proving the moment estimate.
For a large or bent support, partition \(S\) by overlay squares of
side \(cs/16\). At most \(C(1+A/c)^2\) such squares meet
\(S\). In each nonempty piece choose \(z\in S\) in that square.
The piece lies in \(z+B_{cs/8}\), and \(z+B_{cs/2}\) lies in the
assumed free neighborhood because \(z\in S\). Compare only this
local marginal to the plus law in the latter square. The counting
hypotheses persist with constants depending only on \(A,c\). Apply
the proved bound to each piece and use the triangle inequality in
\(L^{2p}\). Lattice rounding at bounded \(cs\) is absorbed in the
same constants. The same construction works in periodic charts.
\end{proof}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//moments.tex ====
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==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//ratios.tex ====
\section{Pointwise comparison through buffers}\label{fd:section}
We bound the probability of every local spin configuration after changing
prescribed spins, uniformly over all common pins.
We prove a finite-graph estimate in terms of a zero-field FK connection
probability, and then use critical crossings to control that probability
through lattice collars. All constants in the resulting geometric bounds
are independent of the number of lattice vertices.
\subsection{A finite-graph bound from extreme log odds}
We first allow any finite graph, nonnegative pair couplings, and arbitrary
finite fields. On a binary cube, a positive weight $p$ satisfies the
\emph{lattice condition}, or is \emph{log-supermodular}, if
\[
p(x\wedge y)p(x\vee y)\ge p(x)p(y)
\qquad\text{for every }x,y,
\]
where the minimum and maximum are taken coordinatewise.
The preservation of this condition under marginalization is a basic
property of multivariate total positivity \cite{KarlinRinott1980}.
We give the finite binary argument and then show that every log-odds
rectangle is bounded by the rectangle between the four constant terminal
configurations. This reduces the likelihood-ratio problem to two spins.
\begin{lemma}[Extreme rectangles in a ferromagnetic marginal]
\label{fd:mtp2}
Let \(p(a,b)>0\) be a marginal of a finite ferromagnetic Ising law with
arbitrary fields, on two disjoint sets of spins. Set \(f=\log p\) and
\[
L=f(+,+)+f(-,-)-f(+,-)-f(-,+).
\]
For every four configurations \(a,a',b,b'\),
\begin{equation}
\left|f(a,b)+f(a',b')-f(a,b')-f(a',b)\right|\le L.
\label{fd:extreme-rectangle}
\end{equation}
\end{lemma}
\begin{proof}
The Ising weight is log-supermodular: each ferromagnetic pair interaction
has nonnegative mixed difference and each field has zero mixed difference.
We recall why summing out spins preserves this property. It suffices to
check a two-coordinate square after summing one other binary coordinate.
Denote the eight unsummed weights by
\(a,b,c,d,e,f,g,h\), where \((a,b,c,d)\) and \((e,f,g,h)\)
are the two square slices, in order \((--,+-,-+,++)\).
Log-supermodularity gives
\(ad\ge bc\), \(eh\ge fg\),
\(ah\ge bg\), and \(ah\ge cf\).
Moreover \((ah)(ed)=(ad)(eh)\ge(bg)(cf)\).
For positive numbers \(X,Y,U,V\), the conditions
\(X\ge\max(U,V)\) and \(XY\ge UV\) imply
\(X+Y\ge U+V\): assume \(U\ge V\) and use
\(X+UV/X\ge U+V\) for \(X\ge U\).
Applying this with \((X,Y,U,V)=(ah,ed,bg,cf)\) gives
\(ah+ed\ge bg+cf\). Hence
\((a+e)(d+h)\ge(b+f)(c+g)\).
Nonnegative adjacent-coordinate square differences are equivalent to
log-supermodularity on a Boolean cube, by summing those differences along
coordinate paths. Iterating the one-coordinate elimination proves that
the marginal \(p\) is log-supermodular.
For ordered \(u\le v\) and \(s\le t\), put
\[
\Delta(u,v;s,t)=f(u,s)+f(v,t)-f(u,t)-f(v,s)\ge0.
\]
If \(D=f(a,b)+f(a',b')-f(a,b')-f(a',b)\), then the exact identity
\[
\begin{split}
L-D={}&\Delta(-,a';-,b)+\Delta(a,+;b',+)\\
&+\Delta(a',+;-,b')+\Delta(-,a;b,+)
\end{split}
\]
proves \(D\le L\). Interchanging \(b,b'\) proves \(-D\le L\).
No order between \(a,a'\) or between \(b,b'\) is required.
\end{proof}
The extreme rectangle has a direct Ising interpretation: restrict all
spins of each terminal set to a common sign, thereby contracting that set
to one vertex. The resulting two-spin interaction is unchanged by shifts
of the terminal fields. Balancing those fields will let us apply the
arbitrary-field covariance inequality of Ding--Song--Sun.
\begin{lemma}[Pointwise ratios with arbitrary common pins]
\label{fd:ratio}
Consider a finite ferromagnetic Ising graph, disjoint nonempty vertex sets
\(I,J\), and any fixed spins on a third set \(F\).
Delete \(F\), retaining its contribution as fields, and contract \(I\)
and \(J\) separately. Let \(q_0\) be the zero-field FK probability that
the two contracted vertices are connected on this contracted, vertex-deleted
graph. Then every two conditional spin laws on \(I\), obtained by
prescribing arbitrary configurations \(b,b'\) on \(J\), satisfy
\begin{equation}
\exp(-4\operatorname{arctanh}q_0)
\le\frac{\mu(\sigma_I=a\mid\sigma_J=b,\sigma_F)}
{\mu(\sigma_I=a\mid\sigma_J=b',\sigma_F)}
\le\exp(4\operatorname{arctanh}q_0)
\quad\text{for every }a.
\label{fd:pointwise-ratio}
\end{equation}
The same inequalities hold if either conditional law is replaced by any
mixture of these laws. The assertion remains valid in the presence of
additional arbitrary finite external fields; those fields are set to zero
in the definition of \(q_0\).
\end{lemma}
\begin{proof}
A prescribed common spin $\tau_v$, $v\in F$, contributes
$K_{uv}\tau_v$ to the field at each surviving neighbor $u$.
After these fields have been included, remove the vertices of $F$.
Apply Lemma~\ref{fd:mtp2} to the joint marginal on \(I,J\).
Restricting both terminal sets to constant configurations is exactly their
separate contraction. Its two-spin marginal has unnormalized weights
\[
w(s,t)=\exp(C+Kst+As+Bt),\qquad s,t\in\{-1,+1\},
\qquad K=L/4\ge0.
\]
The constants \(A,B\) include all effective fields from the other vertices.
Add external fields \(-A,-B\) at the two contracted vertices. This changes
their marginal to weights proportional to \(e^{Kst}\), without changing
\(K\). Both means now vanish and the covariance is \(\tanh K\).
The Ding--Song--Sun inequality applies to this finite ferromagnetic graph
with these arbitrary mixed-sign fields
\cite[Corollary~1.3]{DSS2023}. It gives
\[
\tanh K\le\langle\sigma_I\sigma_J\rangle_0=q_0.
\]
The last equality is the zero-field Edwards--Sokal identity
\cite{EdwardsSokal1988}.
Internal edges of a contracted terminal contribute constants and parallel
edges have nonnegative summed coupling, so all hypotheses of that
inequality are preserved. Thus \(L\le4\operatorname{arctanh}q_0\).
For fixed \(b,b'\), Lemma~\ref{fd:mtp2} bounds the oscillation in \(a\)
of the logarithm of the conditional density ratio by \(L\).
The ratio has mean one under its denominator law; its minimum is therefore
at most one and its maximum at least one. The ratio lies throughout
\([e^{-L},e^L]\), as claimed. Averaging a pointwise inequality proves the
mixture assertion, first for one law and then for the other.
\end{proof}
\begin{remark}[Symmetric terminals]\label{fd:symmetric-terminals}
The extreme log-odds rectangle and the connection probability $q_0$
are unchanged when $I$ and $J$ are interchanged. Thus the pointwise
comparison applies with the observed and prescribed terminal sets
reversed. Common pins are still deleted from the comparison graph;
oppositely signed common pins are never wired together. This symmetry
will also be used when a small region of changed pins is the source
and the observed set is much larger.
\end{remark}
\subsection{Critical collars, small patches, and local charts}
We now set every retained nearest-neighbor coupling to $\beta_c$.
To bound the connection probability in Lemma~\ref{fd:ratio}, it suffices
to find a dual circuit separating the terminals. A fixed collar provides
a probability bounded away from zero; many disjoint collars give a power
of the ratio of scales. Common prescribed spins still contribute fields
in the Ising law and are deleted only in the zero-field comparator.
We use the random-cluster representation with
\(p_c=1-e^{-2\beta_c}\) and cluster weight \(2\). An FK boundary wiring is
a partition of the boundary vertices. An open crossing here always uses
actual lattice edges; an abstract boundary identification is not itself a
crossing. Free clusters receive independent fair spin signs in the
Edwards--Sokal representation \cite{EdwardsSokal1988}, and a cluster attached to plus pins has sign
plus.
We use the FK domain Markov property, association, and monotonicity
under edge addition and boundary wiring in this finite-graph setting;
these properties are recalled in \cite[Section~2]{DHN2011}. Their
association and ordering principles originate in
\cite{FKG1971,Holley1974}.
Association applies to decreasing events too. Thus overlapping
crossings used to construct a dual circuit can be combined by
association, while disjoint tests can also be combined conditionally.
\begin{lemma}[Conditional crossing tests]\label{fd:conditional-rsw}
Fix \(A\ge1\). There is \(c_A>0\) such that a lattice rectangle with both
side lengths at least one and aspect ratio in \([A^{-1},A]\) has probability
at least \(c_A\) of either specified open crossing, and of either specified
dual crossing, at criticality under any boundary wiring. Changing each
side by a bounded number of lattice spacings only changes the constant.
These lower bounds also apply to a decreasing dual-crossing test in a
vertex-deleted graph, conditional on all bonds outside its full rectangular
test region, provided no terminal identification has vertices in the
interior of that region. Additional vertex deletions preserve the bound.
For finitely many tests with disjoint sets of unexposed edges, the conditional
bounds may be multiplied.
\end{lemma}
\begin{proof}
The rectangle assertion is the critical FK-Ising RSW theorem
\cite[arXiv version~1, Theorem~1]{DHN2011}, with planar duality for dual crossings.
The domain Markov property turns an exterior bond configuration into a
boundary wiring. Restore the deleted vertices of the test rectangle,
restore their edges, and wire its entire outer boundary. The resulting
measure stochastically dominates the conditional original bond law, so a
decreasing dual event is at least as likely in the original law. The
domination follows, for example, from the one-edge conditional probabilities
\(p_c\) and \(p_c/(p_c+2(1-p_c))\), according as the endpoints are or are
not already connected: adding edges or wiring increases these probabilities.
Restoring isolated vertices first and then their edges has the same effect.
The hypothesis about terminals ensures that no identification in the
interior of the restored rectangle is being discarded.
For the last assertion, order the test regions. Conditioning on all edges
outside the next region includes the results of all earlier tests. Applying
the uniform bound and then averaging proves the product lower bound by
induction. The corresponding product upper bound for simultaneous failure
of tests follows in the same way. If a bounded lattice correction makes
a side have bounded length, finitely many prescribed closed or open edges,
using the same one-edge probabilities, supplies a uniform replacement test.
\end{proof}
\begin{corollary}[Buffered and small-patch comparisons]
\label{fd:buffered-ratio}\label{fd:patch-ratio}
Fix \(c>0\). There is \(K_c<\infty\) such that, on the square lattice,
\(I\subseteq z+B_R\) and \(J\subseteq(z+B_{(1+c)R})^c\) imply
pointwise density ratios in \([K_c^{-1},K_c]\) for the conditional laws
in Lemma~\ref{fd:ratio}. Other common pins may occur anywhere off
\(I\cup J\).
There are absolute \(K<\infty\) and \(\delta>0\) such that, if
\(I\subseteq z+B_w\), \(J\subseteq(z+B_R)^c\), and \(R\ge8w\ge8\),
the logarithm of every such conditional density ratio has absolute value
at most
\begin{equation}
K(w/R)^\delta.
\label{fd:patch-power}
\end{equation}
In particular the total variation distance is at most
\(K'(w/R)^\delta\), and the expectation difference of a bounded test
function is at most \(2K'\|h\|_\infty(w/R)^\delta\).
No free neighborhood of the source patch \(I\) is required.
The buffered comparison also holds between laws from different domains,
including plane, torus, or cylinder laws, whenever their common collar
is an isometric square-lattice chart and all differing data lie beyond it.
It holds after applying the same conditional randomization kernel to local
spins, in particular internal Edwards--Sokal bond sampling.
\end{corollary}
\begin{proof}
An annulus of fixed radius ratio separating the two terminals contains a
dual circuit with probability at least \(c_1(c)>0\). Cover a middle
subannulus by a fixed number of overlapping rectangles, require dual
crossings in both directions of their overlap squares, and use
Lemma~\ref{fd:conditional-rsw}. The crossings join to an actual dual
circuit, which separates the terminals even after their separate internal
wiring. Extra common pins are deleted in the FK comparator; restoring
them only increases connectivity. Thus \(q_0\le1-c_1(c)\), giving the
first assertion by Lemma~\ref{fd:ratio}.
Between \(B_{2w}\) and \(B_{R/2}\), choose annuli with radius ratio, say,
four and disjoint test edges. There are at least
\(c_2\log(R/w)-c_3\) such annuli. A connection must avoid the dual
circuit in each, so Lemma~\ref{fd:conditional-rsw} gives
\(q_0\le C_1(w/R)^{\delta_1}\).
The fixed annulus between the terminals also gives \(q_0\le1-c_4\).
On \([0,1-c_4]\), \(\operatorname{arctanh}q\le C_2q\).
This proves the logarithmic bound. The total variation and expectation
bounds follow by integrating the density difference.
For different domains, condition on the exterior neighbor spins of a
square contained in the common chart. Each local law is a mixture of
the same finite-volume conditional laws, so the mixture assertion applies.
Finite-volume approximation gives the plane and cylinder statements;
the bounds are uniform in the approximating volume. Finally, applying
the same Markov kernel preserves pointwise domination of measures.
\end{proof}
\begin{corollary}[Local charts and finite patch covers]
\label{fd:finite-cover}
The preceding buffered and patch bounds remain valid in a chart clipped
by a free or a fixed physical boundary, provided the local graph and
the common prescribed boundary spins agree in the compared laws.
Deleted vertices and missing edges may occur in the chart. For laws on
different domains, assume that conditioning on spins beyond the common
chart gives the same local interaction and pinning rule.
More generally, suppose the observed set is partitioned into at most
$m$ patches. If each patch has a square-lattice chart separating it
from every changed datum by a fixed positive relative distance, the
whole observed law has a pointwise density comparison with constant
equal to the product of the patch constants. The same conclusion holds
if instead the changed source set has such a partition, with each
source patch separated from the whole observed set. Constants depend
on $m$ and the fixed chart clearances. Polygonal and rotated collars
are therefore allowed whenever they admit such a finite cover.
\end{corollary}
\begin{proof}
First fix a common finite local conditional graph. In its zero-field
FK comparator restore any vertices and edges missing from the full
square test rectangles. Restoration increases connectivity. The
decreasing dual tests in Lemma~\ref{fd:conditional-rsw} therefore give
the same upper bound on $q_0$ as before. Prescribed physical-boundary
vertices are among the common pins: delete them in the comparator
before this restoration. No identification is added in a test
rectangle's interior. Lemma~\ref{fd:ratio} proves the asserted
pointwise bounds for this conditional graph.
For different volumes or continuations, condition at the edge of the
common chart. Both laws are mixtures of the same local kernels, with
possibly different mixing measures. Apply the mixture assertion of
Lemma~\ref{fd:ratio}. The restoration argument bounds the auxiliary
connection probability; the common conditional kernel is what permits
the comparison between volumes.
For a cover of the observed set, reveal its patches in a fixed order
and factor each configuration probability by the chain rule. In the
comparison for the next patch, previously revealed patches are common
pins. They can lie anywhere off the two terminals. Each conditional
ratio has its stated bound, and multiplication gives the product
constant. For a cover of the source, telescope the changes in its
patches; all other source patches are common pins. Apply
Remark~\ref{fd:symmetric-terminals} to place the small patch in the
inner terminal. Mixtures over source assignments and common
randomizations preserve the resulting bounds.
\end{proof}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//ratios.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//regularization.tex ====
\section{Regular boundaries and ordered erosion}\label{rg:section}
The visible boundary from Section~\ref{if:section} is a Jordan base
with finitely many signed slits. We now replace it by one deterministic
regular boundary. The replacement must work for every sufficiently
nearby realizable stopping transcript, even though those transcripts
can have different examined sets. This uniform local statement is what
will permit the finite probability cover in Section~\ref{de:section}.
\subsection{Local replacement and the reduction to spin means}
Choose a source contour $S_0$ and a square test contour $J$ in the free
region between the entire permitted stopping band and the target.
Choose their positive collars so that $J$ is an admissible cut between
$S_0$ and the target, in the sense of Definition~\ref{tr:admissible}.
All modifications of the facing boundary occur on the band side of
$S_0$. For each fixed distant-wall assignment, the two laws in an
ordered insertion therefore have the same conditional continuation
from $S_0$ toward the target. Write
\begin{equation}\label{if:line-mean}
M_J^\delta=\delta^{7/8}\sum_{x\in J^\delta}\sigma_x.
\end{equation}
Lemma~\ref{fd:moments} bounds its second moment uniformly. The fixed
macroscopic reference scale implicit in Corollary~\ref{tr:ordered-tv}
only changes its constant.
\begin{lemma}[Local ordered erosion]\label{if:erosion}
Fix an admissible facing datum $d$, the band geometry, a target beyond
the band with a free collar, and $\varepsilon>0$. There exist a
neighborhood $U_d$ of $d$ in $\mathcal X_{\mathrm{band}}$, a regular
Jordan contour $\Gamma_d$ on the fresh side of the datum with finitely
many signed arcs, and $\delta_d>0$ with the following property.
For every $0<\delta<\delta_d$ and every realizable stopping transcript
whose datum belongs to $U_d$, the lattice boundary layer following
$\Gamma_d$ is free in the original conditional law. Let $\mu^0$ be
that law, let $\mu^+$ be obtained by inserting its prescribed plus
pins, and let $\mu^{\mathrm f}$ be obtained by then inserting its
prescribed minus pins. The two changes are ordered, and
\begin{equation}\label{if:erosion-tv}
\|\mu^0|_{\mathrm{target}}-\mu^+|_{\mathrm{target}}\|_{\mathrm{TV}}
+\|\mu^+|_{\mathrm{target}}-\mu^{\mathrm f}|_{\mathrm{target}}\|_{\mathrm{TV}}
<\varepsilon.
\end{equation}
On the fresh side of the full contour, $\mu^{\mathrm f}$ is exactly
the Gibbs law determined by this contour and, in the outward case,
the unchanged distant wall. In particular it depends on the chosen
contour, the mesh, and the unchanged distant wall, and not on the transcript within $U_d$.
The contour can be chosen polygonal or smooth, with its sign switches
on smooth portions. Its positive clearances, the neighborhood, and
the mesh threshold may depend on $d$ and $\varepsilon$, but not on
the remaining conditional spins or on the chosen transcript.
\end{lemma}
To see what the construction must establish, consider any candidate
barrier on the band side of $S_0$ whose layer is free in the original
law, and use the three laws in the statement. Put $m_j=\mathbb E_{\mu^j}M_J^\delta$ for
$j\in\{0,+,\mathrm f\}$. Inserting plus pins is upward and inserting
minus pins afterward is downward, so $m_+\ge m_0,m_{\mathrm f}$.
Corollary~\ref{tr:ordered-tv} gives
\begin{equation}\label{if:erosion-mean-reduction}
\begin{split}
&\|\mu^0|_{\mathrm{target}}-\mu^+|_{\mathrm{target}}\|_{\mathrm{TV}}
+\|\mu^+|_{\mathrm{target}}-\mu^{\mathrm f}|_{\mathrm{target}}\|_{\mathrm{TV}}\\
&\hspace{35mm}\le C\bigl[(m_+-m_0)+(m_+-m_{\mathrm f})\bigr].
\end{split}
\end{equation}
The constant depends only on the fixed free collars.
We will construct two deterministic regular signed boundaries, one
upper and one lower, that approximate the same slit datum. Write
$m_{\mathrm u}$ and $m_{\mathrm l}$ for their test-contour means.
After changing pins only in small neighborhoods of the finitely many
roots and tips, boundary monotonicity will place every nearby
transcript law between these two regular laws. If the resulting
mean error is at most $e$, this gives
\[
m_{\mathrm l}-e\le m_j\le m_{\mathrm u}+e
\qquad(j=0,+,\mathrm f),
\]
and the right side of \eqref{if:erosion-mean-reduction} is at most
$2C(|m_{\mathrm u}-m_{\mathrm l}|+2e)$. The small-patch comparison
makes $e$ small. Marked-domain convergence makes the two regular
means approach the same limit.
These are separate requirements. The next subsection proves the
marked convergence for deterministic ribbon outlines, keeping the
two sides of each slit distinct. The incidence criterion then proves
the ordering for actual examined triangles. Their combination will
control every transcript in one neighborhood without requiring a
spin-scaling limit in its irregular domain. For an outward band the
analytic argument first uses a constant distant wall; the common-weight
lemma below restores arbitrary prescribed wall signs with a uniform
change in the error budget.
\subsection{Regular outlines and marked convergence}
An admissible datum bounds a disk with finitely many attached slits in
the inward case. In the outward case it bounds an annulus with slits
attached to its inner boundary; the outer wall is unchanged. Opposite
signs on the two sides of a slit refer to its prime-boundary sides.
They do not prescribe two signs to a single lattice vertex: actual
Gibbs boundaries will be narrow outlines with separate sides.
\begin{lemma}[A chart for a finite slit graph]\label{if:chart}
Let a Jordan base have finitely many disjoint attached simple arcs at
distinct roots. There is a homeomorphism on a neighborhood of this
graph which makes the base a regular circular line and makes the arcs
disjoint straight segments on its fresh side. The chart may be chosen
with a further open collar beyond every tip. A fixed compact target
disjoint from the graph remains outside a sufficiently small
neighborhood used for this construction.
\end{lemma}
\begin{proof}
Subdivide the base at its roots and add further vertices as needed
so that the base and attached arcs form a finite simple plane graph.
Theorem~1 of \cite{Schaefer2021}, with ``isomorphic'' defined there
by a homeomorphism of the plane, gives a polygonal embedding obtained
by an ambient homeomorphism. In particular, the free tip of an attached
arc has a neighborhood in which it is a straight half-edge and can
be prolonged a short distance in the complement. This supplies no
common metric modulus. The Jordan--Schoenflies plane extension
theorem \cite[Section~10, Theorem~4]{Moise1977} also supplies a collar of the
base. We describe the compatibility construction for all the arcs.
Prolong each slit from
its free tip to a surrounding curve on the fresh side, choosing the
prolongations disjoint. This can be done successively: an arc attached
at just one endpoint to a boundary does not separate the adjacent open
region, so its free endpoint can be connected to the surrounding curve
without crossing the other attached arcs. A connecting path can be
simplified to a simple arc. After one prolongation has been chosen,
cut along it and repeat in the appropriate Jordan region. Choose
distinct endpoints on the surrounding curve in the cyclic order of
the roots. This preserves disjointness at each step.
The prolonged arcs cut the collar into finitely many Jordan cells.
Map their boundary arcs, compatibly at common edges, to the sides of
rectangular cells in a circular strip, sending each marked tip to an
interior point of the corresponding straight side. Apply the same
plane extension theorem in each cell and glue the extensions.
The glued map and its inverse are continuous because there are
finitely many closed cells and their boundary maps agree. This gives
the required chart. When there are no slits the initial collar
suffices. The surrounding curve and all prolongations can be chosen
in a neighborhood of the finite graph disjoint from the target;
equivalently take a sufficiently small regular neighborhood after
the successive extensions. Compactness and the positive distance
to the target provide the required final clearance.
\end{proof}
Hereafter a displacement parameter such as $q$ or $\rho$ is measured in
this chart. We do not assume that the chart is Lipschitz. Once finitely
many chart parameters have been fixed, disjoint compact pieces have
positive physical separation. The tolerance for perturbing data and
the lattice mesh will be chosen below that separation. For a lattice
transcript, undoing the recording map $R_\delta$ from
Section~\ref{if:section} moves the curves by $O(\delta)$ in the
physical plane. Thus a small recorded-data tolerance, followed by a
fine mesh, gives any prescribed chart tolerance for the actual wall.
All separation and incidence statements below concern that physical
wall and absorb this displacement into their positive slack.
\begin{lemma}[Small regularization of an outline]
\label{if:regularization}
A Jordan outline, finitely many marked points on it, and a sufficiently
small surrounding Jordan collar admit a polygonal or smooth replacement
through an ambient homeomorphism supported in that collar. Its
displacement can be made arbitrarily small. The marked points retain
their cyclic order and can be placed on straight grid-aligned portions
of the replacement. Finitely many disjoint outlines can be treated
simultaneously.
\end{lemma}
\begin{proof}
Use a Schoenflies parametrization of a closed annular collar of the
outline. Uniform continuity permits a sufficiently thin collar and
a cyclic subdivision into small quadrilateral strips, with small
open overlaps between consecutive strips and with nonconsecutive
strips disjoint. Take marked points away from the overlaps. In
each open overlap choose a joining point. Join successive points
by polygonal arcs in the intervening open strip, erasing loops.
Within an overlap, truncate the two neighboring arcs before their
first intersection and join them by a simple polygonal bridge.
The overlaps are disjoint, so this operation is local. This gives
a Jordan polygon following the prescribed cyclic order. All
polygonal endpoints used in this construction are in open sets;
no straight approach to an arbitrary point of a rough cell boundary
is required. At a marked strip insert a short horizontal or
vertical segment and put its mark there. Smooth corners, if
desired, away from the marks.
Choose transverse crosscuts in the small overlaps to subdivide the
old and new collars, respectively. They give corresponding Jordan
cells above and below the old outline and the new polygon. Their
outer boundary maps are the identity; on their middle boundaries
map the old arc onto the new arc and match the marked points.
Choose agreeing maps on the corresponding transverse crosscuts.
The plane extension theorem extends these maps cell by cell.
The extensions glue to a homeomorphism equal to the identity outside
the collar. Corresponding old and new cells lie in the same small
strip and its adjacent overlaps, so displacement is bounded by
that neighborhood's diameter. These diameters were made arbitrarily
small. The inverse has the same displacement bound. Disjoint
collars permit the finite simultaneous construction.
\end{proof}
For the next lemma, normalize a disk uniformizer by $F(0)=z_*$ and
$F'(0)>0$, where $z_*$ is a fixed interior point. For an annulus,
preserve the inner and outer labels and normalize the inverse map by
$g(z_*)>0$. Marked convergence additionally tracks the preimages of
the finitely many sign switches on the boundary circles.
\begin{lemma}[Marked convergence of ribbon outlines]
\label{if:marked-convergence}
Let $D$ be a disk or annulus bounded on one side by an admissible slit
datum and, in the annular case, on the other side by a fixed disjoint
Jordan wall carrying a fixed finite collection of nondegenerate signed
arcs. Replace each slit by a ribbon, with width and displacement
tending to zero in the chart of Lemma~\ref{if:chart}, and offset the
base by a quantity tending to zero. Use straight strips away from
nodes and fixed polygonal templates, scaled to the offset size, at
the finitely many nodes. The templates are chosen with no narrowing
neck beyond a fixed fraction of their scale. Approximate the resulting
outlines by smooth or polygonal Jordan curves through homeomorphisms
whose physical displacement tends to zero, as in
Lemma~\ref{if:regularization}. Preserve boundary-component
labels and the cyclic order of the side signs. Put one sign-switch
mark on each cap, tending in the chart to its designated slit tip.
One may also retain finitely many additional sign-switch marks on
the base away from slit roots, tracking them by their boundary
parametrization.
Then the resulting domains $D_q$ converge to $D$ in the marked
Carath\'eodory sense. Normalized uniformizing maps converge uniformly
up to the boundary, after radial identification of their source annuli
when necessary. Every cap mark converges to the prime end at its slit
tip. If $m_q$ and $m$ are the corresponding continuum critical Ising
one-point functions, then, for every $K\Subset D$,
\begin{equation}\label{if:mean-continuity}
\sup_{z\in K}|m_q(z)-m(z)|\longrightarrow0.
\end{equation}
\end{lemma}
\begin{proof}
We spell out the boundary control because unmarked kernel convergence
alone would not identify the cap marks. It suffices to prove the
assertion along an arbitrary null sequence of positive parameters
$q_n\to0$. In the proof all statements uniform in $q$ concern this
sequence, with its index suppressed.
\emph{Connectivity of the filled complements.}
In the straightened chart the filled complementary component is a
base region with finitely many thin rectangular teeth. Two nearby
points in one straight tooth can be joined by a segment or a
two-segment path of diameter at most a fixed multiple of their
distance. The same is true in the base region. Near a junction,
the finitely many fixed polygonal templates have a finite local
quasiconvexity constant, which is unchanged on scaling. Nonincident
pieces away from their junctions have fixed positive separation.
Consequently there are $C,t_0>0$, independent of $q$, such that points
at distance $t<t_0$ in the same model filled complementary component
can be joined there by an arc of diameter at most $Ct$.
Let $H$ be the fixed homeomorphism from the model to the physical
chart. Write the polygonalization or smoothing map as $J_q$, so the
physical complementary components are images under $H_q=J_q\circ H$.
The maps $J_q$ and $J_q^{-1}$ tend uniformly to the identity. Thus
$H_q$ and $H_q^{-1}$ have common moduli of continuity $\mu$ and $\nu$
on the relevant compact neighborhoods: uniform convergence controls
the tail of the sequence, and continuity controls its finite head.
Two physical points at distance
$t$ in the same complementary component can be joined there by an arc
of diameter at most
\begin{equation}\label{if:complement-modulus}
\theta(t)=\mu(C\nu(t)),\qquad \theta(t)\longrightarrow0.
\end{equation}
The fixed other complementary component in the annular case has the
same property by its Jordan parametrization, enlarging $\theta$ if
necessary. We require this property for the filled components, not
for their boundary curves; the latter need not be uniformly locally
connected when a ribbon narrows.
\emph{Interior uniformization.}
For the classical disk and annulus uniformization theorems, including
uniqueness of the annular radius ratio, see
\cite[Theorems~1.2.1 and~1.3.1]{Papamichael2008}.
The labeled complementary components converge in Hausdorff distance,
and compact subsets of $D$ are eventually contained in $D_q$.
The normalized kernel theorem gives interior convergence for disks
\cite[Definition~2.8 and Theorem~2.9]{CarmonaFedorovskiy2024}.
Use the modulus convention $m(A(r,R))=(2\pi)^{-1}\log(R/r)$.
For annuli, write the source annulus as $A(r_q,1)$, preserving the
boundary-component labels; existence of these uniformizers follows
from \cite[Theorem~1.3.1]{Papamichael2008}. A fixed smooth essential annulus compactly
contained in $D$ is eventually contained in every $D_q$. Also every
$D_q$ is contained in a fixed bounded annulus whose inner disk lies in
the inner complementary component. Both inclusions are essential:
the intervening annuli separate the same two complementary components.
Monotonicity of annular modulus
\cite[Proposition~4.3.5 and Section~4.3.4]{Beliaev2015}
therefore gives
\[
0<r_-\le r_q\le r_+<1.
\]
To identify a subsequential limit $r_q\to r\in(0,1)$, write the inverse
uniformizers as $g_q:D_q\to A(r_q,1)$ and the forward maps as
$F_q=g_q^{-1}$, with the inverse normalized at a fixed $z_*\in D$.
Take simultaneous locally uniform normal limits $g$ on $D$ and $F$
on $A(r,1)$. The inverse maps have winding number one on a fixed
essential curve and are bounded away from zero. Their limit cannot
be constant, since a nonzero constant has winding number zero.
The maximum and minimum principles give $r<|g|<1$ on $D$.
Consequently local convergence in the identity
$F_q(g_q(z))=z$ gives $F(g(z))=z$ for every $z\in D$, so $F$ is
nonconstant.
We must also exclude values on the limiting slit boundary. For every
finite $a\in\mathbb C\setminus D$, Hausdorff convergence of the
labeled filled complementary components supplies
$a_q\in\mathbb C\setminus D_q$ with $a_q\to a$.
The functions $F_q-a_q$ are zero-free. Hurwitz's Theorem and the
nonconstancy of $F$ imply $F(w)\ne a$ for every $w\in A(r,1)$.
Thus $F(A(r,1))\subset D$. Applying local convergence now to
$g_q(F_q(w))=w$ gives $g(F(w))=w$. The two limits are therefore
conformal inverses onto their domains. The modulus of $D$ is unique,
which identifies $r$; normalization identifies the maps. Every
subsequence has these same limits, proving interior convergence of
the moduli and maps.
\emph{Boundary equicontinuity.}
Let $F_q$ denote a forward map from the unit disk or the source
annulus. Disk uniformization and the Jordan boundary-extension theorem
are recalled in \cite[Theorems~1.2.1--1.2.2]{Papamichael2008}.
For each positive $q$ the annular target boundaries are also Jordan;
the annular extension preserving boundary-component labels is
\cite[Theorem~1.3.2]{Papamichael2008}.
Thus each map extends continuously to its boundary. Its image lies in
a fixed bounded set, and univalence gives a uniform area bound
\[
\int |F_q'(z)|^2\,dA(z)\le A_0.
\]
Fix a source boundary point $\zeta$. Choose $R>0$ uniformly smaller
than the source collar width and the distance of the source normalizing
point from the boundary. For $0<r<R$, the circular arcs
$C_t=\{|z-\zeta|=t\}$ in the source domain, $r<t<R$, are crosscuts
with both endpoints on the same boundary component. If
$\ell_q(t)$ is the length of $F_q(C_t)$, Cauchy--Schwarz and polar
integration give
\[
\int_r^R \frac{\ell_q(t)^2}{t}\,dt\le 2\pi A_0.
\]
For some $t\in(r,R)$ it follows that
\begin{equation}\label{if:short-crosscut}
\ell_q(t)\le s(r):=
\left(\frac{2\pi A_0}{\log(R/r)}\right)^{1/2}.
\end{equation}
The image endpoints can be joined in their common filled complementary
component by a simple arc of diameter at most $\theta(s(r))$.
This arc meets the image crosscut only at its endpoints. Their union
is a Jordan curve in a ball of radius $s(r)+\theta(s(r))$ about one
endpoint. For small $r$ the ball excludes $z_*$, whose distance from
all target boundaries is bounded below. In the annular case take it
also smaller than the separation of the complementary components.
The source cap cut off by $C_t$ does not contain the source normalizing
point. Its image must be on the small bounded side of this Jordan
curve: the other component contains $z_*$, and the curve meets the
domain only in the crosscut. Therefore
\begin{equation}\label{if:boundary-modulus}
\operatorname{diam}F_q\bigl(B(\zeta,r)\cap\text{source domain}\bigr)
\le 2\bigl[s(r)+\theta(s(r))\bigr]\longrightarrow0
\end{equation}
uniformly in $q$ and $\zeta$. Interior convergence and this estimate
prove uniform convergence on the closures. For annuli use the radial
maps between $A(r,1)$ and $A(r_q,1)$; since $r_q\to r\in(0,1)$ these
maps and their inverses have uniformly bounded Lipschitz constants
and tend uniformly to the identity.
\emph{Marks.}
A tip of a finite simple slit has exactly one prime end above it.
In the chart, a small disk about a tip is cut along a segment ending
there; nested crosscuts around the endpoint give its unique approach
class. A root can have two preimages in the prime boundary, but no
sign change is placed at a root. If $b_q$ is a cap mark, any limit of
$F_q^{-1}(b_q)$ maps to the designated tip by uniform convergence on
the closures. Uniqueness of its prime-end preimage identifies the
limit. An additional mark on the base away from the roots has the
same uniqueness property because the base is Jordan and no slit is
incident there. Distinct prescribed marks give distinct limiting
prime ends, so all sign arcs remain nondegenerate.
\emph{Spin means.}
Theorem~1.3 of \cite{CHI2021}, with the convergence convention of its
Definitions~3.1--3.2, applies to the fixed smooth or polygonal domains
here and to the limiting marked disk or annulus. For a slit domain,
boundary arcs are interpreted on the prime boundary through the
conformal pullback in its Definition~3.6. The preceding mark argument
preserves the number of boundary components, the cyclic arc order,
and the nondegeneracy of every sign arc. The rough-boundary local
normalizations of its Lemma~3.34 cancel in the correlation ratios in
the proof of Theorem~1.3. Its bulk convergence is
uniform on compact sets. Continuity of the continuum correlations
under the marked convergence above follows directly by a diagonal
argument: if Equation~\eqref{if:mean-continuity} failed, choose an
offending sequence of continuum outlines and then a lattice
approximation to each so fine that its normalized mean is within the
reciprocal of the sequence index of that continuum mean. Choose the
approximations also sufficiently close in the marked topology. The
combined lattice sequence approximates the limiting marked domain,
contradicting the same bulk convergence theorem. This proves
Equation~\eqref{if:mean-continuity}, including uniformity over $K$.
\end{proof}
\subsection{The boundary-incidence criterion}
Marked convergence controls the means for the regular comparison
outlines. To compare an irregular realized wall with these outlines,
we still need a discrete ordering valid for every nearby transcript.
The next two lemmas provide that ordering from the signs encountered
by the common target component.
We next separate the discrete ordering statement from its geometric
verification. All models are represented on a common finite
nearest-neighbor graph by declaring some vertices pinned. Vertices
beyond a complete boundary layer can be assigned arbitrarily, because
that layer separates them from the target.
\begin{lemma}[Ordering by boundary incidences]\label{if:incidence-order}
Let $\mu_0,\mu_1$ be two ferromagnetic Ising laws with the same
interactions and possibly different pin sets. Let $C$ be the component
of the vertices free in both models which contains a specified connected
target. Suppose that, for every edge $xy$ with $x\in C$ and $y\notin C$,
at least one of the following holds:
\begin{enumerate}
\item $y$ is pinned minus in model $0$;
\item $y$ is pinned plus in model $1$;
\item $y$ is pinned to the same sign in both models.
\end{enumerate}
Then $\mu_0|_C\preceq\mu_1|_C$. The same assertion holds for a union
of target components with the corresponding incidence condition.
\end{lemma}
\begin{proof}
Condition separately on all spins outside $C$. A boundary neighbor
cannot be free in both models, since its edge to $C$ would put it in
that common free component. Each of the three alternatives gives
$\sigma_0(y)\le\sigma_1(y)$ for every pair of exterior assignments
consistent with the pins. Thus at each $x\in C$ the induced boundary
field in model $0$ is at most that in model $1$. Their internal
couplings agree and are ferromagnetic. A monotone coupling of the
finite heat-bath chains on $C$, or equivalently the ferromagnetic
boundary monotonicity, gives domination for each pair of exterior
assignments. Mixing them proves the marginal assertion.
\end{proof}
\begin{lemma}[Examined triangles preserve the facing sign]
\label{if:triangle-incidence}
Fix an admissible limiting datum and disjoint small neighborhoods of
its roots and tips. For sufficiently close resolved data and sufficiently
fine mesh, an edge from a free vertex in the target component to an
examined neighbor, outside those neighborhoods, sees the prescribed
sign of the corresponding accessible face of the ideal datum.
The short approach to the realized wall has length $O(\delta)$ in
the physical plane. Its identification with a limiting wall face
also uses the datum tolerance and the fixed chart's modulus of
continuity.
\end{lemma}
\begin{proof}
An edge between free vertices cannot cross a traced zero segment:
the traversed triangle containing such a crossing would have caused
both endpoints to be inspected. The same observation applies to
the open part of an edge whose last endpoint is examined. If that
endpoint was inspected in a traversed triangle, append an approach
of length $O(\delta)$ in that triangle, on the side of the zero
segment carrying its spin sign. Affine interpolation gives a
constant vertex sign on each side of that segment. If instead it
belongs to the initial revealed side, the edge lies within
$O(\delta)$ of the starting polygon; approach the accessible arc
of that polygon directly. Every initial sign switch produces a
zero crossing of the starting contour and is explored by the
algorithm. Hence no additional switch is present on a retained
base arc away from its roots. In either case this constructs an
approach to the realized wall from the target-accessible side,
carrying the examined endpoint's sign.
Outside the node neighborhoods, use the chart of
Lemma~\ref{if:chart}. The limiting edge lies in a strip disjoint from
all nonincident graph pieces. Uniform parametrized convergence
places the perturbed strand in a narrow middle substrip and makes
it a crosscut between the two plugged ends. A simple crosscut
separates the long sides even if it repeatedly backtracks. The
approach just described cannot change from one side to the other
without crossing this wall. Its sign is consequently the sign of
that accessible face. The positive physical separation after the
chart and node neighborhoods are fixed absorbs the $O(\delta)$
detour. Only near a stopping tip could the short approach continue
around an untraced end; these are exactly the excluded neighborhoods.
\end{proof}
\subsection{A common weight for the distant wall}
\begin{lemma}[Changing a distant wall by a common weight]
\label{fd:outer-wall-weight}
Let $C$ and $W$ be nested lattice square contours, with a fixed positive
macroscopic collar between them. The graph in this collar is the critical
nearest-neighbor square lattice. Let $T$ be a set strictly inside $C$,
and let $\nu_1,\nu_2$ be two laws with plus spins prescribed on $W$.
The laws may have different prescribed spins inside $C$, but have the
same interactions and pinning rule on and outside $C$, where $C$ is free.
Write $\nu_i^b$ for the corresponding laws after replacing the plus wall
by any assignment $b$ on $W$.
There is a positive function $w_b$ of the spins on $C$, independent of
the data strictly inside $C$, such that
\[
d\nu_i^b\big|_{T\cup C}
=\frac{w_b\,d\nu_i\big|_{T\cup C}}{\nu_i(w_b)},
\qquad
\frac{\sup w_b}{\inf w_b}\le K.
\]
The constant $K$ depends only on the fixed collar geometry, uniformly
in the mesh and in $b$. Consequently
\[
\|\nu_1^b|_T-\nu_2^b|_T\|_{\mathrm{TV}}
\le 2K\,
\|\nu_1|_{T\cup C}-\nu_2|_{T\cup C}\|_{\mathrm{TV}}.
\]
\end{lemma}
\begin{proof}
For a contour assignment $j$, sum all spins strictly between $C$ and
$W$, keeping $j$ on $C$ and $b$ on $W$. Denote this exterior partition
weight by $A_b(j)$. Use the same allocation of energies along $C$ in
every law. The quotient
\[
w_b(j)=\frac{A_b(j)}{A_+(j)}
\]
does not involve any prescribed spin or interaction strictly inside
$C$. Factoring the Gibbs density at $C$ proves the stated common-weight
identity, including its separate normalization for each $\nu_i$.
Consider the Ising law on the collar alone with the spins on both
contours free. Its joint marginal on $(\sigma_C,\sigma_W)$ is
proportional to $A_b(j)$. For any $j,j'$, the quantity
\[
\log w_b(j)-\log w_b(j')
=\log A_b(j)+\log A_+(j')
-\log A_+(j)-\log A_b(j')
\]
is therefore a log-odds rectangle. Lemmas~\ref{fd:mtp2}
and~\ref{fd:ratio} bound its absolute value by
$4\operatorname{arctanh}q_0$. The fixed square collar bounds this
quantity by a geometric constant, exactly as in
Corollary~\ref{fd:buffered-ratio}. Exponentiating proves the oscillation
bound on $w_b$. Energies involving only wall spins multiply a partition
weight by a scalar and hence do not affect this bound.
Rescale $w_b$ so that $1\le w_b\le K$. Put
$d=\|\nu_1|_{T\cup C}-\nu_2|_{T\cup C}\|_{\mathrm{TV}}$.
For an event $A$ of these joint spins, the layer-cake formula gives
$|\nu_1(w_b\mathbf1_A)-\nu_2(w_b\mathbf1_A)|\le Kd$ and
$|\nu_1(w_b)-\nu_2(w_b)|\le Kd$. Hence
\[
\left|
\frac{\nu_1(w_b\mathbf1_A)}{\nu_1(w_b)}
-\frac{\nu_2(w_b\mathbf1_A)}{\nu_2(w_b)}
\right|\le 2Kd.
\]
Take the supremum over $A$ and then restrict to $T$.
\end{proof}
\paragraph{Application to the outward erosion lemma.}
It suffices first to prove Lemma~\ref{if:erosion} for a constant plus
distant wall. Indeed, choose a square contour $C$ between the entire
compact target $T$ and the distant wall $W$, with positive clearances
from both and enclosing the full permitted stopping band. All three
inner laws $\mu^0,\mu^+,\mu^{\mathrm f}$ have the same free collar
from $C$ to $W$. Apply the constant-wall erosion construction to the
enlarged compact target $T\cup C$, with its sum of errors smaller than
$\varepsilon/(2K)$. The extra contour remains a positive distance from
the stopping band and from $W$, so this target satisfies the same
geometric hypotheses.
For any assignment $b$ on $W$, Lemma~\ref{fd:outer-wall-weight} now
transfers each of the two joint errors to $T$, and their sum is less
than $\varepsilon$. The weight is the same for every transcript and
for both insertions. Thus the fully inserted law is still determined
only by the selected barrier, the mesh, and $b$, independently of the
transcript. Adding plus and then minus pins retains the same two
orders under this prescribed wall. Every interior transcript remains
realizable under the plus reference wall, since finite Gibbs weights
are positive.
The constants and the choice of the barrier are uniform in $b$; no
limiting boundary description for a mesh-dependent assignment is used.
The marked-domain convergence argument above is therefore applied only
with a constant distant wall. More generally its continuum mean assertion
requires fixed finitely many nondegenerate signed or free arcs on every
boundary component. Arbitrary lattice wall assignments enter through
the common weight proved above.
\subsection{Construction and proof of local erosion}
\begin{proof}[Proof of Lemma~\ref{if:erosion}]
By the preceding common-weight reduction, take the distant wall
constant in the outward case. Keep the source and test contours fixed
as above. We construct the barrier and the two mean bounds used in
Equation~\eqref{if:erosion-mean-reduction}. The choices will be made in
the order in which their uniformity is needed: node neighborhoods,
comparison outlines, erosion contour, and finally the datum tolerance
and lattice mesh.
\emph{Node neighborhoods and their mean error.}
Use the chart of Lemma~\ref{if:chart}. Choose disjoint small closed
neighborhoods $P_1,\ldots,P_k$ of the roots and tips. Include one
auxiliary node on the base if needed to cut its circular strip into
finitely many coordinate strips. Each neighborhood is the image of
a small rectangle in the chart. Choose a smaller rectangle strictly
inside each one, so that every incident ideal edge is monotone in
its local strip coordinate until it leaves the larger rectangle.
The incident edges use separate prescribed strips; the nested
rectangles leave positive slack for the joins constructed below.
Compact parts of nonincident arcs have positive separation.
In a \emph{plugged} law, fix every lattice spin in these neighborhoods
to the same assignment, in every law under comparison, overwriting
any old prescription there. Denote this operation by a superscript
$P$. These plugs will close the channels between displaced boundaries.
Their assignments need not agree with the old side signs.
For a single plug and any of the laws $\lambda$ considered below,
Corollary~\ref{fd:patch-ratio} and the moment bound give
\begin{equation}\label{if:plug-error}
|\mathbb E_{\lambda}M_J^\delta
-\mathbb E_{\lambda^P}M_J^\delta|
\le \omega(\operatorname{diam}P),\qquad \omega(t)\longrightarrow0.
\end{equation}
The estimate is uniform in all other pins and for sufficiently fine
meshes. Indeed, the contracted connection probability in
Lemma~\ref{fd:ratio} is symmetric in its terminal sets. Its small-patch
bound controls the density on the remote contour $J^\delta$ when
the assignment in $P$ changes. If that density ratio is $R_P$, then
$e^{-b_P}\le R_P\le e^{b_P}$, where $b_P\to0$ as the physical plug
diameter tends to zero. Hence
\[
\left|\mathbb E_{\lambda}[(R_P-1)M_J^\delta]\right|
\le (e^{b_P}-1)
\left(\mathbb E_{\lambda}(M_J^\delta)^2\right)^{1/2}.
\]
When some spins in $P$ were free, first condition on all their values,
apply the uniform assignment comparison, and average. To overwrite
old pins, first remove their prescriptions while retaining all vertices
and edges, and use the same conditional comparison. Pins outside $P$
are common. No free collar is needed around the plug, whereas $J$
retains its fixed free collar and therefore has a uniform moment bound.
Sum \eqref{if:plug-error} over the finitely many plugs. Choose their
sizes so that the sum is less than a small number $a>0$, for every
law used in the proof. We will choose $a$ according to the fixed
transmission constant and the requested total-variation error.
\emph{Regular upper and lower boundaries.}
We next construct deterministic comparison laws indexed by a small
chart displacement $q>0$. On a one-sided base arc, move a plus face
towards the fresh side and a minus face away from it by amounts
comparable to $q$. On a slit, move a thin ribbon entirely towards
the plus side by a distance comparable to $q$, keeping the respective
plus and minus face signs. Its thickness is a small fixed fraction
of $q$. These choices give the upper outline. Reverse both moves
for the lower outline. Join the pieces near roots and tips using
fixed scaled polygonal templates, with one sign-switch mark on
each cap and no neck narrower than a fixed fraction of the template
scale. For sufficiently small $q$, all joins lie in the smaller node
rectangles. Regularly approximate each outline by
Lemma~\ref{if:regularization}, within a still smaller fraction of its
clearances. Let $\nu_q^{\mathrm u}$ and $\nu_q^{\mathrm l}$ be the
unmodified Gibbs laws, with the same distant wall in the annular case.
Lemma~\ref{if:marked-convergence} makes their continuum one-point
functions converge to the same limit as $q\downarrow0$. For each
fixed $q$, Theorem~1.3 of \cite{CHI2021} gives convergence of the
normalized lattice means uniformly on $J$. Since $J^\delta$ has
$O(\delta^{-1})$ vertices,
\begin{equation}\label{if:envelope-line-gap}
\lim_{q\downarrow0}\limsup_{\delta\downarrow0}
|\mathbb E_{\nu_q^{\mathrm u}}M_J^\delta
-\mathbb E_{\nu_q^{\mathrm l}}M_J^\delta|=0.
\end{equation}
Choose $q$ and its regular approximations so that this gap is small
for all sufficiently fine meshes. This choice is made after the plug
sizes, and keeps every join strictly inside its plug.
\emph{The erosion contour and its free neighborhood.}
Now choose $0<\rho\ll q$. At chart distance $\rho$ on the fresh
side, put an outline parallel to the base and around both sides and
the tip of each slit. Prescribe the face signs, with one switch on
each cap. This is a Jordan contour. A sufficiently accurate regular
approximation gives $\Gamma_d$. Choose its switches on grid-aligned
straight portions. For fine mesh, a simple lattice cycle following
it, with a boundary layer if necessary, has the prescribed cyclic
arc order and separates the target from the old boundary.
After $\rho$ and this approximation are fixed, the new layer has
positive physical separation from the ideal datum. Choose the datum
neighborhood and mesh small enough to preserve it. The recorded
lattice datum uses the map $R_\delta$ from Section~\ref{if:section};
undoing that map moves its curves by only $O(\delta)$ in the physical
plane. Uniform continuity of the fixed chart and its inverse therefore
turns a sufficiently small recorded-data tolerance, followed by a
sufficiently fine mesh, into the required chart tolerance for the
actual wall. Absorb this displacement and the examined triangles'
$O(\delta)$ thickness in the remaining clearance. The entire new
layer is then free in the original conditional law.
In the inward case the same choice preserves more than the layer.
Let $D$ be the Jordan base interior with its attached slits removed.
The outline wraps around every slit, so
$\overline{\operatorname{Int}\Gamma_d}\subset D$. It has a thin
exterior collar whose union with this closed interior is a compact
set $C\Subset D$. Uniform parametrized convergence of the base
preserves its winding number about every point of $C$, uniformly
because $C$ has positive distance from the base. The nearby slits
stay away from $C$ as well. Thus $C$ lies in the accessible domain
of every sufficiently close realized datum. Nonvisible traces and
hidden pockets lie in its blocked complement. All examined vertices
are on the initial side or within $O(\delta)$ of the traced boundary.
It follows that the entire component kept inside $\Gamma_d$, together
with a positive exterior collar, is free of old pins, uniformly over
the chosen datum neighborhood.
\begin{figure}[!htbp]
\centering
\input{figures/ordered-slit}
\caption{Ordered comparison in a topological slit chart, and the common
continuation used for each insertion. The old slit has plus on its left
and minus on its right; the black tested wall has chart coordinate zero.
The displayed inequalities concern the laws restricted to the common
target component. In (a) the upper ribbon moves towards the plus
side; in (b) the lower ribbon moves towards the minus side. Common root
and tip plugs seal each interwall channel. All ends extend into the
plugs with slack; dotted cap traces show geometry overwritten by the
common plug assignment. The common free target component sees an upper
plus or tested minus face in (a), and a tested plus or lower minus face
in (b), or identical plug data. The exterior sectors may communicate
outside the chart. Dashed erosion layers may supply the tested frontier
after insertion; the opposite layer is screened. The lower strip shows
the shared source contour $S_0$, an intermediate cut $J$, and the target
$T$: all changed facing-boundary pins precede $S_0$, and for each fixed
distant wall the conditional continuation beyond it is the same. The
coordinates are chart coordinates, with $q\gg\rho\gg\xi$; no metric
regularity of the chart is assumed.}
\label{if:slit-figure}
\end{figure}
\emph{Incidence ordering for every nearby transcript.}
Let $\mu^0,\mu^+,\mu^{\mathrm f}$ be the three laws in the statement.
We compare their plugged versions to the same two plugged regular
laws. Figure~\ref{if:slit-figure} shows the local geometry. First
consider a slit strip outside the node plugs. Give the old slit plus
on its left and minus on its right, and put it at chart coordinate
$x=0$. The actual perturbed strand lies in $|x|<\xi$, where
$\xi\ll\rho$. The upper ribbon is near $x=-q$, the erosion plus
layer near $x=-\rho$, and the erosion minus layer near $x=\rho$.
Parametrized convergence also controls visits to the strip ends.
For each ideal edge, its part between the smaller node rectangles
lies in a strip separated from all nonincident edges. On the matched
perturbed edge, select its last exit from the first smaller rectangle
before its first entry into the second. This gives a simple crosscut.
The ideal edge is monotone near each node, so closeness of the
parametrizations places these exit and entry parameters near the
corresponding ideal parameters. The positive gap between the smaller
rectangle and the actual plug therefore keeps the entire discarded
prefix and suffix inside their respective plugs. For example, in a
local edge coordinate with smaller-rectangle exit at $s=2h$ and plug
exit at $s=3h$, a tolerance $e<h/3$ puts every point of the discarded
prefix at $s\le2h+2e<3h$.
Retain the entire selected crosscut, including any later visits to
an outer plug. Every portion of the original edge outside the actual
plugs is then retained. Turns and repeated plug visits do not change
the crosscut's separation of its two long sides: filling the plugs
can remove components but cannot join the two sides. Its orientation
identifies the accessible face signs, as in
Lemma~\ref{if:triangle-incidence}. This argument applies to every
slit and to each base arc between consecutive nodes. There are only
finitely many edges, so one tolerance works for all of them. Taking
the mesh below the remaining physical slack prevents a lattice edge
from passing between a wall end and its plug.
The upper ribbon and this old-wall crosscut, joined through the
wholly pinned root and tip plugs, close the channel between them.
That channel cannot meet the target component of the sites free in
both laws. Additional examined pieces cannot reopen it. The common
target component can consequently see only the plus face of the
upper ribbon or the minus face of the old wall. In the fully eroded
tested law, the latter frontier may instead be the erosion minus
layer. In the plus-only tested law, its extra plus pins lie in the
screened channel. The same upper comparison therefore works for
all three laws. Lemma~\ref{if:triangle-incidence} identifies each
accessible examined-neighbor sign with the appropriate wall face;
the recorded-data displacement was already absorbed into its
physical slack. At plug incidences the signs agree by construction.
For a base arc the verification is one-sided. If its sign is plus,
the upper outline is nearer the fresh side, so the upper model
supplies a plus pin at the common component's frontier. If its sign
is minus, the tested wall is nearer and supplies a minus pin. The
sign-changing joins are inside the common plugs. Thus every
incidence satisfies Lemma~\ref{if:incidence-order}, and
$ (\mu^j)^P\preceq(\nu_q^{\mathrm u})^P$ on the common target
component for $j\in\{0,+,\mathrm f\}$.
For the lower comparison move the ribbon to the minus side. The
accessible incidences are the lower minus face or the tested plus
face. Additional tested plus pins reinforce this order; the erosion
minus pins lie in the screened channel. The base-arc argument is
reversed in the same way, and the plugs again give identical signs.
Applying the incidence criterion proves the full sandwich
\begin{equation}\label{if:plugged-sandwich}
(\nu_q^{\mathrm l})^P
\preceq (\mu^j)^P
\preceq (\nu_q^{\mathrm u})^P,
\qquad j\in\{0,+,\mathrm f\}.
\end{equation}
\emph{Mean bounds and the completed law.}
Removing the plugs changes each mean by less than $a$. Thus
\eqref{if:plugged-sandwich} places every
$\mathbb E_{\mu^j}M_J^\delta$ between the two unmodified regular
means with error at most $2a$ at either end. Each of the two ordered
mean differences in \eqref{if:erosion-mean-reduction} is at most
the absolute regular mean gap plus $4a$. Choose $a$ first and then
$q$ so that Equation~\eqref{if:envelope-line-gap} and this error
make the sum of target total-variation changes less than
$\varepsilon$. All later choices of $\rho$, regularization,
neighborhood, and mesh preserved these bounds. No spin-scaling
limit in a realized transcript's irregular domain has been used.
Finally, the completed layer separates the target from every old
examined vertex. Conditional factorization of the Gibbs weights
therefore leaves exactly the law with the chosen new boundary and
the unchanged distant wall. It is independent of the transcript.
The argument conditions on the full transcript, including the
actual auxiliary radii when they are random. Omitted initial spins
and nonvisible traces lie on the blocked side of the new barrier.
A stopping radius producing no visible tip adds no boundary piece.
Thus the same completed law works even when the datum does not
record its numerical radii. Averaging over these radii or other
omitted information preserves the two total-variation bounds.
\end{proof}
\begin{lemma}[Moving a completed barrier outward]
\label{if:barrier-buffer}
In the inward setting, let a regular signed contour $\Gamma$
have already been fully pinned, and let $V^\delta$ be the entire
free component it encloses, with its pinned boundary layer excluded.
Assume that a positive exterior collar of $\Gamma$ contains no other
pins, as supplied by Lemma~\ref{if:erosion}.
For any compact target on its inner side and any $\varepsilon>0$,
one can choose a regular surrounding contour $\widehat\Gamma$
at a positive distance from $\Gamma$, with corresponding signs,
so that the following two steps on the marginal on $V^\delta$
each change the target law by less than $\varepsilon$:
\begin{enumerate}
\item remove the old plus pins on $\Gamma$;
\item remove the old minus pins on $\Gamma$.
\end{enumerate}
Here the farther pins on $\widehat\Gamma$ are present from the start.
They initially have no effect on $V^\delta$. The first step is
downward and the second upward. After both, the kept volume
$V^\delta$ is separated from every fixed pin by a positive
macroscopic distance, uniform for sufficiently fine meshes.
\end{lemma}
\begin{proof}
Choose $\widehat\Gamma$ as a sufficiently thin outward offset of
$\Gamma$, matching the finite sign arcs. Put the new pins there
before changing the old ones. The full old layer disconnects the
kept volume, so these farther pins are initially harmless. We
retain the same $V^\delta$ throughout the argument; in particular
we do not first discard a free strip inside the old layer.
Use the construction of Lemma~\ref{if:erosion} without slits.
Put small identical plugs around the finitely many sign switches,
make upper and lower regular outlines by moving plus and minus
arcs in their respective ordered directions, and compare by
Lemma~\ref{if:incidence-order}. The test contour lies strictly
inside $\Gamma$, with a common free collar. As the offset tends
to zero, both unmodified envelope means have the same limit by
Lemma~\ref{if:marked-convergence}; the plug errors tend to zero
by Corollary~\ref{fd:patch-ratio}. This controls the means for the
original law, the law with only the old minus pins remaining,
and the law with all old pins removed. The two actual removals
are ordered, so Corollary~\ref{tr:ordered-tv} proves the claimed
target estimates.
Fix the sufficiently small offset after obtaining these estimates.
Its physical distance from $\Gamma$ is positive, and mesh rounding
preserves a fixed fraction of it. The kept volume is inside
$\Gamma$; the effective boundary pins are on the farther contour,
and the original pins are farther away, beyond the available collar.
This proves the final buffer assertion.
\end{proof}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//regularization.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//resolutions.tex ====
\section{Complete interface resolutions}\label{re:section}
The stopping construction uses a fixed face triangulation, whereas the
published loop limit uses leftmost interfaces. We must compare complete
collections, including the number and orientation of their macroscopic
loops: changing one face pairing can merge two loops or split one.
Throughout this section, locators for \cite{BH2019} refer to
arXiv version~3. We use its oriented-loop convention in Section~2.5: an
Ising circuit is a closed dual path, with no repeated edge, having plus
spins on its left and minus spins on its right. It may visit a dual
vertex twice. At a checkerboard face there are two noncrossing ways of
pairing the two incoming and two outgoing disagreement edges. A
\emph{complete resolution} chooses a pairing at every such face and
uses every disagreement edge exactly once. Fixing a diagonal in each
face and taking the zero set of the affine spin interpolation on its
two triangles gives one such resolution, up to a displacement of order
$\delta$.
A loop is \emph{leftmost} when the adjacent plus spins along its left
side are connected successively by nearest-neighbor steps; this is the
strong-path convention of \cite[Section~2.5]{BH2019}. The canonical
leftmost collection consists of all loops satisfying that convention.
For oriented curves, let $d_{\mathrm{u}}$ denote uniform distance modulo
orientation-preserving parametrization; for loops also allow a change
of starting point. Distances between loop collections are obtained by
matching loops and charging the diameters of unmatched loops, as in
\cite[Section~2.7]{BH2019}.
\begin{lemma}[Convergence of every complete resolution]
\label{if:resolution-convergence}
In a plus Jordan domain whose lattice boundary converges uniformly
modulo parametrization, every complete resolution of the spin
disagreement edges converges to the same oriented nested
$\mathrm{CLE}_3$ collection. This assertion is uniform over the choice
of the resolution. In particular it applies to a fixed choice of
diagonal in every square face.
\end{lemma}
\begin{proof}
Theorem~6 of \cite{BH2019} gives convergence of the leftmost loops in
the collection metric and the following simultaneous approximation.
For every $e>0$, with probability tending to one as $\delta\downarrow0$,
every Ising circuit $\ell$ of diameter greater than $e$ has a leftmost
circuit $\ell^{\mathrm L}$ such that
\[
d_{\mathrm u}(\ell,\ell^{\mathrm L})\le e,
\qquad
\operatorname{diam} C<e
\quad\text{for every component }C\text{ of }
(\ell\cup\ell^{\mathrm L})\setminus(\ell\cap\ell^{\mathrm L}).
\]
The paragraph following that theorem explains the common-edge content:
the shared edges become dense in each macroscopic loop. In particular,
above a fixed positive diameter cutoff every circuit shares edges with
its matched leftmost circuit, with probability tending to one. These
assertions hold for all circuits on the same event, including circuits
selected after the spin configuration has been seen.
We explain why this one-sided approximation also supplies coverage and
the correct multiplicities for a complete resolution. From any vanishing
mesh sequence, first choose errors $e\downarrow0$ slowly enough that the
simultaneous events above have probabilities tending to one. Pass to a
subsequence and couple the leftmost collections to their limit so that
both their collection-distance error and the all-circuits error tend to
zero almost surely. Choose a diameter
cutoff which is not the diameter of a limit loop. There are finitely
many limit loops above it. They are simple and pairwise disjoint, and
distinct members of this finite family have positive mutual distance.
Any two complete resolutions are joined by changing the checkerboard
pairings one at a time. Both possible pairings match incoming to
outgoing oriented edges. Each change either merges two edge-simple
circuits or splits one into two edge-simple circuits. Consequently all
intermediate circuits remain in the class covered by the all-circuits
assertion; no crossing circuit or repeated edge is introduced.
Suppose, along a sequence of finer meshes, that some such switch changes
the number of circuits matched near a fixed positive-diameter limit
loop. Select one offending switch on each mesh. The all-circuits
assertion holds for these selected circuits, even though the selection
depends on the configuration. After passing to a subsequence, each
constituent either has vanishing diameter or converges, modulo
parametrization, to a simple loop in the limiting collection. Two
vanishing constituents cannot merge to macroscopic size because they
meet at the switched vertex. Adding a vanishing constituent to a large
one does not change the latter's parametrized limit. If both
constituents are large, their meeting vertices force their limit loops
to meet. The limits must therefore be the same simple loop.
The last alternative is impossible. Choose a point strictly inside
that simple loop. Uniform parametrized convergence preserves the
winding number about the point. Each constituent has winding number
$1$ or $-1$, whereas the merged traversal has their sum, namely $0$ or
$\pm2$. The merged circuit is itself covered by the simultaneous
assertion. Its image approaches the same loop, so uniqueness of the
nearby limiting loop makes its \emph{oriented parametrized} limit that
loop as well. Its winding number must therefore be $\pm1$, a
contradiction. Splits
are the reverse operation. It follows that no switch can change the
macroscopic count near any fixed limit loop, for all sufficiently fine
meshes on this coupled subsequence. This argument selects an offending
switch if one exists and therefore does not accumulate an error over
the number of switches.
A specified leftmost circuit can be included in a complete resolution:
fix its pairings at every visited vertex and resolve the unused edges
arbitrarily. At a once-visited checkerboard vertex the unused incoming
and outgoing edges have the remaining noncrossing pairing. At a
twice-visited vertex the circuit already specifies both pairs. Thus its
oriented noncrossing traversal gives consistent local choices. In this
resolution a second circuit
matched to the same limit loop would have to share edges with the
specified leftmost circuit. Uniqueness of the nearby macroscopic
leftmost loop and the common-edge consequence above give this
conclusion. It contradicts use of every edge only once. The count is
thus one in this resolution, and the switching argument makes it one
in every resolution. The all-circuits assertion also excludes
unmatched positive-diameter circuits. This proves convergence in the
collection metric.
Finally, colors are recovered by nesting from exterior plus. Any
ancestor of a loop of diameter at least $a>0$ also has diameter at least
$a$. There are only finitely many such ancestors. Their nesting and
orientations are stable under convergence to disjoint simple loops,
so the colors needed at positive scales converge as well.
\end{proof}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//resolutions.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//stopping.tex ====
\section{Stopped interfaces and signed slit data}\label{if:section}
We now turn the complete loop limit into the boundary information needed
for a conditional target law. The exploration retains its full spin
transcript, while the geometric datum records only the boundary visible
from the target. The next section will replace that boundary uniformly
over nearby transcripts. All distances in this and the next two sections
are macroscopic: the lattice is $\delta\mathbb Z^2$, with
$\delta\downarrow0$.
\subsection{Band geometry and the information revealed}
Write $Q_r=\{z\in\mathbb R^2:\|z\|_\infty<r\}$ and
$\gamma_r=\partial Q_r$. Fix $0<u<v$ and positive clearances from the
band $\overline Q_v\setminus Q_u$ to the target and to the boundary of
the ambient chart. There are two versions of the construction.
\begin{enumerate}
\item In the inward version, initially reveal the spins outside $Q_v$,
and explore from $\gamma_v$ towards $\gamma_u$. The target is a
fixed compact subset of $Q_u$; in applications it is contained in
a strictly smaller concentric square.
\item In the outward version, initially reveal the spins in $Q_u$,
and explore towards $\gamma_v$. There is a more distant fixed
square wall, and the target lies between $\gamma_v$ and that wall,
with positive clearances from both. The distant wall has prescribed
signs, including the choice of a constant minus wall.
\end{enumerate}
The statement also allows a compact interval of choices for each radius,
provided the intervals are separated and all the stated clearances hold
uniformly. For a requested radius $r$, choose deterministically a nearest
$r_\delta\in\delta\mathbb N$, with ties resolved by a fixed rule, and
use the contour $\partial Q_{r_\delta}$. For sufficiently small $\delta$
these rounded radii preserve all the prescribed separations, and their
contours follow primal grid edges. The
interpolated zero set meets such a contour only at edge midpoints:
it never passes through a spin vertex, and the adjacent triangles
lie on opposite sides of the contour edge. Thus these meetings are
crossings, including near the square corners.
In either version include the vertices of the rounded starting contour
in the initial revealed layer.
For measurability statements, the \emph{band spins} include the vertices
of triangles meeting the closed rounded band. This convention adds only
an $O(\delta)$ layer and includes the starting-contour vertices.
\begin{remark}[Thin fixed collars]\label{if:thin-collars}
No universal lower bound on $v/u-1$ is required. Each collar has
strictly positive width fixed before the mesh limit, and the constants
and mesh threshold may depend on that width. For example, collinear
segments of length $\ell$, separated by gaps $g>0$, fit in disjoint
squares with containing halfside $\ell/2+g/8$ and farther-wall
halfside $\ell/2+g/4$. There is room between these squares for any
fixed finite list of nested contour levels with positive clearances.
When pieces come from a polygonal contour, first remove neighborhoods
of corners and boundary intersections, then take segment lengths
smaller than those separations. The remaining straight pieces admit
the same construction. Its constants can depend on $g/\ell$; a
fixed enlargement factor independent of $g/\ell$ is unnecessary.
\end{remark}
When radii vary, reserve one fixed larger permitted window containing
their compact ranges, and choose all common source and test contours
with positive clearance from the entire range of possible stopping
bands. The regular barriers constructed below stay on the fresh side
of the realized visible wall, with the prescribed target clearance;
they need not lie between each realization's numerical radii $u,v$.
The reserved window provides room for their offsets and tip caps.
We encode lattice boundary data on the requested continuum contours.
Choose disjoint fixed intervals about the two permitted radius ranges,
with positive endpoint clearances. Let $f_\delta:[0,\infty)\to[0,\infty)$
be the increasing piecewise-linear map that sends $u_\delta$ to $u$ and
$v_\delta$ to $v$, fixes the interval endpoints, and is the identity
outside those intervals, linear between consecutive specified knots.
Define
\[
R_\delta(0)=0,\qquad
R_\delta(z)=\frac{f_\delta(\|z\|_\infty)}{\|z\|_\infty}z
\quad(z\ne0).
\]
This homeomorphism and its inverse move every point by $O(\delta)$,
uniformly over the permitted radii: on each affine interval the error
$f_\delta(r)-r$ interpolates endpoint errors of size at most
$\delta/2$. It maps the closed rounded band onto the exact band and
each rounded contour onto its requested counterpart. We apply
$R_\delta$ only when recording the geometry of the exploration below.
The physical spins, reveal, and Gibbs laws remain on the original lattice.
For the probability limit, sample spins in a fixed larger square with
plus boundary conditions. Other buffered laws will enter only through
Corollary~\ref{fd:buffered-ratio}. The sign-reversed reference square may
equally well be used. A finite collection of band observations is always
computed from one common spin configuration, so that its joint law has
an unambiguous meaning.
Fix one diagonal in every square face, without adding interactions to
the Ising Hamiltonian. Affinely interpolate the spins in each triangle.
The zero set consists of noncrossing polygonal strands. After the initial
reveal, inspect every strand meeting the starting contour. Follow it on
the fresh side until its first return to the starting contour or its
first hit of the other contour. In the latter case also follow the
strand from its other starting endpoint, when there is one, stopping by
the same rule. Every time a triangle is traversed, inspect its vertex
spins. The order of exploring the starting endpoints is fixed
deterministically. Repeated inspections reveal no additional information.
From the target, the visible boundary consists of a base formed by the
starting contour and returning excursions, together with strands stopped
at the second contour and their two side signs. This is the
target-facing datum formalized in Definition~\ref{if:datum}.
\begin{lemma}[Stopping property]\label{if:stopping}
For each $\delta>0$, this is a finite adaptive spin reveal. Conditional
on its complete information, the unexamined spins have the Ising Gibbs
law with the examined spins fixed. The revealed set lies on the starting
side and in an $O(\delta)$ enlargement of the band. The traced strands,
their side signs, and the target-facing boundary datum defined below
can be computed from the band spins alone.
\end{lemma}
\begin{proof}
There are finitely many starting edges and finitely many triangles.
At a step, the next triangle is determined by the previously exposed
zero segment and the fixed diagonal convention. Its spins determine
the continuation. Hence the event that a specified reveal transcript
occurs depends only on the spins recorded in that transcript. In the
finite Gibbs density, conditioning on such a transcript fixes precisely
those spins and leaves the original interaction weights on all remaining
variables. This proves the stopping assertion directly, including the
case of a random examined set.
The trace stops at the second contour and examines only adjacent
triangles, which proves the location assertion. Although the initial
reveal contains an entire side of the band, the algorithm deciding a
strand within the band uses only the spins adjacent to the starting
contour and in the band itself. The target-facing datum is determined
by these traces and their local signs. No spins in the rest of the
initially revealed side are needed to compute it.
\end{proof}
\subsection{The visible Jordan base and its slits}
The following definitions refer first to an oriented collection of
simple disjoint continuum loops. Restrict a loop to the fresh side of
the starting contour. A maximal such portion returning to that contour
before reaching the second contour is a \emph{returning chord}. A
portion reaching the second contour is retained only up to its first
hit there and is a \emph{stopped strand}. Cut the fresh region along
these curves and take the component containing the untraced region
beyond the second contour. Retain only boundary pieces seen from this
component. The same definition is made for the resolved lattice walls.
The boundary is read from the remaining domain: a traversal follows
one side of each slit to its tip and returns along the other.
Thus the two sides of a slit are distinct boundary arcs even though
they have the same geometric image. This is the prime-boundary
traversal used below.
\begin{definition}[Admissible facing datum]\label{if:datum}
A facing datum consists of an oriented Jordan base, finitely many
pairwise disjoint simple slits attached to it at distinct roots, the
root and tip marks, and signs on the two sides of each slit and on
the intervening base arcs. Signs on a slit are opposite. In the
prime-boundary traversal the signs change at its tip and continue
unchanged through each occurrence of its root. The whole datum lies
in the closed band, and its tips lie on the second contour. If there
are no slits, the base has one constant side sign.
The lattice datum is the image of its physical boundary record under
$R_\delta$, retaining roots, tips, orientation, and side signs.
When radii vary over permitted windows,
$\mathcal X_{\mathrm{band}}$ denotes the union of admissible data from
those radius choices. The numerical radii are not extra marks; for
example, a datum without slits need not determine its second radius.
Within a fixed finite combinatorial type, use the maximum uniform
distance modulo increasing parametrizations of the base and slits,
respecting root attachments, side labels, and orientation. Cap this
distance at one and put distance one between different types. Denote
this metric space by $\mathcal X_{\mathrm{band}}$.
For approximation centers we also use an ambient space
$\widehat{\mathcal X}$ of finite marked continuous walks of these
combinatorial types. Retain the orientations, side labels, root and tip
marks, and shared coordinates of attached roots, but drop simplicity,
disjointness, and the exact band-containment and contour-incidence
requirements. Use the same matching-distance formula, identifying
representatives at distance zero. Then
$\mathcal X_{\mathrm{band}}\subset\widehat{\mathcal X}$ has its
stated metric as the induced metric. A ball with an ambient prototype
as center always means its intersection with
$\mathcal X_{\mathrm{band}}$ when used as a set of facing data.
\end{definition}
\begin{lemma}[Visible data and their continuity]\label{if:visible-data}
For almost every choice of the two contour radii, the continuum
construction from nested $\mathrm{CLE}_3$ yields an admissible facing
datum. Its base may contain infinitely many returning chords but is
a Jordan curve. The lattice data converge to it in
$\mathcal X_{\mathrm{band}}$. These assertions hold jointly for any
fixed finite collection of bands.
\end{lemma}
\begin{proof}
We give the deterministic continuity argument on the probability-one
event of simple, disjoint loops and local finiteness at every positive
diameter. For a continuous real function on a compact parameter
interval, its local maximum and minimum values form a countable set:
each is the maximum or minimum on an interval with rational endpoints
contained in a witnessing neighborhood. Apply this to the square
radius along each loop. There are countably many loops, so the union
of exceptional levels is countable for each collection. Fubini's
theorem permits a deterministic full-measure set of radius choices for
the random collection. At the selected levels there is no one-sided
contact or constant interval.
Only finitely many excursions have diameter greater than a prescribed
$a>0$. Indeed, only finitely many loops have that diameter; on each,
uniform continuity of a parametrization gives a positive minimum
parameter length for such an excursion. Their parameter intervals
are disjoint. A stopped strand has diameter at least the positive
distance between the contours, so there are finitely many stopped
strands. Their tips are distinct. The two stopped ends of the same
deep excursion could have the same tip only if that tip were a
one-sided contact at the second level, which was excluded.
Distinct excursion endpoints on the starting contour are also
distinct. Endpoints on different loops cannot coincide by
disjointness. A shared endpoint for two excursions of one simple
loop would give a one-sided level contact. Excursion interiors miss
the starting contour, and simplicity and disjointness exclude all
other intersections.
Each visible returning chord deletes the open interval of the starting
contour which is cut off from the target component. Visible deletion
intervals have disjoint interiors: in a pair of nested deletions the
inner chord is screened and is not visible. A small chord deletes a
small interval. To see this, join its endpoints by the short arc of
the starting square. As the chord diameter tends to zero, uniform
continuity of the inverse parametrization of that square makes this
arc small. The resulting small Jordan pocket cannot contain the
target beyond the second contour, which is separated by a fixed
positive distance. The erased interval must therefore be this short
arc, not the complementary long arc. This argument works also on
the exterior side; equivalently, compactify the exterior by a fixed
inversion away from the band.
Parametrize the starting contour and replace every deleted interval
by a parametrization of its chord with the same endpoints. Retain
the original parametrization elsewhere. At an accumulation of
replacement intervals, the interval diameters and chord diameters
tend to zero, by the preceding finiteness and small-pocket facts.
The resulting map is consequently continuous. It is injective:
chord interiors miss the starting contour and one another, their
endpoints are distinct, and retained points of the original contour
remain distinct. A continuous injection of the compact circle into
the plane is a Jordan parametrization. This proves, in particular,
that infinitely many small returning chords do not create an
accumulating pinch. Adding the finitely many visible stopped strands
gives the asserted base and slits.
For continuity, first choose $a>0$ outside the countable set of excursion
diameters and retain the finitely many excursions of diameter greater
than $a$. Uniform convergence of the parent loops
gives convergence of these excursions, their endpoints, and their
first hits of the second contour. Here is the role of the generic
levels. A compact subarc strictly on one side of a level stays there
under a small uniform perturbation. Arbitrarily near an excursion
endpoint on its other side in parameter time, the limiting loop has
points on the opposite side of the level; otherwise the endpoint
would be a one-sided contact. These points bracket the perturbed endpoint
between arbitrarily nearby parameter values. The same argument at
the cutoff brackets its first hit. Splitting, merging, or losing a
retained excursion would contradict one of these bracketing
properties.
The circular order of the finitely many retained endpoints is stable.
Their visibility is stable as well. An additional chord of small
diameter cannot screen a retained chord: its deleted pocket has small
diameter and cannot contain that retained chord. After matching the
retained pieces, match the remaining intervals of the bases in their
circular order. All discrepancies on these intervals have diameter
tending to zero as $a\downarrow0$. This constructs parametrizations
with uniformly small distance. Include the finitely many slit roots
in the matching; their stopped strands already converge by the same
bracketing argument. In particular the fixed band width and the generic
second level give eventual equality of the finite stopped-strand list,
with its cyclic root and tip order. Only the returning chords require
the subsequent limit $a\downarrow0$. The recording map $R_\delta$ adds
an error of $O(\delta)$ and ensures that the lattice records have the
exact contour incidences required by the data space.
At the lattice level, a returning chord replaces an interval whose
two surviving side signs agree. The only sign changes in a traversal
of the visible prime boundary occur at stopped tips. Thus microscopic
returning chords introduce no additional alternations between tips.
When slits are present, their oriented sides determine the signs on
all intervening base arcs, so these signs converge by
Lemma~\ref{if:resolution-convergence}.
To recover the constant sign when there are no tips, we use one further
property of nested $\mathrm{CLE}_3$: a deterministic interior point is
almost surely surrounded by infinitely many nested loops
\cite[Section~1, before Proposition~1]{SSW2009}. Their diameters tend to
zero by local finiteness. Choose such a point on the starting contour.
A sufficiently small surrounding loop misses the second contour and
has a returning excursion into the fresh side. If this chord is hidden,
any chord screening it has diameter bounded away from zero by the
small-pocket argument. There are only finitely many such screening
chords, so one is visible. Its oriented parent loop supplies the constant
side sign. That loop and its finitely many ancestors converge with their
orientations by Lemma~\ref{if:resolution-convergence}. This proves sign
convergence also in the no-slit case. The additional fixed-point nesting
property is used only for this color anchor, not for the preceding
deterministic construction of the Jordan base.
The deterministic argument applies simultaneously to any fixed finite
list. Lemma~\ref{if:resolution-convergence} and the continuous mapping
theorem finish the proof.
\end{proof}
\begin{remark}\label{if:metric-separable}
The ambient space $\widehat{\mathcal X}$ in Definition~\ref{if:datum}
is separable. Approximate finitely many parametrized walks by rational
piecewise-linear walks, approximating shared endpoints and roots
together. These prototypes need not be simple or satisfy the exact
band constraints; they form a countable dense set in the ambient
matching metric. The countable union over finite combinatorial types
remains separable. An ambient countable base restricts to a countable
base on $\mathcal X_{\mathrm{band}}$, so that subspace is also
separable. Only realizable admissible data, not the prototypes, are
used to construct Gibbs boundaries.
\end{remark}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//stopping.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//transmission.tex ====
\section{Ordered transmission across a cut}\label{tr:section}
The pointwise comparison controls the effect of changing one small patch
of a prescribed boundary. We now turn this into a bound for an ordered
change along an entire separating cut. The quantity on the right will be
the change of its normalized magnetization. This lets a one-point
estimate control the law of every bounded observable beyond the cut.
\subsection{The cut geometry and the transmission estimate}
For the lattice statements, each local chart carries the critical
nearest-neighbor Ising model. No external fields are added beyond those
induced by prescribed spins, whose common signs remain arbitrary. We state the geometry
in a form which records all uniformity requirements.
A source $S$ and target $T$ are disjoint vertex sets. Common pins away
from them are allowed. Delete the common pinned vertices from the graph,
retaining $S$ as boundary vertices. A finite set $J$, disjoint from the
source, target, and common pins, is a \emph{separating cut} if every path
from $S$ to $T$ in this graph visits $J$. The continuation on the target
side of $J$ is then independent of the prescribed spins on $S$.
We will partition $J$ into small patches $b_i$. The pointwise comparison
bounds the effect of changing one patch; a straight counting segment
$L_i$ in the same neighborhood will record the corresponding magnetic
change. The next definition provides free space for this local count
and limits how often one spin can be counted.
\begin{definition}[An admissible cut at scales $w,s$]
\label{tr:admissible}
Fix constants $A,K\geq1$ and $a\in(0,1/(16A))$. For $0<w\leq s$, a separating cut
$J$ is admissible if it has a partition into patches $(b_i)$ and associated
straight counting segments $L_i\subseteq J$, with centers $z_i$ and
scales $\ell_i\in[aw,Aw]$, satisfying:
\begin{enumerate}
\item $b_i\cup L_i\subseteq z_i+B_{\ell_i}$ and
$|L_i|\in[a\ell_i,2\ell_i]$;
\item every $z_i+B_{16\ell_i}$ is a pin-free planar lattice chart,
including freedom from source pins;
\item each site of $J$ belongs to at most $K$ counting segments;
\item $T$ is at distance at least $s/A$ from every $z_i$,
$16\ell_i\leq s/A$, and the
patch comparison of Corollary~\ref{fd:patch-ratio} is applicable in a chart
up to a fixed positive multiple of $s$ at every patch.
\end{enumerate}
All constants in the following statements may depend on $a,A,K$ and the
fixed chart geometry, but not on $w,s$, the number of source pins, their
values, or the mesh. Decreasing $a$ or subdividing the patches by a fixed
factor preserves this convention.
\end{definition}
The principal examples are concentric square contours in a common free
collar of width comparable to $w$, separated from the target by a fixed
multiple of $s$, and parallel periodic rows with the same properties.
For a square, partition each side into intervals of length a sufficiently
small fixed multiple of $w$. Near a corner, choose a counting segment on
one of its adjacent sides. Taking these intervals smaller than the collar
width by a fixed factor gives the free squares in
Definition~\ref{tr:admissible}; a bounded number of neighboring intervals
can use the same segment, giving a uniform overlap bound. Fixed finite
unions of these cuts are allowed. The hypotheses are also appropriate
when $w/s$ is small: the constants just described do not change with that
ratio.
Write
\begin{equation}
\label{tr:magnetization-definition}
M_J=s^{-7/8}\sum_{x\in J}\sigma_x.
\end{equation}
The normalization always uses the displayed reference scale $s$, not
the individual patch scale.
\begin{proposition}[Transmission]
\label{tr:transmission}
There is an exponent $\delta\in(0,1/8)$, chosen by decreasing the exponent
of Corollary~\ref{fd:patch-ratio} if necessary, with the following property.
Prescribe two ordered configurations on $S$, use identical common pins
elsewhere, and let $\Delta$ denote higher-law expectation minus lower-law
expectation. For an admissible separating cut at scales $w,s$, a bounded
observable $h$ supported on $T$, and $w$ sufficiently large,
\begin{equation}
\label{tr:transmission-eq}
|\Delta h|
\leq C\|h\|_\infty (w/s)^{-7/8+\delta}\Delta M_J.
\end{equation}
Equivalently, if $\|h\|_\infty\leq H$ for fixed $H>0$,
\begin{equation}
\label{tr:transmission-lower}
\Delta M_J\geq c_H (w/s)^{7/8-\delta}|\Delta h|.
\end{equation}
The constants are uniform as $w/s$ decreases within the stated geometry.
\end{proposition}
The exponent in \eqref{tr:transmission-eq} comes from two estimates.
A changed patch of diameter comparable to $w$ changes the conditional
expectation of $h$ by at most $C\|h\|_\infty(w/s)^\delta$, by the
small-patch comparison. We construct an ordered coupling in which,
conditional on a disagreement reaching that patch, the expected number
of disagreements on its counting segment is at least $cw^{7/8}$. Bounded overlap sums these contributions over
$J$. The normalization $s^{-7/8}$ then gives the factor
$(w/s)^{-7/8+\delta}$.
We prove the second estimate first. An auxiliary FK representation
will identify ordered disagreements with connections to the changed
pins. A local domination statement will let us test for magnetic mass
inside the pin-free charts of Definition~\ref{tr:admissible}. We will
then return to the cut and prove the displayed transmission estimate.
\subsection{An auxiliary FK model for ordered disagreements}
\label{tr:doubled-subsection}
Begin on a finite graph $\mathcal G=(V,E)$ with couplings $K_e\geq0$
and a pin set $P\subseteq V$. For a retained free set $H\subseteq V\setminus P$,
at specified couplings $J_e$ let $Z_{P,\tau}(H)$ denote the Ising
partition sum on the induced graph $P\cup H$, with signs $\tau$ on $P$. Partition sums include edges between
pinned vertices. Their contribution is constant in the free spins.
In the planar applications the original couplings are $K_e=\beta_c$.
The FK bonds introduced below instead have doubled couplings on a
random induced set; the critical FK crossing theorem is not applied
directly to these auxiliary bonds.
Take two independent Ising configurations $\eta,\xi$ whose signs on the
same pin set $P$ are ordered, with the higher signs assigned to $\eta$.
Let $U\subseteq P$ be the disagreement pins and $A=P\setminus U$ the
agreement pins, whose common signs are denoted by $\tau$. Let $W=V\setminus
P$, and let $G\subseteq W$ be the set of free sites where the original
replicas disagree. Thus the full auxiliary disagreement set is $G\cup U$.
\begin{lemma}[Doubled representation and ordered coupling]
\label{tr:doubled-coupling}
With partition sums at couplings $J_e=2K_e$, the law of $G$ is
\begin{equation}
\label{tr:doubled-weight}
\mu_\tau(G)\ \propto\ Z_{U,+}(G)\,Z_{A,\tau}(W\setminus G).
\end{equation}
Conditional on $G$, the spins $\eta=-\xi$ on $G\cup U$ form the first
Ising law in \eqref{tr:doubled-weight}, independently of the common spins
on its complement. Put FK bonds $\omega$ on $G\cup U$. There is a coupling
of the original two Ising marginals, denoted again by $(\eta,\xi)$, for
which $\eta\geq\xi$ and
\begin{equation}
\label{tr:mean-connection}
\eta_x-\xi_x=2\mathbf1_{\{x\longleftrightarrow U\}},
\qquad
\mathbb E\eta_x-\mathbb E\xi_x
=2\mathbb P(x\longleftrightarrow U).
\end{equation}
Here and below a connection to $U$ uses actual open edges, and is false for
$x\notin G\cup U$.
\end{lemma}
\begin{proof}
For an edge with endpoints in the same one of $G\cup U$ and its complement,
the two replica energies add to twice the energy of their common
orientation. For an edge between these sets they cancel. This proves
\eqref{tr:doubled-weight} and the conditional independence. Edges with
both endpoints pinned contribute constants independent of $G$, so cause
no exception to the factorization.
Every FK cluster in $G\cup U$ avoiding $U$ has an independent fair
orientation. On each such cluster replace the pair of opposite signs by
two copies of the first replica's sign. The first replica is unchanged;
the second replica's collection of independent fair cluster signs has been
negated, so its conditional marginal is unchanged as well. Clusters
meeting $U$ retain the pair $(+,-)$, and agreement sites retain equal
signs. This proves both the ordering and \eqref{tr:mean-connection}.
The auxiliary set $G$ need not equal the disagreement set after this
operation: the latter is precisely the union of clusters meeting $U$.
\end{proof}
Order pairs $(G,\omega)$ by inclusion in both coordinates, padding the
bond configuration by closed edges off $G\cup U$. An event or function of
this pair is called increasing in this order. The \emph{plus-agreement
reference} is the law in which all signs on $A$ have been changed to plus,
while the disagreement pins $U$ are retained.
The coupled replicas now differ exactly on clusters meeting $U$.
The auxiliary set $G$ still records the original independent replicas,
and can contain clusters on which the coupled replicas agree. To obtain
lower bounds on its local connections, we compare its law with the
plus-agreement reference. The density between the two auxiliary set laws
is, up to normalization,
\[
r(G)=\frac{Z_{A,\tau}(W\setminus G)}{Z_{A,+}(W\setminus G)}.
\]
We will show that this density increases when $G$ grows. The needed
partition-sum fact is a comparison of the factors gained by adding one
free vertex. We prove it for arbitrary nonnegative couplings $J_e$,
then apply it with $J_e=2K_e$.
\begin{lemma}[Spin products and addition of vertices]
\label{tr:site-gain}
Fix a set $P$ of pinned vertices, couplings $J_e\geq0$, and let
$Z_{P,\tau}(H)$ be the Ising partition sum on the induced graph with vertex
set $P\cup H$, with signs $\tau$ on $P$. Here $H\subseteq V\setminus P$ is
the retained free set. Then the following assertions hold.
\begin{enumerate}
\item With all pins plus, every spin-product expectation is nonnegative and
increases when nonnegative couplings are increased, free vertices and their
incident ferromagnetic edges are added, or additional spins are pinned plus.
The product in question involves only vertices present before the change.
\item For every finite set $Q\subseteq P\cup H$,
\begin{equation}
\label{tr:mixed-product}
\left|\mathbb E_{P,\tau,H}\prod_{x\in Q}\sigma_x\right|
\leq \mathbb E_{P,+,H}\prod_{x\in Q}\sigma_x.
\end{equation}
\item The function $H\mapsto Z_{P,+}(H)$ is log-supermodular. Moreover,
for $v\notin P\cup H$, the site gains satisfy
\begin{equation}
\label{tr:gain-comparison}
\frac{Z_{P,\tau}(H\cup\{v\})}{Z_{P,\tau}(H)}
\leq
\frac{Z_{P,+}(H\cup\{v\})}{Z_{P,+}(H)}.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
Use the FK representation with edge parameters $p_e=1-e^{-2J_e}$. In
descriptions of sign compatibility, an \emph{actual cluster} means a
component formed by open edges of the original graph: virtual wires between
pins do not count as open paths. Write $k_0(\omega)$ for the number of actual
clusters avoiding $P$. Apart from a constant, the all-plus bond weight is
\begin{equation}
\label{tr:pinned-fk-weight}
\prod_{e\in E(P\cup H)}p_e^{\omega_e}(1-p_e)^{1-\omega_e}
2^{k_0(\omega)}.
\end{equation}
This is the usual FK law with all pins wired. Conditional on the bonds,
clusters touching a pin have sign plus, and the other clusters have
independent fair signs.
Let $\mathcal F_Q$ be the event that each actual cluster avoiding $P$
contains an even number of vertices of $Q$. Conditional sign averaging gives
\begin{equation}
\label{tr:parity-event}
\mathbb E_{P,+,H}\prod_{x\in Q}\sigma_x
=\phi_{P,+,H}(\mathcal F_Q).
\end{equation}
The event $\mathcal F_Q$ is increasing. Indeed, merging two unpinned
clusters whose parities are even preserves their even parity, whereas
merging any cluster with a pinned cluster removes its parity restriction.
Adding an initially isolated vertex outside $Q$ adds a cluster of even
parity zero. Thus FK monotonicity under edge addition, increases of edge
parameters, and increases of the boundary wiring proves the first
assertion. Pinning an additional spin plus is covered by the same argument:
its cluster ceases to have a parity restriction and the FK wiring increases.
Positive fields, when needed, may equivalently be represented by edges to
one additional plus vertex.
For mixed signs, put
\[
\mathcal D_\tau=
\{\text{no actual cluster contains two pins with opposite signs}\}.
\]
The mixed-sign bond weight is exactly
\eqref{tr:pinned-fk-weight} multiplied by $\mathbf1_{\mathcal D_\tau}$.
In particular, its law is $\phi_{P,+,H}(\,\cdot\mid\mathcal D_\tau)$.
This identity also handles edges between opposite pins: such an edge must
be closed. The event $\mathcal D_\tau$ is decreasing and has positive
probability. Conditional sign averaging in the mixed law has absolute
value at most $\mathbf1_{\mathcal F_Q}$. Positive association of the wired
FK law therefore gives
\[
\left|\mathbb E_{P,\tau,H}\prod_{x\in Q}\sigma_x\right|
\leq \phi_{P,+,H}(\mathcal F_Q\mid\mathcal D_\tau)
\leq \phi_{P,+,H}(\mathcal F_Q),
\]
which proves \eqref{tr:mixed-product}.
For the last assertion, summing the spin at a newly added vertex gives the
exact identity
\begin{equation}
\label{tr:site-gain-formula}
g_{P,\tau,H}(v):=
\frac{Z_{P,\tau}(H\cup\{v\})}{Z_{P,\tau}(H)}
=2\mathbb E_{P,\tau,H}
\cosh\!\left(\sum_{u\in P\cup H}J_{vu}\sigma_u\right).
\end{equation}
Zero couplings may be omitted from the sum. If $N$ is its neighbor set,
the integrand has the expansion
\begin{equation}
\label{tr:cosh-expansion}
\cosh\!\left(\sum_{u\in N}J_{vu}\sigma_u\right)
=\prod_{u\in N}\cosh J_{vu}
\sum_{\substack{Q\subseteq N\\ |Q|\ {\rm even}}}
\prod_{u\in Q}\tanh J_{vu}\prod_{u\in Q}\sigma_u.
\end{equation}
All coefficients are nonnegative. Under plus pins, enlarging $H$ increases
the old product expectations, adds nonnegative terms, and increases the
prefactor. Hence $g_{P,+,H}(v)$ increases with $H$. Increasing site gains are
equivalent to log-supermodularity: add the vertices of $H_1\setminus H_2$
one at a time, comparing their gains above $H_1\cap H_2$ and above $H_2$,
to obtain
\[
Z_{P,+}(H_1\cup H_2)Z_{P,+}(H_1\cap H_2)
\geq Z_{P,+}(H_1)Z_{P,+}(H_2).
\]
Finally, apply \eqref{tr:mixed-product} term by term in
\eqref{tr:cosh-expansion} to prove \eqref{tr:gain-comparison}.
\end{proof}
\begin{lemma}[Association and increasing conditioning]
\label{tr:increasing-conditioning}
The plus-agreement reference law of $(G,\omega)$ is positively associated.
The mixed-agreement law has an increasing density, depending only on $G$,
relative to this reference. Consequently, if $\mathcal E$ is any increasing
event of positive probability and $f$ is any increasing function, then
\begin{equation}
\label{tr:conditional-reference}
\mathbb E_\tau[f\mid\mathcal E]\geq\mathbb E_+[f].
\end{equation}
There is no restriction on the number or placement of the pins in this
finite-graph assertion.
\end{lemma}
\begin{proof}
By Lemma~\ref{tr:site-gain}, both factors in
\eqref{tr:doubled-weight} with plus pins are log-supermodular functions of
$G$. Complementation exchanges unions and intersections and hence
preserves this property. Their product is log-supermodular, so the
reference set law is positively associated. Conditional on $G$, the FK
law is positively associated and increases stochastically with $G$.
Indeed, one may first add isolated vertices and then increase their edge
parameters from zero. Thus conditional expectations of increasing
functions of $(G,\omega)$ increase with $G$. The decomposition of a
covariance into its conditional covariance and the covariance of its
conditional expectations proves joint positive association.
The mixed law has the density $r(G)$ displayed above.
By \eqref{tr:gain-comparison}, the quotient $Z_{A,\tau}(H)/Z_{A,+}(H)$
decreases as $H$ grows. Therefore $r$ increases with $G$. The conditional
law on the left side of \eqref{tr:conditional-reference} has density
proportional to $r\mathbf1_{\mathcal E}$ against the associated reference.
That density is increasing, and association gives
\[
\frac{\mathbb E_+[f r\mathbf1_{\mathcal E}]}
{\mathbb E_+[r\mathbf1_{\mathcal E}]}
\geq\mathbb E_+[f].
\]
This argument requires neither association of the mixed law nor
preservation of a lattice condition after increasing-event conditioning.
\end{proof}
\begin{proposition}[Local domination]
\label{tr:local-domination}
Let $Q\subseteq W$ contain no pins. In $Q$, form the doubled model from two
independent Ising configurations with plus signs on every exterior
neighbor of $Q$, with the original couplings $K_e$. Call its auxiliary
set-and-bond law $\mathbb P_Q^{++}$. For every increasing function $f$ of
the auxiliary set and bonds internal to $Q$, and every positive-probability
increasing event $\mathcal E$ in the original doubled model,
\begin{equation}
\label{tr:local-domination-eq}
\mathbb E_\tau[f\mid\mathcal E]\geq\mathbb E_Q^{++}[f].
\end{equation}
The same statement holds with $\mathcal E$ the sure event.
\end{proposition}
\begin{proof}
By Lemma~\ref{tr:increasing-conditioning}, it suffices to compare the
plus-agreement reference with $\mathbb P_Q^{++}$. Condition its exterior
auxiliary set $G\setminus Q$. Fix $v\in Q$ and the membership of all other
sites of $Q$. Write $D$ for the full disagreement vertices excluding $v$,
including the pins $U$, and $H$ for the full agreement vertices excluding
$v$, including the pins $A$. The conditional odds for membership of $v$
are the quotient of two site gains:
\begin{equation}
\label{tr:local-odds}
\frac{\mu_+(v\in G\mid G\setminus\{v\})}
{\mu_+(v\notin G\mid G\setminus\{v\})}
=\frac{g_{U,+,D\setminus U}(v)}
{g_{A,+,H\setminus A}(v)}.
\end{equation}
Here $g$ is the gain in \eqref{tr:site-gain-formula}.
The numerator is at least the gain on the induced disagreement set in
$Q$: adding the exterior disagreement vertices, their edges, and their
plus pins only increases it. For the denominator, first pin every exterior
agreement vertex plus. The parity argument in Lemma~\ref{tr:site-gain}
and the nonnegative expansion \eqref{tr:cosh-expansion} show that this
increases the gain. Then restore every missing exterior neighbor of $Q$
as a plus pin, again increasing the gain. The resulting denominator is
exactly the agreement-site gain for the box-alone model: partition
factors involving only the now-pinned exterior cancel in the site ratio.
Thus the odds
in \eqref{tr:local-odds} are at least the corresponding odds of
$\mathbb P_Q^{++}$ at every fixed interior set configuration.
Equivalently, the density of the reference interior-set law relative to
the box-alone set law is increasing: the ratio increases under every
single-site addition. The box-alone set law is associated by
Lemma~\ref{tr:increasing-conditioning}. Tilting it by this increasing
density therefore proves stochastic domination of the sets.
Given a full auxiliary set, conditioning the exterior FK bonds induces
some wiring on its part in $Q$. This wiring dominates the free boundary
condition of the box-alone disagreement-bond system; its orientation has
no disagreement pins in $Q$ or on its exterior. The free FK law on an
induced set also grows with that set. First perform this bond comparison
and then the set comparison just obtained. Averaging over the exterior
set proves the claim for every increasing $f$.
\end{proof}
\subsection{Mass carried by a disagreement connection}
\label{tr:mass-subsection}
Local domination remains valid after conditioning on a connection from
the disagreement pins. We now construct, in the doubled plus-boundary
reference, an increasing event joining many sites of a segment to such
a connection. This is where the correlation and moment estimates of
Section~\ref{mo:section} enter.
For the remainder of the section the underlying graph in each local
chart is the square lattice at $\beta_c$. Bounded integer roundings of
rectangles and segments are understood. Every lower bound below holds
once its scale is sufficiently large; the threshold depends only on the
fixed geometric constants. This is sufficient when macroscopic geometry
is fixed before taking the mesh limit.
The bonds in this subsection belong to the auxiliary set sampled before
the recoloring in Lemma~\ref{tr:doubled-coupling}. Equal plus boundary
assignments can therefore yield auxiliary bond connections, even though
the resulting ordered spin coupling has identical replicas.
\begin{lemma}[Many sites joined across a rectangle]
\label{tr:many-sites}
Consider two independent plus-boundary Ising configurations in a rectangle
of scale $w$ and fixed aspect ratio, and their auxiliary disagreement
bonds. Let $L_1,L_2$ be straight lattice segments of lengths comparable to
$w$, in fixed compact bulk subsets of the rectangle, mutually separated
by a distance comparable to $w$. There are constants $c,p>0$, depending
only on this geometry, such that
\begin{equation}
\label{tr:many-sites-eq}
\mathbb P_Q^{++}\bigl(
|\{x\in L_1:x\longleftrightarrow L_2\}|\geq c w^{7/8}
\bigr)\geq p.
\end{equation}
In particular a disagreement-bond connection between the segments has
probability bounded below.
\end{lemma}
\begin{proof}
In the original independent pair put $Y_x=(\eta_x-\xi_x)/2$. There are no
disagreement pins. Conditional cluster signs give
\begin{equation}
\label{tr:pair-identity}
\mathbb P_Q^{++}(x\longleftrightarrow y)
=\mathbb E[Y_xY_y]
=\tfrac12\operatorname{Cov}_Q^+(\sigma_x,\sigma_y).
\end{equation}
More generally,
\begin{equation}
\label{tr:four-bound}
\mathbb P_Q^{++}(x\longleftrightarrow y,
x'\longleftrightarrow y')
\leq\mathbb E[Y_xY_yY_{x'}Y_{y'}].
\end{equation}
Indeed, if both pairs connect, all auxiliary clusters contain an even
number of these four indexed factors. Conditional sign averaging of the
product is the indicator of this even-parity event, with the additional
requirement that all factors lie in the auxiliary set. This explanation
also proves \eqref{tr:four-bound} when indices repeat.
Set
\[
X=\sum_{x\in L_1}\sum_{y\in L_2}
\mathbf1_{\{x\longleftrightarrow y\}}.
\]
The bulk connected-correlation lower bound in
Lemma~\ref{fd:static-correlations} gives $\mathbb EX\geq c_1w^{7/4}$.
By \eqref{tr:four-bound}, Cauchy--Schwarz, and the fourth line moments in
Lemma~\ref{fd:moments},
\[
\mathbb EX^2
\leq\mathbb E\left[
\left(\sum_{L_1}Y_x\right)^2
\left(\sum_{L_2}Y_y\right)^2\right]
\leq C_1 w^{7/2}.
\]
The moments for $Y$ follow from those for either replica by the triangle
inequality in $L^4$. Paley--Zygmund therefore gives constants $c_2,p_2>0$
such that $\mathbb P(X\geq c_2w^{7/4})\geq p_2$.
Let $L$ be the largest number of sites of $L_2$ in one auxiliary cluster.
Then
\[
L^2\leq\sum_{\mathcal C}|\mathcal C\cap L_2|^2,
\qquad
\mathbb E\sum_{\mathcal C}|\mathcal C\cap L_2|^2
=\mathbb E\left(\sum_{L_2}Y_x\right)^2
\leq C_2w^{7/4}.
\]
Choose $K$ so large that $C_2/K^2<p_2/2$. With probability at least
$p_2/2$, both $X\geq c_2w^{7/4}$ and $L\leq Kw^{7/8}$ hold. Every site
of $L_1$ reaching $L_2$ contributes at most $L$ to $X$. On that
intersection at least $(c_2/K)w^{7/8}$ sites reach $L_2$, proving
\eqref{tr:many-sites-eq}. Although the event involving $L$ was used to
establish its probability, the event in \eqref{tr:many-sites-eq} itself
is increasing.
\end{proof}
\begin{lemma}[A surrounding disagreement circuit]
\label{tr:circuit}
In the doubled plus-boundary model on $B_{16w}$, a circuit of actual
disagreement bonds in $B_{4w}\setminus B_{3w/2}$ surrounding $B_{3w/2}$ has
probability at least a constant $p_0>0$ for all sufficiently large $w$.
\end{lemma}
\begin{proof}
Consider the four strips
\[
[-4w,4w]\times[2w,4w],\quad
[-4w,4w]\times[-4w,-2w],\quad
[2w,4w]\times[-4w,4w],\quad
[-4w,-2w]\times[-4w,4w].
\]
For the top strip, extend the long coordinate to $[-5w,5w]$ and put
agreeing plus boundary conditions on this extended rectangle. Apply
Lemma~\ref{tr:many-sites} to vertical segments at $x=-4w$ and $x=4w$,
each with its $y$ coordinate in $[5w/2,7w/2]$. The segments are in its
bulk. Their connection stays inside the extended rectangle, since the
auxiliary disagreement set cannot contain its agreeing boundary pins.
A subpath of this connection gives a left-to-right crossing of the
unextended top strip. Proposition~\ref{tr:local-domination} transfers
its positive lower bound to the doubled model on $B_{16w}$. The same
construction applies to the other three strips.
The four crossing events are increasing, so their intersection has
probability at least the product of their four lower bounds. On this
intersection, adjacent crossings intersect inside their corner overlap:
one crosses that overlap in one coordinate and the other in the
perpendicular coordinate. The union contains a circuit surrounding the
central square. One way to see the last assertion is to join successive
intersection points along the four crossings, erase portions lying beyond
the adjacent corner overlaps, and obtain a closed walk in the annulus
with winding number one about the origin. Removing closed subwalks
decomposes it into simple circuits; at least one retains nonzero winding
number. All its edges lie in the four strips. This proves the claim.
\end{proof}
\begin{proposition}[Mass at a source-connected patch]
\label{tr:mass}
Fix $a>0$. Let $b_0\subseteq z+B_w$, and let $L\subseteq z+B_w$ be a
horizontal or vertical lattice segment of length at least $aw$ and at
most $2w$. Suppose $z+B_{16w}$ is a free square-lattice neighborhood:
it contains no common or disagreement pins. Then in the doubled model
with arbitrary ordered pins outside this square,
\begin{equation}
\label{tr:mass-eq}
\sum_{x\in L}\mathbb P(x\longleftrightarrow U)
\geq c_a w^{7/8}\mathbb P(U\longleftrightarrow b_0).
\end{equation}
The constant and lower threshold for $w$ depend only on $a$. The same
statement holds inside an isometric planar chart of a torus or cylinder.
\end{proposition}
\begin{proof}
Translate the picture so that $z=0$. Choose a translate $L'$ of $L$
by $6w$ in a perpendicular coordinate. Both segments lie in fixed bulk
subsets of $B_{16w}$, while $L'$ lies outside $B_{4w}$. Under its doubled
plus-boundary law, Lemma~\ref{tr:many-sites} gives positive probability
that at least $c w^{7/8}$ sites of $L$ reach $L'$ by paths using bonds
internal to $B_{16w}$. Intersect this increasing local event with the
increasing circuit event of Lemma~\ref{tr:circuit}.
Association makes the probability of their intersection at least a
positive constant $p$.
If $\mathbb P(U\longleftrightarrow b_0)=0$, the claim is immediate.
Otherwise condition on that increasing event and use
Proposition~\ref{tr:local-domination}. The circuit and the many-sites
event still occur jointly with conditional probability at least $p$.
Every path from $U$, which is outside $B_{16w}$, to $b_0\subseteq B_w$
meets the surrounding circuit. Every counted path from $L\subseteq B_w$
to $L'$ also meets it. By planarity all the counted sites are therefore
connected to $U$. The conditional expected number of such sites is at
least $pc w^{7/8}$. Multiplying by the probability of the conditioning
event proves \eqref{tr:mass-eq}.
\end{proof}
\subsection{From local mass to transmission}
\label{tr:transmission-subsection}
Proposition~\ref{tr:mass} provides the local estimate promised at the
start of the section. At each patch of an admissible cut, its scale
$\ell_i\asymp w$ converts the probability of reaching that patch into
expected disagreement mass on $L_i$. We now sum these estimates and
use the common target-side continuation to complete the proof.
\begin{proof}[Proof of Proposition~\ref{tr:transmission}]
Use the monotone coupling of Lemma~\ref{tr:doubled-coupling}. Let
$\eta_J\geq\xi_J$ be its cut configurations. Put
$k(j)=\mathbb E[h\mid\sigma_J=j]$, where the conditional expectation
uses the common target-side continuation. The spatial Markov property
and the separating property imply that this function does not depend
on which source configuration was prescribed. Consequently
\[
\Delta h=\mathbb E[k(\eta_J)-k(\xi_J)].
\]
In this telescope, the other cut spins are common pins for the pointwise
comparison. The local mass estimate is applied to the original coupling,
whose counting charts remain pin-free.
Interpolate between the two cut configurations one patch at a time,
using any deterministic ordering of their partition. Intermediate
configurations are legitimate cut configurations; spins at all other
patches are fixed for each comparison. Corollary~\ref{fd:patch-ratio}
bounds the oscillation of $k$ in one patch by
$C\|h\|_\infty(w/s)^\delta$. It applies with these arbitrary other cut
spins as common pins. If the two sampled cut configurations agree on
the patch, the corresponding summand is zero. By
\eqref{tr:mean-connection}, disagreement occurs somewhere in $b_i$
exactly when $U\longleftrightarrow b_i$. Therefore
\begin{equation}
\label{tr:telescope}
|\Delta h|\leq C\|h\|_\infty(w/s)^\delta
\sum_i\mathbb P(U\longleftrightarrow b_i).
\end{equation}
Proposition~\ref{tr:mass}, applied with scale $\ell_i\asymp w$, and the
overlap bound on the counting segments give
\[
\sum_i\mathbb P(U\longleftrightarrow b_i)
\leq C w^{-7/8}\sum_i\sum_{x\in L_i}
\mathbb P(U\longleftrightarrow x)
\leq C K w^{-7/8}\sum_{x\in J}
\mathbb P(U\longleftrightarrow x).
\]
The last sum is $\tfrac12s^{7/8}\Delta M_J$ by
\eqref{tr:mean-connection}. Substitute it in \eqref{tr:telescope} to
obtain \eqref{tr:transmission-eq}. All local constructions use the same
fixed aspect ratios and overlap constants, independently of $w/s$.
\end{proof}
\begin{definition}[Order-positive signed source]
\label{tr:order-positive}
For a probability law $\mu$ on the source spins, a real function $q$ is
an order-positive centered density if $\mathbb E_\mu q=0$ and
$\mathbb E_\mu[qf]\geq0$ for every increasing source function $f$.
The word density here refers to a signed measure; $q$ need not be
nonnegative or itself increasing.
\end{definition}
\begin{corollary}[Random and signed sources]
\label{tr:signed-transmission}
The conclusions of Proposition~\ref{tr:transmission} hold after averaging
over any ordered coupling of two random source configurations, with the
appropriate common Gibbs continuation given each source. In particular,
let $\mu$ be a Gibbs law with common pins and let $q=q(\sigma_S)$ be an
order-positive centered density for its source marginal. Then
\begin{equation}
\label{tr:signed-transmission-eq}
\left|\mathbb E_\mu[qh]\right|
\leq C\|h\|_\infty(w/s)^{-7/8+\delta}
\mathbb E_\mu[qM_J].
\end{equation}
The constants do not depend on the mass of the positive part of $q$.
\end{corollary}
\begin{proof}
For an ordered random pair of sources, apply
\eqref{tr:transmission-eq} to each pair and integrate. The magnetic
differences are nonnegative. The triangle inequality bounds the absolute
value of the integrated target difference by the integral of its absolute
value, proving the first assertion.
For the signed assertion put $m=\mathbb E_\mu q_+=\mathbb E_\mu q_-$.
If $m=0$ there is nothing to prove. Otherwise let
$\mu_\pm=m^{-1}q_\pm\mu_S$ be probability laws on source configurations.
Order positivity says $\mu_+(F)\geq\mu_-(F)$ for every increasing set $F$.
These laws admit a coupling $(s_+,s_-)$ with $s_+\geq s_-$. For
completeness, this finite transport fact follows by sending mass
$\mu_-(x)$ from each lower configuration $x$ to upper configurations
$y\geq x$ with capacities $\mu_+(y)$. For a set $B$ of lower
configurations, the required cut inequality is
\[
\mu_-(B)\leq\mu_-(\mathord\uparrow B)
\leq\mu_+(\mathord\uparrow B),
\]
so the finite max-flow/min-cut criterion transports all the mass.
Apply the random-source assertion to this coupling and multiply by $m$.
The Gibbs continuation changes the resulting difference of source
integrals into $\mathbb E_\mu[qh]$ and $\mathbb E_\mu[qM_J]$ respectively,
which proves \eqref{tr:signed-transmission-eq}.
\end{proof}
\begin{corollary}[Ordered changes of domains and total variation]
\label{tr:ordered-tv}
Suppose two ordered Ising laws, possibly defined by different sets of
pins or different volumes, have a common free source contour $S_0$ such
that their conditional continuation from $S_0$ toward a target $T$ is the
same. Let an admissible cut $J$ lie between $S_0$ and $T$. Then
\eqref{tr:transmission-eq} holds for their expectation differences.
In particular, with total variation defined as the supremum of event
probability differences,
\begin{equation}
\label{tr:ordered-tv-eq}
\|\mu_T^+-\mu_T^-\|_{\rm TV}
\leq C(w/s)^{-7/8+\delta}
\bigl(\mathbb E_{\mu^+}M_J-\mathbb E_{\mu^-}M_J\bigr).
\end{equation}
At fixed positive $w/s$, convergence of the magnetic difference to zero
therefore implies convergence in total variation on the buffered target.
\end{corollary}
\begin{proof}
Restrict an ordered coupling to $S_0$ and then use the common Gibbs
continuation. Corollary~\ref{tr:signed-transmission} in its random-source
form applies. Taking $h$ to be an arbitrary target-event indicator proves
\eqref{tr:ordered-tv-eq}. All differing pins must be on the source side of
$S_0$; the common-continuation hypothesis explicitly excludes a changed
pin in the target component.
\end{proof}
==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//transmission.tex ====
==== BEGIN Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//references.bib ====
@article{DHN2011,
author = {Duminil-Copin, Hugo and Hongler, Cl{\'e}ment and Nolin, Pierre},
title = {Connection probabilities and {RSW}-type bounds for the two-dimensional {FK Ising} model},
journal = {Communications on Pure and Applied Mathematics},
volume = {64},
number = {9},
pages = {1165--1198},
year = {2011},
doi = {10.1002/cpa.20370},
eprint = {0912.4253},
archivePrefix = {arXiv},
primaryClass = {math.PR},
note = {Crossing and one-arm locators refer to \href{https://arxiv.org/abs/0912.4253v1}{arXiv:0912.4253v1}. The spatial-mixing locator refers to Section 5.3, Proposition 5.11 of the \href{https://unige.ch/~duminil/publi/RussoSeymourWelsh.pdf}{author-hosted version}. Published version: \url{https://doi.org/10.1002/cpa.20370}},
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archivePrefix = {arXiv},
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==== END Buffered-comparison-and-stopping-band-resolution-in-critical-Ising//references.bib ====