===============================================================================
THE CRITICAL CORRELATION EXPONENT OF THE PLANAR XY MODEL
A complete proof, written to be read from the top
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WHAT IS THE PLANAR XY MODEL, AND WHAT IS THE 1/4?
Picture a huge square grid, and on each grid site a tiny arrow (a "spin")
that can point in any direction in the plane. Neighbouring arrows would
rather point the same way: the energy of a configuration is minus the
inverse temperature b times the sum of cos(angle between the two arrows)
over nearest-neighbour pairs. Free boundary: no spins outside the box, no
field, no twist.
At low temperature every arrow moves with its neighbours and the model is
magnetically ordered. At high temperature the arrows jiggle
independently. In TWO dimensions there is a middle regime, the
Berezinskii-Kosterlitz-Thouless phase: the arrows are still correlated at
every distance, but the correlation weakens like a power law. Write
C_b(n) = E[ cos(theta_0 - theta_{n e_1}) ]
for the correlation between two spins at lattice separation n (with the
thermodynamic limit taken first at fixed n). In the middle regime
C_b(n) behaves like n^{-eta} for some exponent eta > 0 that does not depend
on n. This document proves the sharp value
eta = 1/4
at the exact critical inverse temperature b_c where the middle regime
meets the hot phase. The number 1/4 is what the whole argument computes.
THE ONE OUTSIDE PAPER. Throughout, [BKT] means the companion paper
"BKT universality for height and planar spin fields",
OpenAI math release, 24 September 2026.
A citation "[BKT 3.14]" means Proposition 3.14 of that paper. [BKT]
supplies ONLY generic Gaussian analysis: a Gaussian free field with a
real quadratic energy, on rectangles and tori (its covariance limits,
localization estimates, and a renormalisation-group map for small
activities). The XY model itself is NOT in [BKT]. Everything specific to
XY -- the Bessel edge weights, the vortex/winding sector comparison, the
hard-pin transfers, and the openness at criticality -- is proved here.
That set of transfers is the actual content of this paper.
"Complete" therefore means: every logical step reducing the XY model to a
Gaussian field is written out, with the order of all limits tracked. The
generic Gaussian lemmas we import are quoted precisely (with their [BKT]
numbers) and their role is explained; re-deriving them would be a second
paper and would only add pages. If you accept the named Gaussian lemmas,
you have the entire proof of the exponent.
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1. WHAT WE PROVE, AND WHY THE ANSWER IS 1/4
===============================================================================
1.1 The model.
Lattice: Z^2 with standard basis e_1=(1,0), e_2=(0,1). Square box
Lambda_L = { x in Z^2 : |x|_inf <= L }.
Edges:
E_L = { {x,y} : x,y in Lambda_L and |x-y| = 1 },
every unordered nearest-neighbour edge counted once. On each site sits a
spin, an angle theta_x in R/(2 pi Z). Boundary is free: no site outside
Lambda_L, no fixed spins, no identification of opposite sides. For an
inverse temperature b > 0 the probability law is
mu_{L,b}(dtheta)
= (1/Z_{L,b}) * exp{ b * sum_{ {x,y} in E_L } cos(theta_x-theta_y) }
* prod_{x in Lambda_L} dtheta_x / (2 pi) . (1.1)
This is the ordinary nearest-neighbour cosine (plane-rotator) law. b is
exactly the usual beta. Each edge enters once.
1.2 Correlation and mass.
For an integer n >= 0 let the two-point function and the mass be
C_b(n) = lim_{L -> inf, L >= n} E_{mu_{L,b}}[ cos(theta_0 - theta_{n e_1}) ],
m(b) = lim_{n -> inf} -(1/n) log C_b(n) . (1.2)
Why these limits exist:
* Inner limit: ferromagnetic monotonicity in the coupling (Ginibre 1970);
adding sites/edges with free boundary can only help alignment.
* Outer limit: the multiplicative correlation ineuality
C_b(n+m) >= C_b(n) C_b(m) (van Engelenburg-Lis, arXiv v2 Lemma 19,
after free exhaustion and translation invariance). A submultiplicative
sequence has a limit of -(1/n) log C_b(n) by Fekete's lemma.
Order of limits, kept everywhere below: FIRST the infinite-volume limit
L -> inf at each fixed integer separation n, THEN n -> inf. This matters
and is never swapped.
1.3 Critical point.
b_c = inf{ b > 0 : m(b) = 0 }. (1.3)
The exponential-versus-polynomial dichotomy of van Engelenburg-Lis gives
0 < b_c < inf, and the threshold b_c itself lies in the massless
(power-law) regime.
1.4 The theorem.
THEOREM 1.1 (main). For the model (1.1) and the threshold (1.3),
lim_{n -> inf, n >= 2} log C_{b_c}(n) / log n = -1/4 ,
equivalently: for every eps > 0 there is n_0 with
n^{-1/4-eps} <= C_{b_c}(n) <= n^{-1/4+eps} for all n >= n_0 .
No claim is made that n^{1/4} C_{b_c}(n) has a limit; multiplicative
logarithmic corrections are allowed.
One external, deliberately non-sharp input (a published result, so it
costs us nothing new):
PROPOSITION 1.2 (critical polynomial lower bound; vEL). For every n>=1,
C_{b_c}(n) >= 1/(8n).
Proof. Theorem 1 of van Engelenburg-Lis is stated for free-boundary XY
spins in the same convention as (1.1); with sigma_x = e^{i theta_x} its
density is exactly ours. It gives exponential decay below a threshold and
the bound 1/(8|x-y|) at and above it. Those alternatives are precisely
"positive mass" and "zero mass", so their threshold is (1.3). Put
x=0, y=n e_1. Aligning the real parts by reflecting all angles removes the
imaginary part. QED (1.2).
1.5 Why the answer is 1/4.
(a) DUAL HEIGHTS. The XY model is exactly dual to a random integer height
h (in units of 2 pi) on the faces of the lattice. The weight of an edge
difference 2 pi j is the Bessel weight p_b(j) = e^{-b} I_j(b), the law of
the difference of two independent Poisson(b/2) variables. All the action
is in the gradients of h.
(b) CORRELATION = VORTEX PAIR. The spin correlation at separation n is a
ratio of two height partition functions: numerator twists h by +2 pi
around one endpoint and -2 pi around the other (a "vortex pair"),
denominator has no twist. So the whole problem is the response of h to
one inserted vortex pair.
(c) ONE COEFFICIENT. On scales large compared with the lattice, h behaves
like a Gaussian free field whose covariance is a(b) times the inverse
lattice Laplacian. The single number a = a(b) in [0,inf) is the entire
fluctuation strength of h.
(d) COST OF A VORTEX PAIR. Measure |z| in SCALED coordinates where the
two points are one unit apart (a lattice separation of n becomes 1 here).
Near a unit vortex the phase winds once around, so |grad arg z| = 1/|z|.
Its Dirichlet energy in an annulus 1/n < |z| < 1/2 is
int |grad arg z|^2 dA = 2 pi log n + O(1),
and a vortex pair costs twice that: 4 pi log n + O(1). In a Gaussian
height field of strength a the total energy enters the partition
function with a prefactor 1/(2a), so the numerator picks up
exp( -(1/(2a)) * 4 pi log n ) = n^{-2 pi / a} .
Hence C_b(n) = n^{-2 pi/a + o(1)} whenever a > 0.
Upper bound (Part C): split the vortex pair into disjoint dyadic
annuli; each annulus contributes a real tilt charged against a.
Lower bound (Part E): pin little microscopic holes at the sources
and compare the twisted and centered partition functions sector by
sector; the twist shows up as the Gaussian energy above.
(e) THE GAP. The coefficient a is either exactly 0, or at least 8 pi.
No positive value strictly below 8 pi is possible. The tool is an
amplitude z_m in (0,1], a characteristic function of the height
field's four outer-face averages on an m-by-m square (so it behaves
like a probability of alignment). Theorem B.2 compares an m-square to
a G-times-larger square, for a large fixed integer G, and finds
z_{Gm}/z_m >= c_b G^{2 - a/(4 pi)}. If 0 < a < 8 pi the exponent is
positive, so some integer G makes the right side exceed 2; iterating
m -> Gm -> G^2 m -> ... then forces z to grow without bound, which a
quantity living in (0,1] cannot do. The constant 8 pi is exactly where
the growth turns off. See Part B, Theorem B.2.
(f) PINNING a AT b_c. Two independent facts trap a(b_c):
* a(b_c) = 0 is impossible: then part (d) gives decay faster than any
power, contradicting the polynomial lower bound Proposition 1.2.
* a(b_c) > 8 pi is impossible: the openness theorem (Part F) says then
a > 0 on a whole interval around b_c, in particular at some
b < b_c, which by (d) makes m(b) = 0, contradicting the very
definition of b_c.
So a(b_c) = 8 pi and the exponent is -2 pi/(8 pi) = -1/4.
That is the entire argument.
1.6 Roadmap and the [BKT] dictionary.
PART A The dual height picture; Bessel weights; the finite Gaussian
comparison rules and the reference-cost bounds. These are the
only tools we need from the Gaussian side, imported through
[BKT 2.1].
PART B The coefficient a(b): existence (free limits), and the gap
a(b) in {0} union [8 pi, inf).
PART C Spin upper bound C_b(n) <= n^{-2pi/a+o(1)}.
PART D Hard pins: bands, cable exploration, vanishing reach, the
separated pin interaction, and bond restoration. This is the
technical heart that lets us localize.
PART E Spin lower bound C_b(n) >= n^{-2pi/a-o(1)} when a>0, via the
winding-sector comparison on geometric annuli.
PART F Openness: a(b) > 8 pi implies a stays positive on a
neighbourhood of b.
PART G Assembly at b_c.
Named [BKT] results used (quote them, do not re-derive):
[BKT 2.1] finite Gaussian comparison ineualities (Regev-
Stephens-Davidowitz lattice ineuality), for quadratic
lattice laws: precision up, covariance down; ratios <= 1.
[BKT 2.2] free-rectangle Gaussian limit for a quadratic field
(cutting/folding argument we mirror in Part B).
[BKT 2.3] uniform fractional mean for a quadratic field.
[BKT 2.4] face-ratio growth for a quadratic field.
[BKT 3.1] band density bounds for a quadratic field.
[BKT 3.2] path-Poincare / anchoring for a quadratic field.
[BKT 3.6] size-bias tilt bound for a quadratic field.
[BKT 3.7] vanishing reach for a quadratic field.
[BKT 3.11] geometry conventions (face-connected pieces, no
opposite-only junctions).
[BKT 3.12] real Dirichlet-form approximation and Poincare.
[BKT 3.13] mixed-domain scaling for a quadratic field.
[BKT 3.14] Gaussian localization / separated-pin determinant.
[BKT 4.3-4.4] RG block supports and regulator.
[BKT 8.3] analytic renormalisation-group map for small activities.
[BKT 9.1-9.34] entry theorem and its finite-expansion ingredients.
[BKT 10.3,10.10,10.11] Gaussian curves, gapped ineualities, and the
terminal gradient diagnostic.
For every such import we say exactly which hypothesis it needs and check
it in the Bessel setting; that checking is the model-specific work.
===============================================================================
PART A. DUAL HEIGHTS, BESSEL WEIGHTS, AND THE GAUSSIAN TOOLS
===============================================================================
Summary of this part. We rewrite the spin model (1.1) exactly as a random
integer height. The only probabilistic input it needs -- how a single edge
difference is distributed -- is the Bessel weight. We then record three
tools, all of which are comparisons between Gaussian (quadratic) laws and
all of which survive the passage from a quadratic approximant to the true
Bessel weight: the finite comparison rules, the reference-cost bounds, and
the restoration sandwich. After Part A the Gaussian machinery of [BKT] can
be applied to the XY heights, and Parts B-G use nothing else.
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A.1 The dual height law: Bessel weights.
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Start from the one-variable identity behind every duality. For real theta,
exp( b cos theta ) = sum_{ j in Z } I_j(b) e^{ i j theta }, (A.1)
where I_j is the modified Bessel function of order j. (This is the
generating series of the I_j's; I_{-j}=I_j, and at theta=0 it gives the
standard sum_j I_j(b) = e^b.) Normalise:
p_b(j) := e^{-b} I_j(b), sum_{ j in Z } p_b(j) = 1, p_b(j)=p_b(-j)>0. (A.2)
p_b is the Skellam(b/2,b/2) law: it is the law of P1 - P2 with P1,P2
independent Poisson(b/2). Indeed, for real t,
sum_j p_b(j) e^{t j} = e^{-b} sum_j I_j(b) e^{t j}
= e^{-b} exp(b cosh t)
= exp( b ( cosh t - 1 ) ) (A.3)
= exp( (b/2)(e^t - 1) ) exp( (b/2)(e^{-t} - 1) ),
which is the MGF of P1-P2. Because p_b is even, every height law below is
inversion symmetric, so the spin correlation below is real and all
characteristic functions are even and positive at the origin.
The dual height model. In a finite spin box, view the sites of Z^2 as the
VERTICES of the primal graph and let the FACES be the elementary squares.
Put an integer height h in 2 pi Z on each face (one representative, or mod
common translations -- see below). Weights: to every dual edge crossing a
primal edge, give the increment h_f - h_g = 2 pi j the weight p_b(j); the
overall height law is
dmu^H(h) proportional to prod_{ dual edges {f,g} } p_b( (h_f-h_g)/(2 pi ) )
* (a fixed sum over a common-translation representative). (A.4)
"Averages on a finite graph use counting measure." An UNORIENTED primal
edge is counted ONCE, so the product runs over unoriented dual edges; this
matches "b cos(theta_x-theta_y) with each {x,y} once" in (1.1).
A free connected height component carries an overall shift h -> h + 2 pi c.
We identify heights modulo 2 pi Z (a pinned representative) whenever we
evaluate only translation-invariant tests; when a confining observation
controls the constant we instead SUM over its translation copies against
that observation (Section A.4 and Part D). Either convention is stated
where used.
The lattice Laplacian A. On the retained primary dual graph,
< g , A g > := sum_{ primal edges {x,y} } ( g_x - g_y )^2 , (A.5)
the unit-conductance graph Dirichlet form. With mesh n^{-1} this
converges to the continuum Dirichlet integral int |grad g|^2. A is the ONLY
Laplacian we use for heights; the "precision A/a" of the Gaussian field of
strength a means: covariance a A^{-1}, energy <v,Av>/(2a). Distinguish A
from the unit-conductance nearest-neighbour Laplacian "A_cell" on coarse
averaging cells used in Part F; they are different graphs.
-------------------------------------------------------------------------------
A.2 Duality: a spin correlation is a vortex-pair partition ratio.
-------------------------------------------------------------------------------
LEMMA A.1 (free-boundary duality; faithful to [vEL Section 2] and
van Engelenburg-Lis duality 2025). In a finite free spin rectangle, the
two-point spin correlation
E_{mu_{L,b}}[ cos(theta_0 - theta_{n e_1}) ]
equals the ratio (twisted height partition sum) / (centered height
partition sum), both with weights p_b and exterior face height fixed at
zero. The twist is an integral connection with circulations +2 pi around
the source vertex 0 and -2 pi around the sink vertex n e_1, and zero
circulation around every other vertex and around the outer boundary (for
interior sources it can be taken to vanish near the outer boundary). After
cutting along paths, the same description holds chart by chart, WITHOUT
identifying heights across a cut.
Proof.
(1) Fourier-expand each edge factor via (A.1). Summing over every spin
theta_x in R/(2 pi Z) makes the total exponent's theta_x-coefficient vanish,
i.e. it forces the integer edge current (the set of labels j on edges) to
be DIVERGENCE FREE at every vertex. That is the denominator: height
partition = sum over divergence-free integer currents of prod p_b.
(2) Insert the two-point factor e^{ i (theta_0 - theta_{n e_1}) } (its real
part is the cosine; and the insertion with the opposite sign gives the
same sum because reversing all currents maps one to the other, so taking
the real part changes nothing). This shifts the divergence condition to
"divergence = unit source at 0 minus unit source at n e_1". That is the
numerator.
(3) The factors e^{b} cancel between numerator and denominator under the
normalization (A.2).
(4) Current -> height. Rotate currents onto dual edges. In a planar
rectangle a divergence-free integer current is a difference of integer
face heights with exterior height zero; this is true because summing the
rotated current along dual paths is path-independent exactly when the
enclosed primal vertices have zero divergence. For the numerator, subtract
one fixed unit current on a path from 0 to n e_1; the remainder is again
divergence free and is a height gradient. The subtracted fixed unit current
IS the twist: its circulation is +2 pi around 0, -2 pi around n e_1, and 0
elsewhere.
(5) Changing the path of the subtracted current is an integer change of
height variables, so the partition sum is independent of the chosen path.
Equivalently the numerator's heights are multivalued with monodromy +2 pi
at 0 and -2 pi at n e_1; they are single-valued on any simply connected
chart, and releasing identifications across a cut gives the stated cut
version. QED (A.1)
Consequence, used all the time. C_b(n) is a "response of the height field
to one inserted vortex pair". If the height field on large scales is
Gaussian with strength a, the response is governed by the minimal energy
of that vortex pair, which is where n^{-2 pi/a} will come from. No spin
physics remains: the rest is the analysis of the height field subject to a
twist.
-------------------------------------------------------------------------------
A.3 The finite comparison rules: from quadratic approximants to Bessel.
-------------------------------------------------------------------------------
The Bessel weight (A.2) is a limit of honest quadratic (Gaussian) lattice
laws. That is the whole bridge: any comparison ineuality that is valid for
Gaussian lattice laws and is stable under the limit passes to the Bessel
law. We make that precise.
Chain approximation. On one primary edge, replace p_b by N bonds in series,
each carrying a normalized integer increment weight
q_N^{ j^2 }, q_N := b / (2N). (A.6)
For all large N this is a genuine positive-definite Gaussian weight on Z.
If a primary edge carries the sum of its N chain increments, then for every
fixed real t, uniformly on compacts of t,
( sum_j q_N^{j^2} e^{t j} ) / ( sum_j q_N^{j^2} )
= 1 + (b/N)(cosh t - 1) + O_t(N^{-2}). (A.7)
So the increment of a full chain has exponential moments converging to
those of p_b, and its law converges to p_b. Match the chain endpoints, and
impose the cycle and period constraints, to get a quadratic lattice law on
the enlarged (subdivided) graph. The twisted numerator is an affine
translate, parallel to the denominator's lattice.
LEMMA A.2 (finite comparison rules; [BKT 2.1] applied through (A.6)).
On a fixed primary graph, allow additional nonnegative quadratic forms in
real linear observations, centered equalities, and independent Gaussian
observation noises. Then:
(i) Increasing centered precision DECREASES centered covariance and every
centered real moment-generating function. Covariance on any translate
of the field (including after a fixed real linear tilt) dominates the
centered covariance. In particular, for a centered inversion-symmetric
law and a real linear test X,
log E[ e^{X} ] >= (1/2) Var(X). (A.8)
(ii) A shifted-to-centered partition ratio is <= 1, and it DECREASES when
the same positive precision is added to the affine and centered
coordinates. "Shift" may be nonzero equality values, an integral
connection, or the centers of observation penalties. With a fixed
real linear statistic added to the numerator exponent the MONOTONE
direction still holds, but the bound by 1 need not. Equality
constraints are imposed by infinite penalties whenever the affine
constraint set is nonempty.
(iii) Centered characteristic coefficients are nonnegative and INCREASE
with precision; their logarithms have Hessian, at every phase, at
least the Hessian at zero. A phase on a free component is allowed
whenever its coefficient total is an integer; in the invariant
finite-graph cases used below these coefficients are positive.
Released constant modes are handled modulo translations or by a confining
observation; real tests used when releasing a constant annihilate it.
Proof. The enlarged chain law (A.6) is quadratic, so the covariance and
partition comparisons are exactly [BKT 2.1] (built on the Regev-
Stephens-Davidowitz lattice ineuality). They hold on a proper affine
subspace by restricting to its parallel span and completing the square in
the remaining fixed coordinates. For clarity, with quadratic precision
T + tQ (Q >= 0) the derivative in t of the log shifted-to-centered ratio is
-(1/2) tr[ Q ( Cov_shift - Cov_0 + m_shift (x) m_shift ) ] <= 0, (A.9)
proving not only (ii) but its TILTED form: after a fixed real tilt, complete
the square and the same formula holds with the tilted covariance and mean.
Partition ratios at every listed step converge as we take the fixed primary
graph and then N -> infinity. Domination for the limit uses a spanning
forest: express vertex differences as sums of chain increments, bound the
remaining normalized edge weights by 1, and note forest variables have
uniformly bounded exponential moments; a confined constant is summed
against its confining Gaussian. This also dominates fixed real tilts and
polynomial factors, and the normalizers of the independent chain laws agree
in the two sectors. Infinite centered penalties realize the equalities;
Poisson summation on the enlarged lattice gives (iii). Taking the fixed-
graph limit gives the stated rules for the Bessel law. Vanishing auxiliary
masses justify releasing translation modes; alternatively use integer
gradient coordinates. QED (A.2)
Reading of (A.2). We may now treat the Bessel height as if it were a
Gaussian field for the purpose of COMPARISONS and CHARACTERISTIC FUNCTIONS,
even though it is not quadratic. What is NOT available and is never used:
a uniform-in-subdivision bound on the bare microscopic Gaussian variance
(the inverse precision of the subdivided chain).
-------------------------------------------------------------------------------
A.4 Reference cost: the right norm for Bessel heights.
-------------------------------------------------------------------------------
The comparisons of A.3 need a quantity that stays uniformly controlled no
matter how fine the subdivision. That quantity is not a bare Gaussian
variance; it is the unit-edge flow cost of a test.
LEMMA A.3 (reference-cost bounds). Suppose a primary test h -> X(h) has a
flow representation
X = sum_e w_e (grad_e h), R(w) := sum_e w_e^2 .
Then in any centered height law with any additional centered precision,
Var(X) <= C_b R(w), E|X|^r <= C_{r,b} R(w)^{r/2}, (A.10)
and for each fixed real t,
E[ exp( t X / sqrt(R(w)) ) ]
is bounded uniformly over the graph and over sufficiently fine subdivision.
The constants may be taken uniform for b in a compact subset of (0,inf).
Here R may be minimized over flows with the prescribed divergence and
grounded vertices, and R always refers to the PRIMARY unit-edge graph.
Proof. Keep the independent full-chain increments as variables and release
their endpoint, curl and period constraints. Lemma A.2 bounds the original
centered transform by the product transform of these independent chains.
For one subdivision bond and bounded t, pairing opposite increments j and
-j in its moment generating function gives, for |t| <= R,
log[ ( sum_j q_N^{j^2} e^{2 pi t j} ) / ( sum_j q_N^{j^2} ) ]
<= C_{b,R} t^2 / N . (A.11)
Summing over the N bonds of a chain gives C_{b,R} t^2 for a whole chain. A
flow normalized by sqrt(R(w)) has every coefficient of modulus <= 1, so
independence bounds its log transform at a fixed real argument by a
constant; rescaling gives (A.10). The original Bessel law inherits this by
the fixed-graph limit. A centered primary observation is a real linear
function of the enlarged coordinates, so its squared penalty is a positive
form; cycle, period and endpoint conditions are limits of positive equality
penalties, removed before the independent-chain bound. No uniform estimate
on the inverse microscopic variance is asserted or used. QED (A.3)
For a zero-total profile f the minimum of R(w) over flows is <f, A^+ f>
(the usual split of a flow into its minimizing gradient part plus a
divergence-free part). This identifies reference cost with the Dirichlet
form of the test on the primary graph.
Two consequences we use repeatedly.
(1) PHASE TRANSFER. If a characteristic coefficient phi(s) > 0 is varied by
a zero-total test Y of variance at most V, then
sqrt( -log phi(s + t Y) ) <= sqrt( -log phi(s) ) + |t| sqrt(V/2). (A.12)
Indeed f(s) := log phi(s) <= 0 and f'' >= -V give |f'|^2 <= -2 V f, which
integrates to (A.12). This is precisely the transfer bound [BKT 2.6], now
stated with centered variance, hence (by A.3) with reference cost. An
integer-total phase on a fixed free graph is the same as a zero-total phase
after subtracting an integer point charge; applying (A.12) from the zero
phase proves its strict positivity.
(2) CENTERED RESTORATION SANDWICH. Represent a removed edge by an
independent chain and restore it by the centered equality that matches its
endpoint difference. Adding exterior bonds or centered guard pins only
increases the probability of the required equality. Hence the centered
restoration ratio lies between its free-patch value and its value with a
separating guard pinned to zero. At that pin the interior no longer depends
on the exterior (provided no observation crosses the guard). If every
endpoint of the restored bonds, and the support of any restored form, lies
in a region H, then pinning every primary vertex of H makes restoration a
fixed scalar, also for a chain. Bayes' rule and centered precision
comparison therefore bound the free-to-guard multiplicative gap by
P( h_p = 0 | h_H = 0 ) / P( h_p = 0 ) . (A.13)
Independent-chain scalar normalizers are the same in each member of the
comparison, so they cancel. (A.13) is the "restoration sandwich" used in
Parts D-F; it is valid however small the two pin probabilities are.
===============================================================================
PART B. THE COEFFICIENT a(b): EXISTENCE AND THE GAP a >= 8 pi
===============================================================================
Summary of this part. We define a single number a = a(b) as the large-scale
fluctuation strength of the tilt-gradient of the height, prove it exists
(Theorem B.1), and prove that a nonzero value cannot fall below the
marginal constant 8 pi (Theorem B.2). Both theorems are where the "Gaussian
behaviour on large scales" claimed in Section 1 becomes a theorem.
-------------------------------------------------------------------------------
B.1 The free coefficient exists: a(b) is well defined.
-------------------------------------------------------------------------------
Let B_m = {0,1,...,m-1}^2, and let
g_m(x) := x_1 / m (B.1)
be the normalized horizontal coordinate (an affine "tilt"), so that
A_{B_m} g_m is a unit-flux divergence (the horizontal gradient test). Define
a_m(b) := Var_{B_m,b} ( < h , A_{B_m} g_m > ). (B.2)
There is NO bare quadratic parameter in (B.2): a_m is computed under the true
Bessel law. The tilt <h, A g_m> has a clean reading on the square graph -- it
is (a boundary sum) the difference between the right face average and the
left face average of h, which is why it measures the macroscopic gradient.
THEOREM B.1 (the free coefficient; cf. [BKT 2.2]). There is a finite
a = a(b) >= 0 with a_m(b) -> a. Moreover, let B_n, at mesh n^{-1}, be any
sequence of free axis rectangles whose endpoints converge to those of a
nondegenerate rectangle B. Then for every finite collection of smooth
shifts g defined near closure(B),
log E[ exp( <h, A_{B_n} g(./n)> ) ] -> (a/2) int_B |grad g|^2 . (B.3)
Joint limits are Gaussian, with covariance a int_B (grad g . grad g'), and
all joint moments converge. For zero-total profiles f_n whose scaled
densities converge in L^2(B) to a bounded piecewise continuous f of integral
zero,
log E[ exp( <h, f_n> ) ]
-> (a/2) < f , (-Delta_{B,N})^{-1} f >_{L^2(B)} , (B.4)
with the Neumann Laplacian. Face smears (after subtracting their totals from
a volume smear) are also allowed. Any of these limits may be degenerate when
a = 0.
Idea of proof, and why the constant really exists.
We mirror the cutting/folding proof of [BKT 2.2], but the Laplacian is the
PRIMARY A from (A.5), not a subdivided precision.
Step 1 (cutting an M-square into m-squares). Partition B_M into m-by-m
squares and strips of width at most m (seams). In the affine observation
<h, A g> the only nonzero seam and strip coefficients carry reference cost
O(m^{-1} + m/M) (there are O(M) seam coefficients of size O(M^{-1}) or
O(m^{-1}), by Lemma A.3). Cutting the square into the m-cells removes those
seams; removing edges can only INCREASE covariance (Lemma A.2(i)); and the
retained cell observations are independent across cells. Therefore
sqrt(a_M) <= sqrt(a_m) + C_b ( m^{-1} + m/M )^{1/2} . (B.5)
Let M -> inf, then let m run along a liminf subsequence. The upper bound is
finite, so liminf a_m is finite and, by square symmetry making the covariance
of the two affine observations scalar, (B.5) forces convergence to a single
a. This is existence and finiteness of a.
Step 2 (affine saturation in every rectangle). Cutting B into squares gives
the variance upper bound a int_B |grad g|^2 for AFFINE g. For the reverse
bound, tile an arbitrarily large square by copies of any fixed discrete
rectangle (choose the large side divisible by both side counts) and apply
the same seam estimate, first sending the large side to infinity. If
m_1(n), m_2(n) are the rectangle's side counts and V_n(p) its variance for
g = p.x/n, this yields
sqrt(a) |p| <= ( n / sqrt(m_1(n) m_2(n)) ) sqrt(V_n(p))
+ C_{b,p} / sqrt( min(m_1(n), m_2(n)) ) . (B.6)
So every affine shift attains the upper bound, i.e. it "saturates", along
every admissible rectangle sequence.
Step 3 (smooth shifts via small cells). Cut into cells of small fixed
macroscopic diameter and approximate the gradient on each cell by a constant.
Seams have vanishing reference cost and the interior error is controlled by
the squared L^2-modulus of continuity. Hence
limsup_n Var( <h, A_{B_n} g> ) <= a int_B |grad g|^2 (B.7)
for smooth (and piecewise smooth) g, and for partial gradient observations
on polygonal pieces. Call equality in (B.7) saturation; saturated shifts form
a linear space (on a convergent covariance subsequence the defect is
positive semidefinite with vanishing diagonals on saturated shifts, so every
entry vanishes). Saturation passes to enclosed rectangles and through
gradient-energy approximation (Lemma A.3).
Step 4 (non-affine saturated shifts: reflection + association). Condition on
the whole trace of a site-axis OR diagonal reflection line and pin one of its
vertices. The subdivided quadratic graph is attractive (positive edge
couplings), so its conditioned law is associated, and association passes to
the fixed-primary-graph limit. If a saturated seed is ODD about the line,
the two partial tests on its open sides are reflected negatives; after a
vanishing-cost transport of off-line coefficients to the line, conditional
association makes their conditional cross covariance <= 0, so the
unconditional cross covariance is <= 0. The saturated sum together with the
separate upper bounds (B.7) then forces EACH partial test to saturate. This
produces the axis hinges (x_1 - t)_+ and, with a small extra diagonal-fold
construction, the pieces (F(x) - F(y+t)) 1_{x>y+t} for arbitrary smooth F.
Their mixed derivative is g_{xy}; differences from g are sums of univariate
functions, which already saturate; Riemann sums converge in gradient L^2, so
every smooth g saturates. The only errors are O(n^{-1})-width seams whose
reference cost is O(n^{-1}).
Step 5 (moment generating function on smooth smears). The lower Laplace
bound log E e^X >= (1/2) Var X is Lemma A.2(i). For the upper bound cut into
cells of side comparable to a fixed r, remove seam errors by Holder and the
reference-cost exponential bound (A.10), and use that a cell test has
reference-cost square root O(r), so for fixed p>1,
log E e^{pX} = (1/2) p^2 Var X + O_p(r^3),
the remainder from E[|X|^3 e^{p|X|}] = O_p(r^3) via normalized exponential
moments (NOT a bare Gaussian tail). There are O(r^{-2}) cells; take the
microscopic limit, then r -> 0, then p -> 1. This gives the joint Gaussian
limit (B.3) and moment convergence.
Step 6 (general zero-total smears). Nearest-neighbour Poincare bounds the
reference cost of a density error by a constant times its scaled L^2 norm
squared. Approximate f by finite sums of Neumann cosines, use their smooth
inverse-Laplacian shifts, then remove the approximation. For a shift
n^2 A_{B_n} g converging in density to -Delta g, the boundary coefficients
are O(n^{-2}) on O(n) sites; using the actual rectangle endpoints before
letting them converge handles arbitrary admissible sequences. This proves
(B.4). A face probability can be moved to a strip of width r at reference
cost O(r + n^{-1}) by parallel normal flows, giving the face extension.
QED (B.1)
-------------------------------------------------------------------------------
B.2 The forbidden range: a(b) = 0 or a(b) >= 8 pi.
-------------------------------------------------------------------------------
Write u_m for the uniform probability on B_m, and define the phase
characteristic
w_m(s) := E_{B_m,b} [ exp( i <h,s> ) ], provided sum_x s_x is an integer.
The definition does not depend on the representative of h mod 2 pi Z, and
0 < w_m(s) <= 1. The following theorem is the "gap": it says a positive
coefficient cannot be small, and that if a > 0 the total height phase is
asymptotically uniform.
THEOREM B.2 (uniform phase and forbidden range; cf. [BKT 2.3, 2.4, 2.23,
2.25]). For every b > 0,
a(b) = 0 or a(b) >= 8 pi . (B.8)
If a(b) > 0, then the fractional mean <h, u_m> mod 2 pi converges to
UNIFORM measure on the circle, independently of any fixed finite collection
of the Gaussian observations of Theorem B.1. More generally, for a fixed
nonzero integer l and any zero-total profiles t_m of bounded reference cost,
w_m( l u_m + t_m ) -> 0 . (B.9)
Proof architecture. The only hard part is a ceiling on how much a soft
("glued but not quite") joining can beat a real one. We do it in five
moves: soft joining; a replica identity for the transform; the transform's
explicit form; the sector argument forcing (B.9); and a radial-flow
counting argument making the exponent 8 pi.
(1) SOFT JOINING. Tile a rectangle by free cells, keep their lattice
translations, and assign each seam edge an independent difference of law
2 pi p_b. Let V be the neutral Euclidean space of cell totals, and let P be
a linear flow right inverse on the neighboring-cell graph,
div P(d) = d for d in V. Define the soft mismatch H in V by
<H,d> = sum_{<i,j>} P(d)_{ij}
( <h_i, rho_{ij}> - <h_j, rho_{ji}> - eta_bar_{ij} ) , (B.10)
where rho_{ij} is the uniform seam-face probability and eta_bar_{ij} the
mean of the independent seam difference. Translating cells by constants c_i
moves H by the orthogonal projection of c onto V. Weight the law by
e^{-|H|^2/2}, summing relative translations and taking the common
translation mod 2 pi. Adding exact seam matching makes H = 0 and recovers
the truly joined rectangle. So the soft law has LARGER internal variances
and SMALLER invariant characteristic coefficients than the joined law -- a
comparison that is made rigorous by first adding the auxiliary precision
eta sum_i (cell mean of h_i)^2, Poisson-summing the translation, letting
eta -> 0, and using cell-refinement dominated convergence. In particular the
soft variance of a sum of two internal affine tests is at most their
independent free-cell variance.
(2) FOURIER FORM OF THE SOFT LAW. For a phase with cell restrictions s_i of
totals xi_i, Poisson summation in the relative translations gives, up to a
common scalar, the numerator
sum_{ n_i in Z, sum n_i = sum xi_i } e^{-|d|^2/2}
prod_i w_i( s_i + sum_{j~i} P(d)_{ij} rho_{ij} ) chi_m(P(d)) ,
(B.11)
with d = n - xi, and the denominator is the same with s = 0. The signs of P
are chosen so the displayed cell totals are n_i. The factor chi_m is a
positive seam characteristic function, <= 1, tending to 1 for fixed cell
fluxes as the cell mesh refines (each seam average has variance O(m^{-1})).
For two cells one may use a scalar mismatch penalty, giving (B.11) with
integer sectors l weighted by e^{-l^2/2}. Everything can first be computed
on the subdivided quadratic graph of Lemma A.2 and then passed to the
Bessel law.
(3) THE REPLICA IDENTITY AND THE TRANSFORM. Let rho be the right-face
probability, and set
X = <h, A(x_1/m)>, Y = <h, rho - u_m> .
On the unit square X is exactly the right-face average minus the left-face
average; Theorem B.1 gives joint Gaussian limits with Var X -> a and, by
reflection, Cov(X,Y) -> a/2. Suppose some nonzero l had w_m(l u_m) bounded
away from zero along a subsequence. The reference-cost exponential bounds
(A.10) give a further subsequence on which
F_m(t,v) := E[ exp( i l <h,u_m> + i t X + i v Y ) ]
converges locally with all derivatives to an entire F, positive on real
arguments (by (A.12) and the assumed positive value at 0). To find its form,
take two independent copies h_1, h_2 of a centered subdivided quadratic
height and put S = h_1 + h_2, D = h_1 - h_2. Conditioned on their common
parity class modulo 4 pi Z, the two copies are independent and identically
distributed. With V_{ij} := E(S_i S_j | parity),
E V_{ij} = 2 Cov(h_i,h_j), Cov(V_{ij}, V_{rs}) = 2 kappa(h_i,h_j,h_r,h_s),
(B.12)
where kappa is the fourth joint cumulant (verified by expanding
E S_i S_j D_r D_s and subtracting E V_ij E V_rs = 4 C_ij C_rs; the leftover
is exactly twice the cumulant). In particular Var(V_XX) = 2 kappa(X,X,X,X)
and Var(V_XY) = 2 kappa(X,X,Y,Y), both of which vanish after the chain and
Gaussian limits. Differentiating the phase variables and using conditional
independence (with the error at most sqrt(Var V_ij)) gives
-2 F F_tt + 2 F_t^2 = 2 a F^2, -2 F F_tv + 2 F_t F_v = a F^2 ,
so, since F > 0 on real arguments and reflection gives F_t(0,0) = 0,
F(t,v) = F(0,v) exp( - a t^2/2 - a t v/2 ) . (B.13)
(4) FORCING THE UNIFORM PHASE. Softly join two horizontal m-squares and
test the sum of their internal affine observations. Its variance is at most
twice the free-cell variance and at least its variance in the fully joined
rectangle; the joined full affine test differs from this sum by a seam test
of reference cost O(m^{-1}), hence limiting variance 2a, so the soft
variance also tends to 2a. In (B.11) every sector function has logarithmic
second derivative at least minus twice the free-cell variance; the log
second derivative of a positive sum is the weighted mean of these plus the
variance of the sector log first derivatives, so the soft variance is at
most twice the free variance MINUS that nonnegative sector variance. The
summability of the differentiated sector expansion is controlled by
psi_l |(log psi_l)'|^2 <= 2 V ( -psi_l log psi_l ) <= 2V/e and by
e^{-l^2/2} chi_m <= e^{-l^2/2}; the sector l stays bounded away from 0
(transfer from l u_m to the seam face costs a bounded amount and chi_m -> 1).
By (B.13) its log first derivative tends to -a l (up to a common
orientation), while the zero sector has derivative 0. If a > 0 this
contradicts the vanishing sector variance, hence w_m(l u_m) -> 0. The phase
transfer (A.12), Fourier approximation on the circle, and the Gaussian
limits give the full assertion (B.9).
(5) THE SHARP CONSTANT 8 pi BY RADIAL FLOW. Let rho_bar_m be one quarter of
the sum of the four face probabilities and z_m := w_m(rho_bar_m) > 0. Tile a
Gm-square into G^2 cells (G fixed). In the outer-face phase the cell totals
xi_i are supported at the boundary, sum to 1, and satisfy
sum_i xi_i^2 <= C/G. The denominator of (B.11) tends to 1: its zero sector
is 1; every other sector has a cell of nonzero integer total and tends to 0
by (B.9); Gaussian sector weights permit dominated convergence at fixed G.
Retain numerator sectors n = e_{i_0} with i_0 at least G/4 from the
boundary. The radial-flow construction [BKT 2.25] supplies outward face
fluxes H_{ie} with divergence e_{i_0}, constant outer normal flux 1/(4G),
and, writing
D_{ij} := ( H_{i,+j} - H_{i,-j} )/2, S_{ij} := H_{i,+j} + H_{i,-j},
the bounds
sum_{i,j} D_{ij}^2 <= (1/(2 pi)) log G + C, sum_{i,j} S_{ij}^2 <= C.
(B.14)
(Continuum field V(x) = (x-x_0)/(2 pi |x-x_0|^2) + E(x),
|E| <= C/G, |grad E| <= C/G^2; the correction fixes the constant outer flux
via the zero-Neumann problem after even reflection; face-sum derivative
errors are summable.) The internal flows have divergences e_{i_0} - xi,
linearly independent because the central coordinates are distinct and xi
lives on boundary cells; prescribe P on them by these flows and extend
linearly, so ONE soft precision serves all retained sectors.
In each cell expand the log of its characteristic coefficient around the
symmetric four-face profile of the same total, either 0 or rho_bar_m; the
linear term vanishes by square symmetry, and the logarithmic Hessian lower
bound gives a product lower bound z_m exp( -(1/2) sum_i Var<h,Y_i> ), with
Y_i the face-profile difference. Coordinate reflections split Y_i into two
mutually orthogonal odd face differences (limiting variance a) and an even
remainder (reference cost bounded by C sum_j S_ij^2). Hence
limsup_m sum_i Var<h,Y_i> <= a sum_{i,j} D_{ij}^2 + C_b sum_{i,j} S_ij^2 .
The soft penalty is bounded below uniformly on these sectors
(|e_{i_0} - xi|^2 bounded) and the seam factor tends to 1 at fixed G. There
are at least c G^2 retained sectors; characteristic monotonicity from the
soft law to the joined law yields
liminf_{m->inf} z_{Gm} / z_m >= c_b G^{ 2 - a/(4 pi) }, (B.15)
with c_b > 0 independent of sufficiently large fixed G. If 0 < a < 8 pi,
pick one integer G making the right side exceed 2. Then z_{Gm} >= (3/2) z_m
for all large m, and iterating along m, Gm, G^2 m, ... gives z -> infinity,
contradicting 0 < z_m <= 1. Therefore a is either 0 or at least 8 pi.
QED (B.2)
Why 8 pi. Trace the exponent in (B.15). The radial unit flow has
squared cost sum_{i,j} D_ij^2 <= (1/(2 pi)) log G. Each odd face
difference Y_i has limiting variance a, so sum_i Var<h,Y_i> is roughly
a (1/(2 pi)) log G. The Gaussian product lower bound z_m exp(-(1/2)
sum_i Var<h,Y_i>) in the proof therefore carries a factor
G^{-(a/(4 pi))}, because (1/2) times (a/(2 pi)) is a/(4 pi). On top of
this, at least c G^2 retained sectors each contribute a bounded amount,
giving a factor G^2. Combining, the ratio z_{Gm}/z_m is at least
c_b G^{2 - a/(4 pi)}. The exponent 2 - a/(4 pi) is positive exactly
when a < 8 pi, so a positive coefficient cannot be below the marginal
value 8 pi. Nothing from (B.13) is used here; (B.13) served only the
uniform-phase statement of Part B step 4.
===============================================================================
PART C. THE SPIN UPPER BOUND
===============================================================================
Summary. We now convert the fluctuation strength a into decay of the spin
correlation, giving the upper half of the exponent. The mechanism is the one
advertised in Section 1.5: insert a vortex pair, decompose it into dyadic
annuli, and charge the twist against the Gaussian strength a in each annulus.
PROPOSITION C.1 (spin upper exponent; cf. [BKT 2.2]). For the free-box
correlation of (1.2),
limsup_{n->inf} log C_b(n) / log n <= - 2 pi / a(b) . (C.1)
When a(b) = 0 the assertion means the limsup is -inf, i.e. every fixed
polynomial upper exponent is eventually available.
Proof.
Work in a sufficiently large free spin box; at the very end take its free
limit, so all cuts are internal.
Step 1 (annuli and a test circulation). Put disjoint circular dyadic
annuli of radii r and 2r around each of the two sources, out to radii of
order n. On the fixed annulus 1 < |z| < 2 choose a smooth, tangential,
divergence-free form p, compactly supported inside that annulus and
approximating d arg z in L^2 (a radial cutoff produces such forms).
Discretize p at annular scale r by taking differences of a sampled stream
function on faces. Its pairing with the twisted height differences is
DETERMINISTIC, by circulation, and tends to
D = int p . grad arg z . (C.2)
Use the opposite sign at the negative source, so the two circulations add.
Step 2 (real tilt on the twisted numerator). Insert in the twisted
numerator the factor exp( lambda * (the sum of these deterministic
pairings) ), lambda > 0 real. Now release the endpoint constraints between
a grid of small rectangular cells at every annular scale, keeping the
independent seam increments, and release all remaining exterior constraints
as well. Lemma A.2(ii), in its TILTED form (A.9), lets us bound this
normalized tilted numerator by the product of the transforms of the
released cells.
Each retained cell is simply connected and avoids the two sources, so its
integral connection vanishes up to an integer gauge change: if the
connection on a chart is alpha = grad kappa, the reindexing
h' = h + kappa replaces grad h + alpha by grad h' in BOTH the weight and
the tilt. So the tilt introduces NO leftover real scalar. The tilt sits on
the affine edge differences, the same coordinates as the edge weights.
Step 3 (per-cell bound). Take cells of side comparable to t r, lying inside
the annulus, with t > 0 small and fixed. Only O_t(r) leftover nonzero edge
coefficients survive, each of size O(r^{-1}); their independent-chain log
moment is o_r(1) (Lemma A.3). On the squares, approximate the coefficients by
constant gradients. Then Theorem B.1, Holder's ineuality, and the
reference-cost bound on the coefficient error give
log(transform) <= (lambda^2 a / 2) int |p|^2 + o_r(1) + o_t(1) . (C.3)
The errors are taken first at fixed t, lambda, p; rounding the cells changes
only vanishing reference-cost terms.
Step 4 (summing the annuli). There are log n / log 2 + O(1) annuli per
source. At any fixed n all cuts are internal to a sufficiently large box, so
we may first take the prescribed free-box limit. Averaging the convergent
estimates (C.3) along the increasing annular scales gives, after t -> 0,
limsup_n log C_b(n) / log n
<= (2 / log 2) ( - lambda D + (lambda^2 a / 2) int |p|^2 ) . (C.4)
The factor 2 = two sources.
Step 5 (optimizing). Let p -> d arg z. Then both D and int |p|^2 tend to
2 pi log 2. For a > 0, the optimal choice lambda = 1/a yields
(2 / log 2) ( - D/a + ( a / 2 a^2 ) int |p|^2 )
= (2 / log 2) ( - 2 pi log 2 / a + (1/(2a)) 2 pi log 2 )
= - 2 pi / a . (C.5)
For a = 0 every fixed lambda > 0 is allowed, giving -4 pi lambda, and letting
this fixed parameter be arbitrarily large proves the stated interpretation.
QED (C.1)
Reading. The upper bound used only the Gaussian limit of Theorem B.1 and
the comparison rules; it did NOT need the gap or the pins. The matching
lower bound (Part E) is the hard direction, because it must compare
partition functions in two different winding sectors -- a global, not a
local, statement.
===============================================================================
PART D. HARD PINS: BANDS, EXPLORATION, PIN INTERACTION, RESTORATION
===============================================================================
Summary of this part. To turn local Gaussian behaviour into global statements
(comparing winding sectors in Part E, and openness in Part F) we need two
technical facts about the height field: (D-1) fixing a region's heights
("a hard pin") does not change the Gaussian description away from that
region, and (D-2) distinct pins interact only through a finite, explicit
Gaussian determinant, and typical guard data transfer cleanly. Both are
proved here for genuine Bessel heights, using primary reference costs
(Lemma A.3) in place of bare Gaussian variances, and using a cable
exploration to localize. Nothing in this part is used before Part E.
Throughout Part D fix b > 0 with a = a(b) > 0.
-------------------------------------------------------------------------------
D.1 Geometry, hard pins, and the two theorems.
-------------------------------------------------------------------------------
Fixed geometry. A fixed geometry is a finite union of bounded rectangular
pieces with nondegenerate faces and widths. Pieces in one component are
joined along full interfaces (possibly subdivided into segments). POINT
contacts do NOT identify components, and opposite sectors are never joined
only through a vertex. Equivalently we use the polygonal Lipschitz
geometries of [BKT 3.11], including its cut geometries. The nearest-
neighbour primary lattice has mesh n^{-1}; aligned boundaries may be
rounded by a bounded number of lattice steps, preserving faces, corners,
and positive-width passages, and all assertions hold along such roundings.
Hard pin. A hard pin on a region means EVERY PRIMARY height in that region
is zero. It does NOT pin the internal subdivision vertices. Pin regions are
aligned unions of positive-area boxes.
Reference form. The ordinary unit-edge real Dirichlet form is the
reference form. On a component U with pin region p its limiting energy
space is
H_{U,p} = { v in H^1(U) : v = 0 a.e. on p }, E_U(v) = int_U |grad v|^2 ,
with the obvious modulo-constants reading for an empty pin. Free walls are
understood in the variational (Neumann) sense. The real form approximation,
Poincare estimates and energy-density statements of [BKT 3.12] apply to
these ordinary nearest-neighbour forms.
THEOREM D.1 (mixed height limits; cf. [BKT 3.12, 3.13, 3.14]).
For every fixed geometry above and every a(b) > 0, the massless height law
with primary hard pins has Gaussian Laplace and moment limits for fixed
bounded piecewise continuous density smears. On a component meeting a pin,
the covariance is the inverse of
a^{-1} E_U on H_{U,p} . (D.1)
On an unpinned component the assertion concerns MEAN-FREE smears; its
fractional mean mod 2 pi is asymptotically uniform and independent of the
mean-free Gaussian field. Adding a fixed finite-rank nonnegative quadratic
penalty in box averages gives the corresponding Gaussian law; if the
penalty controls every unpinned component constant, lattice translations are
SUMMED and the limiting constants integrated, giving an ordinary
unprojected Gaussian law. The statements include energy-approximable shifts
and hold along every admissible primary mesh sequence.
The centered high law. Let mu_n be the "high law" obtained by adding
||B_n h||^2 / 2, where B_n is a fixed finite collection of box averages
controlling all component constants, each defining box lying wholly in one
pre-pin component. For positively separated box regions H, p (initially
without hard pins) put
D_n(H,p) := log[ mu_n( h_H=0 | h_p=0 ) / mu_n( h_H=0 ) ], (D.2)
r_{H,n}(b^p) := mu_n( h_H=0 | h_p=b^p ) / mu_n( h_H=0 | h_p=0 ) . (D.3)
Regions lying in different components contribute independently.
Finite-observation sense. Choose an increasing total list of smooth density
probes inside p. A function of h_p has a GAUSSIAN FINITE-OBSERVATION
APPROXIMATION if, for every tolerance, it is approximated in probability by a
function of a sufficiently long fixed list of these probes whose joint law
and function converge to the Gaussian counterparts. The probe list grows
AFTER the primary lattice limit. No pointwise continuum trace, and no
uniform approximation at all microscopic pin values, is asserted.
THEOREM D.2 (separated pin interaction and conditional laws; cf. [BKT
3.14]). Under the centered high law, D_n(H,p) >= 0 converges to the Gaussian
pin interaction for the precision
a^{-1} E_U(v) + ||B v||^2 . (D.4)
For every fixed finite list of density or box smears X, the conditional
Laplace transforms and conditional weak laws given h_p have the Gaussian
finite-observation approximations above. Moreover
0 < r_{H,n} <= 1, E_{mu_n}[ r_{H,n} ] = exp( - D_n(H,p) ), (D.5)
and r_{H,n} has the corresponding Gaussian approximation, whose limit is
strictly positive almost surely. The convergence persists when data are
sampled under a fixed finite linear exponential tilt, with D_n and r_{H,n}
retaining their CENTERED definitions and the limiting law sampled under the
corresponding Gaussian tilt. The identity in (D.5) concerns mu_n, not its
tilts; conditional transforms under a tilted baseline converge by taking
ratios of centered conditional transforms.
Both theorems are proved in D.5. The Gaussian determinant and localization
facts used there are [BKT 3.14], applied to the ORDINARY real nearest-
neighbour form, never to a subdivided bare form.
-------------------------------------------------------------------------------
D.2 A confining band and uniform primary estimates.
-------------------------------------------------------------------------------
Working in one fixed free component U, choose a positive-width band K
between a starting region and a test region, with positive gaps on both
sides. Partition K into rectangles d of side comparable to ell and bounded
aspect ratio. Write h_d for a primary probability average and |d| for its
scaled counting area, and define the band weight
W_{ell,lambda}(h) := exp( - (lambda/2) sum_d |d| h_d^2 ) . (D.6)
Let P_n be the sigma-finite free measure obtained by normalizing one
representative of each common-translation orbit and summing h + 2 pi m over
m in Z (changing the representative only relabels the sum). The band law is
dP_{n,W} = W_hat dP_n , W_hat := W / P_n(W) ,
with finite positive denominator. All limits in Part D are taken in the
order
n -> inf , then ell -> 0 , then lambda -> inf . (D.7)
The subdivision size N is taken large at each fixed primary graph (or first
through a fixed-graph limsup in the uniform estimates below).
Free limits on a piece. Before pins, the free Gaussian limits on U follow
from Theorem B.1 by the enclosing-rectangle argument of [BKT 3.12]: cut the
enclosing rectangle into the prescribed pieces; smooth-gradient seam errors
have vanishing primary reference cost; each partial observation has the
limiting variance upper bound a times its energy; subtracting the bounds for
the other pieces from the saturated enclosing observation gives saturation
on each piece. Energy density and the primary inverse-form approximation
extend this to the required smears. For a component phase of nonzero integer
total charge, precision monotonicity bounds its characteristic function by
the enclosing-rectangle one, so Theorem B.2 gives the INDEPENDENT uniform
fractional mean on each free component.
LEMMA D.3 (band density bounds; cf. [BKT 3.1]). For every fixed lambda > 0,
q >= 0, and 1 <= p < inf,
limsup_{ell->0} limsup_{n->inf}
P_n( W_hat^p (1+|S|)^q ) < inf , S := (h, u) , (D.8)
u the uniform primary probability on U. In the first two limits of (D.7),
fixed density smears under the band law converge with every fixed moment to
the Gaussian field of precision
T_lambda = -Delta_U / a + lambda 1_K . (D.9)
For a smooth shift g vanishing near K, its weak Neumann Laplacian test has
limiting variance a E_U(g) (including free-face terms), and its limiting
fourth moment is bounded uniformly for lambda >= lambda_0 > 0.
Proof sketch. At a fixed band partition, the free limits and the uniform
fractional mean make translation summation converge to integration of the
constant against dc/(2pi). To justify unbounded insertions let
h_K_bar := |K|^{-1} sum_d |d| h_d. The band energy is at least
lambda |K| h_K_bar^2 / 2, while S - h_K_bar is a mean-free test of bounded
primary reference cost, so Lemma A.3 supplies all its moments and fixed
linear exponential moments; summing the Gaussian translation tail gives
uniform integrability for every fixed polynomial insertion, and convergence
of normalizing constants and weighted moments. After that limit integrate the
constant first: on the mean-free Neumann field the remaining quadratic
penalty is the squared L^2(K) norm of the cell projection after subtracting
its K-average, whose covariance C_ell satisfies
sup_ell ||C_ell||_HS < inf (the square's inverse Neumann eigenvalues are
square summable; cutting into rectangles preserves this by [BKT 3.12]). With
t_j the eigenvalues of lambda C_ell,
(1/2) sum_j { p log(1+t_j) - log(1+p t_j) } <= C_p sum_j t_j^2 ,
and Poincare plus band control of the constant give
||v||_{L^2(U)}^2 <= C_lambda ( a^{-1} E_U(v) + lambda sum_d |d| v_d^2 ) ,
so every fixed moment of S under the continuum p-weighted law is bounded.
This proves (D.8) in its iterated-limsup sense (it does NOT assert a lattice
quadratic-exponential moment). Strong convergence of the cell projections in
L^2(K), weak energy compactness and smooth recovery sequences give
convergence of the inverse forms to T_lambda^{-1} exactly as in [BKT 3.1],
hence the finite Gaussian limits. If g vanishes near K then in the weak
Neumann sense T_lambda(a g) = -Delta_U g, so its test variance is
a E_U(g); Gaussian covariance decreases with lambda, proving the fourth-
moment assertion. QED (D.3)
Cable exploration (why we need it). We must pass from the free limits to
HARD PINS and then to CONDITIONAL laws at random guard data, which ordinary
finite-smear convergence does not give. The tool is a sign-cluster
("cable") exploration. Metric-graph Gaussian sign clusters and their Markov
exploration are in [Lupu 2016, Section 3]; the integer-restricted Gaussian
cable in [AHPS2022, Section 7, Prop 7.1]; for Bessel heights a cable using
conditioned differences of Poisson processes in [vEL, Section 2.2, Remark 1].
We use Gaussian cables on the finite series approximants and prove the
uniform primary-observation estimates that pass to Bessel weights. On the
N-subdivided lattice use the Gaussian chain weights of Lemma A.2. Give each
quadratic bond an independent Brownian bridge conditional on endpoint
heights; the bond is OPEN when its bridge does not hit zero. Conditional on
magnitudes and this open graph, nonzero components have independent fair
signs. Exploration reveals a bridge only until its next vertex or first
zero; the unobserved remainder starts at zero and has a shorter bridge time,
hence increased precision. Conditional on a completed exploration, the
unvisited law before band weighting is a centered Gaussian lattice law with
additional zero pins and tightened outgoing segments -- the finite-dimensional
bridge facts of [BKT 3.8]. The remaining transition kernel at a stopped
segment is proportional to p_t(0,h_y) with t no larger than its original
time; the revealed nonzero value contributes only to the exposed part of an
observation and, after band weighting, to its linear cross-term. Only
primary heights contribute to smears or cluster masses; a cluster carrying
primary mass is counted once (e.g. by its least primary vertex); a cluster of
positive macroscopic diameter contains a primary path with bounded
successive steps.
Uniform chain estimate. With q = b/(2N), for |t| <= R,
log[ ( sum_j q^{j^2} e^{2 pi t j} ) / ( sum_j q^{j^2} ) ]
<= C_{b,R} t^2 / N (N large), (D.10)
by grouping opposite j's and using
cosh(2 pi t j) - 1 <= C_R t^2 j^2 e^{2 pi R |j|}. So extending a unit-edge
flow through a full chain costs at most a constant times its squared
coefficient, and additional internal zero pins or shorter cable segments
only improve this centered exponential comparison. It is NOT the bare
series resistance that is used.
LEMMA D.4 (primary path anchoring; cf. [BKT 3.2]). After exposure of a
cluster of scaled diameter at least r > 0, the centered posterior moments of
every fixed primary density smear are bounded in terms of its unit-edge
reference cost, uniformly in subdivision. The remaining full-component mean
has reference cost at most V_r. If D consists of band cells touched by the
exposed path and ell < r/C, then the reference cost of
sum_{d in D} |d| b_d h_d^U is at most
C ell^2 sum_{d in D} |d| b_d^2 , (D.11)
averages retaining the full primary cell denominator. The same holds for
exploration from an entire filled starting region: a probability average in
a cell reached by a path from a fixed multiple of its diameter away has
uniformly bounded posterior reference cost.
Proof sketch. Extend variations on unvisited primary vertices by zero on
the visited primary set. Apply the ordinary unit-edge path-Poincare argument
of [BKT 3.2]: in a fixed-factor enlargement of a touched cell, stop the
primary path after it spans a comparable cell distance; one coordinate spans
a positive fraction of the lattice levels, giving anchored rows or columns.
One-dimensional telescoping on those rows, followed by rectangle Poincare,
bounds the cell's scaled L^2 norm by C ell^2 times its local unit-edge
energy; enlargements have bounded overlap and duality with flows gives
(D.11). A macroscopic anchored rectangle plus global Poincare control the
full-component mean. At a free wall or permitted junction use the bounded
union of comparable rectangles within the same component; their face
Poincare ineualities control differences of rectangle means, and the
no-opposite-only convention ensures these rectangles are joined through
faces, so one anchored rectangle anchors the union. A filled starting region
provides the initial zeros; for a normalized cell average the squared L^2
density is O(ell^{-2}), canceling the local ell^2. These are ineualities for
the PRIMARY reference network: extend the optimizing flows through chains,
use (D.10), impose the posterior constraints and increased precisions by
Lemma A.2, insert the removed internal vertices at zero, ground them and
tighten the first surviving segment. The resulting centered quadratic law is
exactly the unvisited posterior, and the flow identity involves only the
unvisited test (all inserted and visited coordinates are zero in the
comparison). This transfers the ineualities to centered posterior moments,
uniformly in N. QED (D.4)
-------------------------------------------------------------------------------
D.3 Signless cluster masses, bins, reinsertion, and tilts.
-------------------------------------------------------------------------------
Write the sigma-finite cable measure as nu_n(d omega) R_omega(d sigma),
where omega is the SIGNLESS data (magnitudes and open graph) and R_omega
gives the fair independent cluster signs sigma. For a primary smear f put
M_C(f) := sum_{x in C} |h_x| f_x , m_C := M_C(u) , Q := sum_C m_C^2 .
(D.12)
All nonnegative calculations are made by Tonelli first, before finiteness
of the sigma-finite integrals is known. For a mean-free smear f, let R(f)
denote its minimum unit-edge squared flow cost (Part A.4).
LEMMA D.5 (bins, reinsertion, polynomial tails; cf. [BKT 3.6, 3.7]).
For t >= 1 and every integer p >= 1,
nu_n( Q <= 4 t^2 ) <= C (1+t) , (D.13)
int_{Q<=4t^2} ( sum_C M_C(f)^2 )^p d nu_n
<= C_p (1+t) ( t |sum_x f_x| + R(f - u sum_x f_x)^{1/2} )^{2p} . (D.14)
After exposure of a cluster of diameter at least r, the complement squared-
mass sum Q_U has conditional moments of every fixed order bounded uniformly
by constants depending only on that order and r. Consequently, for every
nonnegative intrinsic completed-cluster cost I(C) and all sufficiently large
t = t(r),
int sum_{diam C >= r} I(C) 1_{m_C<=2t} d nu_n
<= 2 int 1_{Q<=8t^2} sum_{diam C>=r} I(C) 1_{m_C<=2t} d nu_n . (D.15)
For every j > 0 there is a sufficiently large M such that
P_n( (1+|S|)^{-M} 1_{t^2<Q<=4t^2} ) <= C_j (1+t)^{-j} . (D.16)
All estimates are uniform in large subdivision and fine primary mesh on the
fixed geometry.
Proof. Given signless data, E_sigma S^2 = Q. On Q <= 4t^2 Chebyshev gives
high probability of |S| <= Ct. For Y = (h,f), the Rademacher fourth-moment
bound and Paley-Zygmund give a fixed positive probability of
|Y| >= c (sum_C M_C(f)^2)^{1/2}. Choose C so the two events intersect with a
fixed positive probability. On every translation orbit, |S| <= Ct allows at
most C'(1+t) translations, and a mean-free Y is translation invariant with
moments of every order bounded by Lemma A.3. This gives arbitrary polynomial
bin tails; integrating them gives (D.14). For nonzero total charge use
Y = (h, f - u sum_x f_x) + S sum_x f_x on the same intersection.
After a macroscopic cluster is exposed, let S_U be the complement signed
mean. Lemma D.4 gives every conditional moment of S_U, uniformly. Jensen in
the remaining fair signs yields Q_U^p <= E_sigma |S_U|^{2p}, so
P(Q_U <= 4t^2 | F_C) >= 1/2 for large t. If m_C <= 2t this event implies
Q <= 8t^2; multiplying by I(C) 1_{m_C<=2t}, conditioning, and summing over
the least-primary-root disintegration proves (D.15), also for initially
extended nonnegative integrals.
For (D.16), choose a small fixed diameter threshold r and finitely many
separated probability box smears p_j so that every cluster of diameter below
r misses one box. If small clusters contribute at least t^2/2 to Q, assign
each a missed box; since M_C(p_j - u) = -m_C there, one mean-free squared-mass
sum is at least c t^2, and its polynomial bin tail of arbitrarily high order
controls this part. Otherwise the macroscopic clusters supply the majorant
1_{t^2<Q<=4t^2} <= sum_{diam C>=r} (2 m_C^2/t^2) 1_{m_C<=2t} 1_{Q>t^2}.
Expose C and write S = sigma_C m_C + S_U. If m_C >= t/2 separate
|S_U| <= m_C/2 from its complement: polynomial damping handles the first
event, arbitrarily high conditional moments of S_U the second. If m_C < t/2
the condition Q > t^2 forces Q_U > 3t^2/4, again giving arbitrary polynomial
decay. Precisely, with the intrinsic cost
I_t(C) := (2 m_C^2/t^2) 1_{diam C>=r, m_C<=2t}, the conditional expectation of
(1+|sigma_C m_C + S_U|)^{-M} 1_{m_C^2+Q_U>t^2} is at most C_K (1+t)^{-K},
uniformly on this cutoff. Under the unweighted base disintegration,
reinsertion then gives
int sum_C I_t(C) d nu_n
<= 2 int 1_{Q<=8t^2} sum_C I_t(C) d nu_n
<= 32 nu_n(Q<=8t^2) = O(1+t) . QED (D.5)
LEMMA D.6 (weighted cluster moments and tilts; cf. [BKT 3.1]). At fixed
lambda, all fixed moments of squared primary cluster-mass sums, and of
signlessly selected signed partial sums, are bounded in the first two limits
of (D.7). For a band cell d, with D_ell^2 := 1 + log(1/ell),
E_W ( sum_C M_C(d)^2 )^p + E_W |d_*|^{2p}
<= C_{p,lambda} D_ell^{2p} + o_n(1) , (D.17)
where d_* is any signed selection determined by signless data with
nonnegative coefficients bounded by the full cell smear. Fixed observation
boxes have a fixed constant in place of D_ell.
Proof. Jensen in the fair signs bounds the p-th moment of a squared-mass sum
by the Gaussian 2p-th smear moment, which is uniformly bounded for
lambda >= lambda_0 since the Gaussian covariance decreases with lambda
(Lemma D.3's fourth-moment assertion). For the logarithmic enhancement,
partition U into r-cells: a cluster of diameter below r meets only boundedly
many neighbouring cells, so by Cauchy-Schwarz and (D.14) its signed second
moment on a fixed bin is at most C_t r^2 (1+log(1/r)), and higher fixed
moments are bounded there too. Interpolation, followed by the weighted tail
transfer, gives their vanishing; resampling first below a threshold below r
and removing the smaller clusters reduces to Lemma D.3, and fair-sign second
moments are the cluster cross sums, whose summands are nonnegative for
nonnegative smears. QED (D.6)
Relative and bad tilts. For an exploration under the band law write
h_d = d_* + h_d^U. Its remaining linear tilt is L = -lambda sum_d |d| d_* h_d^U,
and Lemma D.4 bounds its reference cost by
eps^2 = C_0 lambda^2 ell^2 sum_d |d| d_*^2. For a single macroscopic cluster C
define eps_C^2 := C_0 lambda^2 ell^2 sum_d |d| M_C(d)^2 and
E_2 := sum_{diam C>=r} eps_C^2. Weighted Jensen and (D.17) give, for every
fixed p, lambda, r,
E_W eps^{2p} + E_W E_2^p <= C_{p,lambda,r} ell^{2p} D_ell^{2p} + o_n(1) . (D.18)
If eps <= 1 the centered posterior mu satisfies
1 <= mu(e^L), mu(e^{2L}) <= e^{C eps^2}, mu((e^L-1)^2) <= C' eps^2 (by (D.10)
and precision comparison); Cauchy-Schwarz with fixed-cost fourth moments
shows tilting changes means and second moments of fixed unexposed smears by
O(eps). For an INVERSION-SYMMETRIC event E, symmetry gives the relative
estimate mu_L(E) >= e^{-C eps^2} mu(E), because mu(e^L | E) = mu(cosh L | E)
>= 1. This estimate does not depend on the size of mu(E). Large tilts are
TRUNCATED rather than estimated by a Gaussian tail: for comparison boxes B, i
put Q_B := sum_{diam C>=r} M_C(B)^2, Q_i := sum_{diam C>=r} M_C(i)^2. Since
at most E_2 clusters have eps_C > 1, pointwise
sum_{diam C>=r} M_C(B) 1_{eps_C>1} (1+h_i^2+M_C(i)^2)
<= sqrt(Q_B E_2) (1+h_i^2+Q_i) , (D.19)
whose expectation vanishes by Holder, the actual band moments, and (D.18).
Good tilts obey sum_C M_C(B) eps_C (1+eps_C)(1+M_C(i))
<= sqrt(Q_B)(1+sqrt(Q_i))(sqrt(E_2)+E_2), whose expectation also vanishes.
No quadratic-exponential moment of a Bessel height, and no subdivision-
dependent bare variance, is used anywhere.
-------------------------------------------------------------------------------
D.4 Vanishing reach: an exposed cluster cannot cross the band cheaply.
-------------------------------------------------------------------------------
Setup. Keep a FILLED starting region S (all of it pinned, or reached from a
pin), the band K of (D.6) separating S from a target region B, with positive
clearances eta > 0 between K and both S and B. A "reaching" path is a
connected cluster meeting S and touching B.
PROPOSITION D.7 (vanishing reach; cf. [BKT 3.2, 3.6, 3.7]). Fix
lambda >= lambda_0 and a compact family of cells B at distance at least eta
from S and from K. In the first two limits of (D.7),
P_W( some cluster meeting S reaches B ) -> 0 , (D.20)
and for a fixed cell i at distance at least eta from S,
E_W [ sum_{C : C meets S and i} M_C(B) ] -> 0 . (D.21)
Here M_C(B) is the primary mass of cluster C tested against B (D.12); the
estimates are uniform in the last limit.
Proof.
(1) TWO SEPARATORS AND A PATH-ANCHORING CONSTANT. First choose two
separator strips, clipped to the free component, with a fixed clearance
eta > 0 from the band K and from the filled starting and target regions.
Cover them by cells of radius delta << eta. A reaching path contains, in a
fixed-factor enlargement of the touched cell, a segment spanning c delta. The
path-Poincare constant on this rescaled patch depends only on the fixed
finite face and corner geometry, NOT on delta (at a wall use the adjacent
face-connected rectangular pieces of the same component). Lemma D.4 then
gives a uniform upper bound C_0 for the centered remaining average in a
reached cell.
(2) LOGARITHMIC TRIAL AND THE GAP c_0. A logarithmic trial supported in a
fixed ball away from the band gives continuum prior variance at least
c a log(1/delta) - C in a cell of radius delta. Cut the trial off at a radius
rho < eta/4 fixed before delta; it pays no band mass. At a free wall or
corner restrict it to the component, still an admissible Neumann energy
function. Its cell average is at least c log(rho/delta) - C and its energy at
most C log(rho/delta) + C, uniformly over the bounded cell shapes. So
shrinking the cells raises the prior lower bound WITHOUT raising the
path-anchoring constant C_0. Choose the cells so small that the prior
variance exceeds C_0 + 2 c_0 for some c_0 > 0, uniformly in lambda; at each
fixed lambda, band convergence (Lemma D.3) gives the strict gap c_0 on fine
meshes.
(3) THE EXTERIOR DEFICIT. Write h_B = e_B + u_B, exposed plus unexposed.
The centered posterior is obtained by zero-pinning visited vertices and
increasing cable precision, so its variance for u_B is at most the full
prior variance v_B, and at most v_B - c_0 on a visit to B. Restoring the tilt
changes the averaged second moment and cross term by o(1) (the good- and
bad-tilt estimates of D.3). The identity E_W h_B^2 = v_B then gives
c_0 P_W( B reached ) <= E_W e_B^2 + o(1) .
All contributing clusters are macroscopic with a threshold fixed by the
separation from S, so sign resampling with the same signless selection gives
c_0 P_W( B reached ) <= E_W sum_{C meets S} M_C(B)^2 + o(1) . (D.22)
(4) THE SIZE-BIASED DEFICIT. Bias the prior by |h_x| for x in B and expose
only its cluster. The pre-exposure second moment at a fixed cell i is at
least v_i under this size-biased prior. The bias is measurable at completed
exploration and cancels from the conditional complement law, so the centered
remaining variance at i is at most v_i, and at most v_i - c_0 when the
cluster visits i. Multiply the total second-moment comparison by E|h_x| and
by the primary averaging weight of x, then sum over x in B: this produces
EXACTLY the weight M_C(B), not a count of microscopic clusters. Only clusters
reaching the band or i cause a nonzero tilt or exposed term, and the
separating gaps make all of them macroscopic. The estimates preceding this
step give
c_0 E_W sum_{C : C meets i} M_C(B)
<= E_W sum_C M_C(B) M_C(i)^2 + o(1) . (D.23)
Points with E|h_x| = 0 contribute zero.
(5) CONCLUSION. In (D.22)-(D.23) the right sides vanish in the first two
limits of (D.7) by the band moments of Lemma D.6 and the good-tilt
estimates, so the left sides do too. QED (D.7)
Reading. D.7 is the "locality" statement behind every comparison in
Parts E and F: with high probability under the band law, exploration from a
filled set stays inside its own neighborhood, so a block of one region cannot
influence a distantly separated block. This is what makes the pin
interaction D_n(H,p) in (D.2) a genuinely LOCAL Gaussian quantity.
-------------------------------------------------------------------------------
D.5 Proofs of the mixed limits and the pin interaction.
-------------------------------------------------------------------------------
PROOF OF THEOREM D.1 (mixed height limits).
On a component meeting p, start with a smooth shift g vanishing near the
whole pin but allowed to meet free walls. Choose a separating band between p
and the support of g, with fixed gaps; it may be a union of strips clipped to
the component. The ordinary mixed-domain geometry is [BKT 3.13], and Lemma
D.4 already checks its reference ineualities at faces and junctions.
For the weak Neumann Laplacian test of g, the band variance is a E_U(g).
Explore from every primary vertex of p. ON SUCCESS the exposed test is zero,
and the centered posterior has at least the desired primary pin, possibly
further zero pins, tighter chains, and restricted band precision; hence its
variance is at most that under the desired massless hard-p law. The tilt
errors vanish as in D.3/D.7. ON FAILURE the actual second moment is at most
(its fourth-moment root) x (sqrt of the failure probability). Lemma D.3
bounds that fourth moment uniformly after the first two limits and toward
large lambda, and Proposition D.7 makes the failure probability vanish.
Therefore, along every primary subsequence,
liminf Var_{p}( <h, A g_n> ) >= a E_U(g) . (D.24)
The free variance upper bound supplies the opposite limsup. The free
logarithmic-transform upper bound and the centered variance lower bound
under a real tilt (A.12) give the same value for tilted tests, hence the
Laplace and moment limits. Energy approximation and the primary inverse-form
density extend to the stated smears. On an UNPINNED component the mean-free
part is the limit of the same exploration from a far-away pin; the fractional
mean modulo 2 pi is asymptotically uniform and independent of that field by
Theorem B.2 (the enclosing-rectangle characteristic function bounds each
component phase). Adding a fixed finite-rank penalty and then integrating its
constant is D.3. Along a primary mesh sequence, take the subsequence given by
compactness of the inverse forms and repeat the same finite-probe
comparisons. QED (D.1)
PROOF OF THEOREM D.2 (pin interaction and conditional laws).
Part 1: the interaction limit. Write U for the original finite box-average
penalty and W for a separating band penalty. Let Z_{M,Q} be the partition
sum with penalty M and primary hard pin Q. Cancellation gives
D_U - D_W
= log Z_{U,H u p}/Z_{W,H u p} + log Z_{U,empty}/Z_{W,empty}
- log Z_{U,H}/Z_{W,H} - log Z_{U,p}/Z_{W,p} . (D.25)
Each of the four ratios changes only a fixed finite-rank confining penalty on
one fixed massless pinned or unpinned domain; its common normalization
cancels, so Theorem D.1 gives its Gaussian limit at each fixed band
partition and strength. The real Gaussian determinant theorem [BKT 3.14]
identifies the corresponding interaction difference, giving D_W -> the
Gaussian pin interaction (D.4).
Part 2: D_W -> 0 in the last two limits (D.7). Under the unpinned band law,
explore all clusters meeting p. On the event Sm that H is untouched, the
centered posterior is obtained from the band law conditioned on h_p = 0 by
further primary or internal zero pins and increased precision; its
probability of h_H = 0 is therefore at least mu_W(h_H = 0 | h_p = 0). For
eps <= delta <= 1 the actual tilted posterior thus satisfies
mu_W(h_H = 0 | F) >= e^{-C delta^2} mu_W(h_H = 0 | h_p = 0). Averaging gives
the RELATIVE estimate
e^{-D_W} >= e^{-C delta^2} P_W( Sm and {eps <= delta} ) . (D.26)
This holds however small the two pin probabilities are. By precision
monotonicity D_W >= 0. First the tilt vanishes at fixed lambda (D.3), then
reach vanishes as lambda increases (D.7). Sending delta -> 0 proves D_W -> 0.
Consequently D_n(H,p) -> its Gaussian limit, and D_n >= 0 by the same
monotonicity.
Part 3: conditional laws by regression. Choose the first J smooth density
probes on p and let L_n^{(J)} be their linear combination with the limiting
Gaussian regression coefficients (remove redundant directions, or invert on
the Gram-matrix range). Conditional variance decomposition gives the exact
identity
E| m_n - L_n^{(J)} |^2
= E| X_n - L_n^{(J)} |^2 - E Var( X_n | h_p )
<= E| X_n - L_n^{(J)} |^2 - v_{p,n} . (D.27)
Theorem D.1 identifies both limiting variances, and as J -> inf totality of
the pin probes makes the right side tend to 0: in the Gaussian energy space
it is the squared distance between the full pin projection and the J-probe
projection of the representer of X. Hence the conditional mean m_n is
approximated in L^2 by the finite Gaussian linear predictions.
Integrating the conditional variance ineuality twice in the tilt variable t
gives Y_n := E( e^{X_n} | h_p ) >= exp{ m_n + v_{p,n}/2 }. Both sides have
limiting expectation e^{v/2} (v the full limiting variance). For the right
side use (D.27) and uniform exponential integrability: E e^{q m_n} <= E e^{q X_n}
by Jensen for every fixed real q, so the nonnegative difference tends to 0 in
L^1. This is the conditional Laplace assertion. Apply the same argument to a
countable dense set of linear combinations of any fixed test list; subsequence
extraction and conditional exponential moment bounds give tightness and
identify all conditional finite-dimensional weak limits. Fixed smooth
quadratic penalties follow by the same device.
Part 4: the pin ratio and the singular-value determinant. Precision
comparison at prescribed b^p shows every normalized conditional positive-
penalty ratio is at most 1 and decreases when the penalty is increased;
letting a full coordinate pin on H tend to infinity yields 0 < r_{H,n} <= 1,
and the definition gives exactly E_{mu_n} r_{H,n} = e^{-D_n}.
Let Q_J be increasing finite smooth quadratic penalties exhausting the pin on
H and put R_{J,n}(b^p) := E(Q_J | h_p = b^p)/E(Q_J | h_p = 0). Then
r_{H,n} <= R_{J,n} <= 1, and for fixed J the conditional weak convergence and
the centered hard-p limit identify R_{J,n} with its Gaussian counterpart.
Moreover
E( R_{J,n} - r_{H,n} ) = E R_{J,n} - e^{-D_n} . (D.28)
The interaction limit and the finite-probe determinant approximation of
[BKT 3.14] make the right side tend to 0 as J -> inf after n -> inf. More
explicitly, let s_j be the singular values of the cross projection between
the Gaussian pin spaces of p and of H. Positive separation gives
sup_j s_j < 1 and sum_j s_j^2 < inf. In canonical standard Gaussian pin
coordinates xi_j,
r_H^g = exp{ -(1/2) sum_j ( s_j^2 / (1 - s_j^2) ) xi_j^2 } > 0 a.s. ,
E r_H^g = prod_j sqrt( 1 - s_j^2 ) = e^{-D^g} . (D.29)
Finite probes exhaust these convergent expressions, so Markov's ineuality
in (D.28) proves the finite-observation approximation of the full ratio. In
particular
lim_{eta -> 0} limsup_{n -> inf} mu_n( r_{H,n} <= eta ) = 0 . (D.30)
The persistence under a fixed finite linear tilt is by Holder and the bounds
for both positive and negative fixed exponential moments. QED (D.2)
-------------------------------------------------------------------------------
D.6 Restoration: the two-way transfer between pinned and unpinned laws.
-------------------------------------------------------------------------------
The statements of Part E compare two graphs that differ by restoring a set
of bonds (or positive forms). The following lemma is the exact bookkeeping,
and it is the reason we defined r_{H,n} with a GUARD region: it lets a
negligible-probability event be moved between the cut and the restored laws
in BOTH directions.
LEMMA D.8 (restoration and two-way transfer; cf. [BKT 3.14]). Let mu_n be a
CUT centered high law and let nu_n restore bonds or positive forms supported
inside a core H, separated from the guard p. Include both primary endpoints
of every restored bond, and the support of every restored form, in the hard
core pin. If Q_Delta is the positive restoration factor and
r_Delta(b^p) := E_{mu_n}( Q_Delta | h_p = b^p )
/ E_{mu_n}( Q_Delta | h_p = 0 ) , (D.31)
then
r_{H,n} <= r_Delta <= 1 , (D.32)
dnu_{n,p} / dmu_{n,p} = r_Delta / E_{mu_n} r_Delta , (D.33)
r_{H,n} <= dnu_{n,p} / dmu_{n,p} <= e^{ D_n(H,p) } . (D.34)
Events concerning random guard data whose probabilities tend to zero
transfer in BOTH directions between the two topologies. This typical-data
transfer also holds when either law is sampled with a fixed finite linear
tilt; the displayed identities and ineualities remain statements about the
CENTERED laws and their centered-defined ratios.
Proof. Bayes' formula gives the middle identity (D.33). In it the
denominator E_{mu_n} r_Delta lies in [ e^{-D_n}, 1 ] by (D.5), and this proves
the two outer bounds (D.34). For the transfer, the interaction D_n has a
finite limit (Theorem D.2), so the upper density bound in (D.34) sends
mu_{n,p}(A_n) -> 0 to nu_{n,p}(A_n) -> 0. Conversely, for eta > 0,
mu_{n,p}(A_n) <= mu_{n,p}( r_{H,n} <= eta ) + eta^{-1} nu_{n,p}(A_n) ,
(D.35)
by bounding the contribution with r_{H,n} > eta using the lower bound in
(D.34). Use (D.30) and send eta -> 0 AFTER the lattice limit. Fixed linear
tilts transfer by Holder and the bounds for both positive and negative
fixed exponential moments. All of this concerns the finite-observation sense
at typical data fixed in D.1. QED (D.8)
Why a GUARD and not just H. The ratio r_Delta is a ratio of conditional
probabilities of an equality, and can be astronomically small; (D.32)-(D.34)
are RELATIVE and remain valid no matter how small these probabilities are.
That is precisely what the four-ratio cancellation (D.25) and the
typical-data statement (D.30) buy us: we never need to control a pin
probability absolutely, only its ratio between two topologies.
===============================================================================
PART E. THE SPIN LOWER BOUND: MAGNETIC SECTORS ON ANNULI
===============================================================================
Summary. Convergence of ordinary height observations does NOT by itself
compare partition functions in different winding sectors. We make that
comparison by, on each of finitely many scales, adding confining
observations on fixed microscopic holes and restoring the bonds they cut.
Local Gaussian limits (Part D) then control the guarded restorations, and
an exact finite expansion bounds the accumulated error. The outcome is that
a-twisted numerator and its centered denominator agree up to exp(+-C log n),
while the twist carries the Gaussian energy (1/2a)(4pi log n + O(1)). That
gives the lower bound.
Throughout Part E fix b > 0 with a = a(b) > 0.
PROPOSITION E.1 (magnetic lower bound; cf. [BKT 3.14]). For the prescribed
free-box correlation,
liminf_{n -> inf} log C_b(n) / log n >= - 2 pi / a . (E.1)
In particular m(b) = 0.
-------------------------------------------------------------------------------
E.1 Coordinates and the localized twist.
-------------------------------------------------------------------------------
Divide the coordinates by n, so the two sources are 0 and e_1. Take the
outer box sufficiently large that its boundary lies well outside [-8,8]^2 in
these coordinates. There is a real MULTIVALUED function v with periods
2 pi, -2 pi at the two sources, which equals a signed polar angle plus a
smooth function near each source, and whose single-valued lift is equal to
zero outside [-2,2]^2. Indeed, the difference of the two polar angles has
zero period on an outer annulus; multiply its lift there by a cutoff. Choose
bounded-value branches, with bounded integer jumps. Whenever a local chart
is changed, apply the SAME integer gauge to the lattice height and to v.
The real Gaussian energy that the twist v will cost is
(int |grad v|^2)/2a. The gradient of a single unit-charge angle contributes
int_{1<|x|<N} |x|^{-2} dx = 2 pi log N + O(1); two opposite sources give
twice that, i.e. 4 pi log n + O(1). Cancellation of the cross terms gives
the exact asymptotic
(1/2a) int |grad v|^2 = (2 pi/a) log n + O(1) . (E.2)
This is the number the whole Part is trying to recover as an EXPONENT.
-------------------------------------------------------------------------------
E.2 Annuli, tiles, and fixed magnetic holes.
-------------------------------------------------------------------------------
Around each source draw disjoint dyadic square rings of radii
r_j = n 2^{-j}, stopped when the ring radius reaches a fixed microscopic
size m_* (so r_q = O(m_*) at the innermost scale). Each ring is partitioned
into tiles much smaller than the ring's distance to the source. Between the
tiles keep COLLARS spanning the full interfaces, so that a tile boundary
never coincides with a source, a hole, or a ring boundary. The construction
trims tile junctions; the number of processed tiles is O_M(1 + log n), while
the unprocessed exterior is kept intact (see the annular schematic: two
sources of opposite sign, geometric rings, a local patch with collar).
Two features make the construction robust to the fact that we stop at
microscopic size m_*:
* The LOGICAL ring radii in primary lattice units are n 2^{-j}. Round each
shared logical face once to a dual-lattice division and reuse it on both
sides; the displacement is O(1). A fixed-radius patch spans only
boundedly many neighbouring ring scales, so its ideal normalized shape
and deletion pattern range over FINITE lists.
* The mesh and face-rounding error in mass-cell units is at most
C_M R / r_j <= C_M R / m_*. For large m_* all widths, incidences, and
collars are preserved. Hence the same estimates hold for EVERY integer n,
including nondyadic separations, not only dyadic ones.
Holes and refinement hierarchy. A magnetic hole is a fixed microscopic
region whose primary heights are pinned ("observed") to zero; the holes are
placed at bounded separation from the sources and from the tile junctions.
The comparison of the twisted numerator and the centered denominator uses
the following order of limits, held fixed through the eventual n -> inf:
first a MESH refinement, then a BAND-CELL (tile) refinement, then an
increasing band strength. The magnetic hole size and the localization
parameters are fixed through the whole separation limit. This is the order
fixed here, the same order as in D.1, and it is what permits the
Gaussian localization estimates of Part D to be applied uniformly in n.
Reading. With (E.1) in view, the task is now purely bookkeeping: prove that
Z^s / Z^0 >= exp( - (1/2a) int |grad v|^2 - C_M (1 + log n) eta_R
- O_{m_*,R,M,b}(1) ) , (E.3)
where Z^s is the a-twisted (vortex-pair) partition sum and Z^0 its centered
denominator, both with the fixed holes and the confining band. Everything
from here to D.7's analogue in Part F is the proof of (E.3): the isolation
expansion, the pattern-gain bound, the finite-gas logarithm, and the
sector-comparison estimate.
-------------------------------------------------------------------------------
E.3 The isolation expansion: proxies, separated marks, and pattern gain.
-------------------------------------------------------------------------------
Idea. A twisted numerator differs from its centered denominator by a
handful of "twists" (integral connections) near its sources. We compare the
two by locally REPLACING each twist on a tiny patch by a centered proxy,
recording the difference. The differences are supported in small separated
patches ("marks"); the rest of the product is identical for the two laws.
The bookkeeping therefore reduces to bounding the product over marks.
Fix an order of the processed tiles. An ORDINARY step deletes every
still-present bond from its tile to all other tiles, including the exterior
object. Around its core choose a free patch of fixed tile radius, containing
a separating guard and collars wider than any later support enlargement. In
the current local topology let U_o and U_c be the patch BEFORE and AFTER this
proposed deletion. Both admit pure-gauge lifts; so a twist restricted to a
simply connected patch is an integer gauge and costs nothing there. Replace
the bond factor by the scalar
D_i^s :=
( Z_{U_o}^0 / Z_{U_c}^0 )
* ( Z_{g,U_o}^s / Z_{g,U_o}^0 )
/ ( Z_{g,U_c}^s / Z_{g,U_c}^0 ) (E.4)
times the cut factor, PLUS their difference. In the centered calculation
omit the second fraction, obtaining D_i^0. For the real Gaussian expansion
use its own centered constants in the same formula. Selecting the
difference MARKS this step.
Marks are separated. After a mark, skip every later step within a fixed
number of tile layers, larger than all patches and guard-testing boxes.
Thus marks are separated, their guard interiors are left untouched, and a
subsequently considered proxy cannot depend on which of the two positive
terms is selected inside a previous difference. This is a finite, EXACT
expansion. All deviations from the unmarked sequence are supported in
bounded halos of marks; at the end every surviving cross-tile bond (other
than a bond wholly within the exterior) lies in such a halo or joins it to
the exterior.
Normalization. Normalize each sector by its no-mark term: the product of
its successive proxies and the final isolated-tile partition functions,
including the unchanged exterior. For a specified set I of j marks, write
w(I) for its contribution divided by this normalization. The pattern-gain
lemma bounds every w(I) uniformly.
LEMMA E.2 (pattern gain; cf. [BKT 3.14]). For either model and either
sector, after increasing m_* at fixed sufficiently large R,
| w(I) | <= e^{-c R j} . (E.5)
Here and below c > 0 may decrease, and all fixed constants may depend on
b, a, M, but not on n or on the outer volume.
Before proving (E.5) we record its two consequences, because they are what
(E.3) needs and they explain why E.2 is the right statement.
CONSEQUENCE 1: FINITE GAS LOGARITHM. The normalized partition is a finite
gas of compatible connected mark sets with activities bounded by
e^{-c R j} for size j. Record the smallness argument for SIGNED activities,
since the expansion is signed. In a graph of bounded degree there are at
most D^j connected sets of size j containing a given site, and their closed
neighbourhoods have at most D' j sites. Let Xi(V) be the gas partition sum
with marks allowed only in a set V (ordinary unmarked steps are still
retained). Removing one allowed site x and summing over the connected
polymer P containing it gives
Xi(V)/Xi(V minus {x}) - 1
= sum_{P ni x, P in V} w(P) Xi(V minus N[P]) / Xi(V minus {x}) . (E.6)
Induct on |V|. If smaller deletion ratios lie in [1 - eps_R, 1 + eps_R], the
absolute value of the right side is at most
sum_{j>=1} D^j e^{-cRj} (1 - eps_R)^{-D'j}. For sufficiently large R this is
at most eps_R = O(e^{-c'R}) < 1/2, after adjusting constants. Induction gives
nonvanishing and positivity, also for all restricted sums, and
| log Xi(V) | <= 2 eps_R |V| . (E.7)
Thus signed cancellations cannot invalidate the logarithmic estimate.
CONSEQUENCE 2: SECTOR COMPARISON. Iterating (E.7) over the O_M(1 + log n)
processed tiles, and bounding the leftover halo costs by the same c R, gives
Z^s / Z^0 = exp( O_M( (1 + log n) eta_R ) ) , (E.8)
with eta_R = O(e^{-c R}) and constants independent of n and of the outer
volume. E.8 is the exact bridge used in E.6.
Observation on scope. Lemma E.2 is stated for BOTH the physical (Bessel)
model and the real Gaussian model, and for either sector (twisted or
centered). The physical fact is the centered sandwich of Lemma D.8: a
guarded restoration is the same in the two topologies up to the guard pin
ratio r_Delta in [r_H, 1]. The Gaussian fact is the localized determinant
estimate [BKT 3.14]. Neither needs the bare microscopic variance. This is
why Parts C, E, F can all use the same machinery.
-------------------------------------------------------------------------------
E.4 Proof of the pattern-gain lemma.
-------------------------------------------------------------------------------
We separate three things that could interfere: the CENTERED normalization,
the RANDOM GUARDS, and the LOCAL gain.
(1) CONDITIONAL DIFFERENCES. Prescribe all primary heights u_i on the
guards of the marks. Let K_o, K_c be the two positive partition kernels
inside one guard, in the corresponding marked step. Set
delta_i(u_i) := | K_o - D_i^s K_c | / ( K_o + D_i^s K_c ) <= 1 , (E.9)
and use D_i^0 in the centered sector. Guard and exterior factors cancel in
this expression. Conditional integration of the product of the differences,
followed by the triangle ineuality, gives a sum of at most 2^j POSITIVE
topologies; in each one the integrand carries the extra factor
prod_i delta_i. We have NOT assumed independence of the random guard data.
(2) CENTERED NORMALIZATION. For a positive topology G, including its
ordinary proxy factors, its contribution divided by the no-mark term is at
most
e^{Cj} ( Z_G^s / Z_G^0 ) E_{G,s}[ prod_i delta_i ] . (E.10)
(Only affected components occur; untouched final tile factors cancel.)
Here is the accounting. A centered restoration supported in a core lies
between its free-patch value and its value with a separating guard pinned to
zero, whether the actual exterior is absent or present; Lemma D.8 gives the
direction. For a chain representation, retain the independent chain
increments and restore endpoint matching; all scalar chain normalizers are
the same across these comparisons. A full primary pin on a region H
containing the restored endpoints makes restoration a fixed scalar, so its
free-to-guarded multiplicative gap is at most
P( h_p = 0 | h_H = 0 ) / P( h_p = 0 ) = e^{ D(H,p) } . (E.11)
Theorems D.1 and D.2 and [BKT 3.14] bound this uniformly, and in fact make
its logarithm exponentially small in R after sufficient microscopic
refinement: gradient energy plus cell-average mass is uniformly coercive in
mass-cell units, the clearance is of order R, and the local cell volumes are
polynomial in R.
TELESCOPING. Enlarge the affected region to include every otherwise
untouched isolated tile in these boxes (their shifted ratios cost at most
another e^{Cj}), and include the exterior object whenever it is met. Sum
(E.10)'s analogue over the guard data, keeping the same delta_i, since
cutting beyond a guard does not change its interior kernel. The outside
shifted-to-centered factor is at most one; inside, the cut boxes are
independent and their own shifted-to-centered factors are also at most one.
All twists within a box are now ORDINARY BOUNDED OBSERVATION SHIFTS after
integer gauge changes. So it remains to prove, in either of its two positive
topologies,
E_{box,s}[ delta_i ] <= e^{-c R} . (E.12)
The exact telescoping identity. Call the induced union U (with free
artificial walls) and let alpha be either deletion history. For an actual
deletion e put
T_e^alpha := Z_{U, e before}^0 / Z_{U, e after}^0 ,
Dtilde_e^alpha := its centered proxy CLIPPED to U . (E.13)
A skipped step or an open marked choice is not a deletion; a cut marked
choice is a deletion and carries its proxy. The exact identity is
Z_{U, final alpha}^0 prod_{e in alpha} Dtilde_e^alpha
= Z_{U, initial}^0 prod_{e in alpha} Dtilde_e^alpha / T_e^alpha .
(E.14)
The initial integral is common to both histories. In their QUOTIENT,
changing an original proxy to a clipped one cancels at the artificial
boundary, where the histories agree through a collar wider than the proxy
radius. Final components outside U are identical and have no surviving bond
to U; including an attached exterior object ensures this also for terminal
connections. Each product on the right has O_M(j) factors, each bounded with
its inverse by the centered sandwich, also up to a new free wall. An exterior
object has no deletion events, and its whole inner collar has only O_M(1)
tiles, so its arbitrary microscopic volume adds no charge. This proves the
e^{Cj} of (E.10) without an error proportional to untouched tiles.
(3) THE LOCAL GAIN AT TYPICAL DATA. Write K_t(y,u), t = o,c, in a common
bounded lift, with y = B v, and cancel the factors depending only on the
guard or its exterior. Put
R(u) := K_o(0,u)/K_c(0,u) , r_Delta(u) := R(u)/R(0) ,
F_t(y,u) := log E_{t,0}( e^{<y, B h>} | h_p = u ) . (E.15)
The common observation constant e^{-||y||^2/2} cancels as well. With
P_g(y) := log[ (Z_{g,U_o}(y)/Z_{g,U_o}(0)) / (Z_{g,U_c}(y)/Z_{g,U_c}(0)) ]
the EXACT decomposition is
log[ K_o(y,u) / ( D_i^s K_c(y,u) ) ]
= log[ R(0)/D_i^0 ] + log r_Delta(u)
+ F_o(y,u) - F_c(y,u) - P_g(y) . (E.16)
The first term is O(e^{-R}) in the refinement limit by the centered
sandwich. The full-core pin ratio r_H of the CUT CENTERED law satisfies
log r_H <= log r_Delta <= 0, with H containing the restored bonds. NO
separate scaling limit for r_Delta is required: this sandwich and the
bounded discrepancy delta_i suffice.
Quantitative Gaussian input. Let u_c be the cut harmonic prediction from
the guard in the centered massive energy, and let chi equal one on the core
and vanish before the guard. With T the gradient-plus-observation energy and
y = Bv, Gaussian localization gives
E || chi u_c ||_T^2 <= poly(R) e^{-R} ( 1 + ||y||^2 ) . (E.17)
This is the localized-operator and Hilbert-Schmidt estimate of [BKT 3.14] /
[BKT 9.23]. Restoration decreases centered guard covariance, and the
deterministic prediction due to the tilt is bounded by the same localized
operator. The Gaussian full-core pin ratio satisfies
-2 log r_H^g <= || chi u_c ||_T^2 , (E.18)
because subtracting this cutoff prediction makes the conditional mean zero
on the core without changing its guard values. Orthogonal projection onto
the restored admissible space also bounds the change of prediction between
cut and restored topologies by || chi u_c ||_T; this remains valid when
restoration matches two previously free traces, since vanishing on the core
is an admissible correction. Thus the linear-in-data part of the conditional
Laplace discrepancy is at most C ||y|| ||chi u_c||_T. In fact, for f = B^* y,
centered Gaussian conditional means m_t and conditional covariances C_t, the
same completion of squares gives the quadratic part the same bound.
Since ||y|| <= poly(R) on the compact target families, all these bounds have
an exponentially small right side after weakening c. Use
| (x-1)/(x+1) | <= min{ 1, |log x| } , x > 0 . (E.19)
Convergence in probability of the finite-observation approximations (Theorem
D.2), the bound delta_i <= 1, and Gaussian localization prove (E.10) with
slack. At fixed R increase m_* so that it holds uniformly over the finite
geometries and compact target families. Applying it in the disjoint boxes
and absorbing the factors 2^j e^{Cj} proves (E.5). QED (E.2)
-------------------------------------------------------------------------------
E.5 The sector comparison.
-------------------------------------------------------------------------------
We must be careful about what a "twist" is. It is NOT an additive constant
shift of the field; it is a change of MONODROMY. Fix the outer box, the two
tiny holes, and the confining observation band. Write U for the punctured
domain (the outer box minus the two holes), and let v_harm be the harmonic
representative of the multivalued twist v of (E.1): the unique gradient field
on U with circulations 2 pi, -2 pi about the two sources and vanishing
tangential derivative on the walls of the holes and the outer boundary
(i.e. the minimizer of int_U |grad w|^2 subject to the monodromy and the
boundary conditions). In the GAUSSIAN model the twisted numerator is the
centered Gaussian measure on the AFFINE space { h_0 + v_harm : h_0 zero on
the boundary and on both holes }.
PROPOSITION E.3 (sector comparison; cf. [BKT 3.14]). Fix the outer box, the
holes, M, and then let R be large. For every integer n >= 2 and every
admissible rounding of the annular tiling,
log( Z^s / Z^0 )
= - (1/2a) int_U |grad v_harm|^2
+ O( C_M (1 + log n) eta_R ) + O_{m_*,R,M,b}(1) , (E.20)
where eta_R = O(e^{-c R}) is the pattern-gain constant of Lemma E.2 and
Z^s, Z^0 are the LATTICE (Bessel) sector partition sums.
PROOF. Two pieces: a Gaussian computation, and a lattice-to-Gaussian
comparison.
Piece 1: the Gaussian ratio, done exactly. Write Q(w) = (1/2a) int |grad w|^2.
For the Gaussian model,
Z^s_gauss = int exp( - Q(h_0 + v_harm) ) dh_0 ,
Z^0_gauss = int exp( - Q(h_0) ) dh_0 . (E.21)
Expand Q(h_0 + v_harm) = Q(h_0) + (1/a) <h_0, v_harm>_E + Q(v_harm), where
<h_0,v_harm>_E = int grad h_0 . grad v_harm. Because v_harm is harmonic in U
and h_0 vanishes on the walls of the holes and on the outer boundary,
integrating by parts gives <h_0, v_harm>_E = 0 EXACTLY (the boundary terms
vanish). Hence the cross term is absent and
Z^s_gauss / Z^0_gauss = exp( - Q(v_harm) ) . (E.22)
Now compute Q(v_harm). On U, v_harm is the gradient-part of a pair of
opposite unit-charge angles with a smooth correction near the sources (E.1).
Its Dirichlet energy is 4 pi log(1/m_*) + O(1) computed between the two
holes, plus O(1) for the smoothing and for the cutoff near the outer
boundary, i.e.
int_U |grad v_harm|^2 = 4 pi log n + O(1) (E.23)
uniformly in n, with the O(1) depending on the outer box, m_*, and R. This
is the computation behind (E.2), now performed on the punctured domain; the
holes change the energy by O(1) because v_harm varies by an O(m_*) amount on
them. Combining (E.22)-(E.23),
log( Z^s_gauss / Z^0_gauss ) = - (2 pi / a) log n + O_{m_*,R,M}(1) . (E.24)
Piece 2: lattice vs Gaussian, tile by tile. Compare each lattice sector to
its Gaussian counterpart using the clipped-proxy telescoping of Lemma E.2.
Process the O_M(1 + log n) tiles of E.2 and, at each step, the exact identity
(E.14) moves the lattice graph one deletion closer to the Gaussian graph,
with a multiplicative cost bounded by the pattern-gain constant (E.5) and its
inverse -- uniformly in the target families by Theorem D.2 and (E.17). There
are O_M(1 + log n) such steps, and the pattern gain is two-sided
(|w(I)| <= e^{-cRj} holds for the inverse step as well, since E.2 is stated
for either model and either sector and the sandwich r_H <= r_Delta <= 1 is
symmetric), so the accumulated ratio satisfies
exp( - C_M (1 + log n) eta_R ) <= ( Z^s / Z^0 ) / ( Z^s_gauss /
Z^0_gauss ) <= exp( + C_M (1 + log n) eta_R ) . (E.25)
Multiplying (E.24) by (E.25) gives (E.20). QED (E.3)
Reading. Up to two small errors -- O_M((1+log n)eta_R), which vanishes as
R -> inf at each fixed n, and O_{m_*,R,M,b}(1), which is a constant once the
holes and the localization are frozen -- the lattice sector ratio is the
Gaussian twist energy. The exponent is (2 pi / a) log n, and (E.20) is
therefore exactly the statement that a vortex pair costs 4 pi log n in
gradient energy and 2 pi / a log n in free energy. This is the mechanism
promised in Section 1.5.
-------------------------------------------------------------------------------
E.6 Filling the holes: from the punctured sector ratio to the spin.
-------------------------------------------------------------------------------
The sector comparison (E.20) is a statement about the PUNCTURED domain U
(the outer box minus the two tiny holes). The spin correlation is a ratio of
partition sums on the FULL dual graph. Two steps relate them.
Upper side. Return to the full dual partition of Lemma A.1. Add the
observation penalties only OUTSIDE the holes: this can only lower the
shifted-to-centered ratio, so it is harmless. Compare the weighted full
partition with its punctured version using normalized edge weights p_b. The
centered filling kernel is bounded ABOVE by one: retain a spanning forest
attaching the finitely many removed vertices to their rim, sum the forest
increments using sum_j p_b(j) = 1, and bound the remaining edge factors by
one. (A fixed harmless normalization convention would instead give a fixed
upper bound.) Hence the punctured sector ratio is at most the full one up to
a factor <= 1, i.e. the punctured Z^s/Z^0 is the more pessimistic object on
the upper side.
Lower side. By Proposition E.3, in the SHIFTED punctured law all adjacent
rim values are bounded by a fixed constant with uniformly positive
probability. Fill both holes with any fixed bounded configuration in the cut
coordinates: there are only finitely many required integer differences,
bounded in terms of the fixed choices, including those on the continued
offset path. All their weights p_b(j) are strictly positive. So, on this
event of uniformly positive probability, the filling factor in the numerator
is bounded BELOW by a positive constant c = c_{m_*,R,M,b} > 0. Averaging
over the rim data gives
E_{mu_{L,b}}[ cos( theta_0 - theta_{n e_1} ) ]
>= c_{m_*,R,M,b} ( Z^s / Z^0 ) (E.26)
uniformly for the large outer boxes under consideration. In adding the two
hole pins and the confining band we have NOT introduced a boundary condition
on the spins: the exterior dual face remains fixed at zero throughout, which
is exactly the original free spin boundary condition. No primal boundary
spin has been fixed.
Reading of (E.26). It says the spin correlation DOMINATES the punctured
sector ratio; together with (E.20) that gives
log E_{mu_{L,b}} cos( theta_0 - theta_{n e_1} )
>= -(2 pi / a) log n - C_M (1 + log n) eta_R
- O_{m_*,R,M,b}(1) . (E.27)
-------------------------------------------------------------------------------
E.7 Assembly of the magnetic lower bound.
-------------------------------------------------------------------------------
PROOF OF PROPOSITION E.1. Choose M, then an arbitrarily large fixed R, then
a sufficiently large fixed m_* as in Proposition E.3. (Recall the order:
mesh refinement, band-cell refinement, increasing band strength, and the
hole size m_* fixed through the separation limit.) Combining (E.20) and
(E.26) gives, uniformly in sufficiently distant outer boundaries,
log E_{mu_{L,b}} cos( theta_0 - theta_{n e_1} )
>= -(2 pi / a) log n - C_M (1 + log n) eta_R
- O_{m_*,R,M,b}(1) . (E.28)
Now perform the operations in the order the model demands.
(i) At each FIXED n, first take the prescribed free-box limit L -> inf.
The infinite-volume correlation C_b(n) exists (Section 1) and inherits
(E.28), because every remaining object (holes, band, insertion
constants) is fixed before this limit.
(ii) Divide (E.28) by log n and take the liminf as n -> inf. The term
C_M (1+log n) eta_R / log n = C_M eta_R + C_M eta_R / log n -> C_M
eta_R, and the hole/insertion constants disappear because they are
O(1) relative to log n.
(iii) Finally let R -> inf. Since eta_R = O(e^{-R}) -> 0, we obtain
liminf log C_b(n)/log n >= -2 pi / a.
This is (E.1). Since 0 < C_b(n) <= 1, this polynomial lower bound also
implies m(b) = 0, completing the proof. QED (E.1)
Where every ingredient entered. (E.20): the Gaussian twist energy, and the
pattern-gain/telescoping comparison of lattice to Gaussian. (E.26): the
filling kernel and the punctured rim law. (E.28): the three limits, taken in
the order fixed in Section 1 and in D.1. Nothing in this Part used the
subdivision size N except through the uniform estimates of Part D.
===============================================================================
PART F. OPENNESS ABOVE THE MARGINAL COEFFICIENT
===============================================================================
Summary. We now exclude a STRICTLY supermarginal coefficient at the mass
threshold. The statement we need is not a continuity of a as a function of b
(we neither prove nor use that); it is that if a(b) > 8 pi, then a stays
positive in a whole neighbourhood. The route is: add a small positive
precision on BLOCK AVERAGES of the height. That leaves the fine-scale
coupling a untouched but replaces the large-scale coefficient by
a' = ( a^{-1} + delta )^{-1} , chosen with a' > 8 pi . (F.1)
The strengthened law is therefore comfortably in the gapped regime where a
generic renormalisation-group map for small activities applies ([BKT 8.3]).
We prove ENTRY into its smallness hypotheses for Bessel weights (this is the
model-specific work), run one RG step, and then transfer back to the
ORIGINAL coefficient by a torus variance diagnostic.
Throughout Part F fix b > 0 with a = a(b) > 8 pi, and choose delta > 0 so
small that both delta and a' in (F.1) are fixed (they do NOT move when the
physical parameter b~ varies), and a' > 8 pi. Put alpha := sqrt(a').
-------------------------------------------------------------------------------
F.1 The strengthened law, the observation scheme, and local limits.
-------------------------------------------------------------------------------
Blocks. For a dyadic s_0 let B_{s_0} average over disjoint square cells of
side s_0, and let A_cell be the unit-conductance nearest-neighbour Laplacian
on these CELLS. Add to the height energy the term
(delta/2) < B_{s_0} h , A_cell B_{s_0} h > . (F.2)
All grids are nested and use the common square-symmetry origins. Put
s = t s_0 with t also dyadic, and observe
Y = B_s h + eta , eta ~ N(0, Id) independent . (F.3)
On tori we work modulo the global height translation by 2 pi; when local
observations confine a component constant, we sum its translations instead.
The comparison observation. It is B_s alpha psi + eta', where psi is the
mean-free Gaussian field with covariance A^+ and its constant is uniform
modulo 2 pi/alpha. In particular this is the ORDINARY nearest-neighbour
Gaussian reference: NO averaged precision is added to it. On a cut graph G
write its minimized observation energy as
E_G(y) = inf_v { a'^{-1} <v, A_G v> + || B_s v - y ||_2^2 } , (F.4)
using free constants on the components. Denote by Z_G(y; b~) the
corresponding Skellam partition sum with the added precision (F.2) and the
observation penalty ||B_s h - y||_2^2 / 2.
Local domains. These are the mixed polygonal patches of Theorem D.1,
aligned with WHOLE s-cells. Whenever microscopic bonds between tiles are
deleted, delete also the bonds of A_cell crossing that cut, so a pre-pin
component has precisely its own subcells and face bonds. A hard core pin used
to dominate a restoration includes ALL primary vertices in every subcell
supporting a restored averaged bond; such supports fit inside the fixed core
collars.
THEOREM F.1 (strengthened local limits; cf. [BKT 3.12, 3.13, 3.14]). Fix a
bounded mixed geometry in observation-cell units, with separated positive-area
core and guard pins and observation masses controlling each unpinned
constant. At the central parameter b, the strengthened laws have the
Gaussian smear, conditional-smear and pin-interaction limits of Theorems D.1
and D.2, WITH COEFFICIENT a', in the iterated limit
lim_{t -> inf} lim_{s_0 -> inf} . (F.5)
The limits hold under each indicated pin set. D and r_H retain their
CENTERED definitions, and finite-probe convergence of these conditional
quantities holds also when their data are sampled under any fixed real
observation tilt. This is convergence in probability under the corresponding
guard laws, not a redefinition of the centered pin ratios. At fixed geometry
the conclusions are uniform on compact sets of observation data.
PROOF. For fixed t the additional energy (F.2) is a finite-rank quadratic
tilt of finitely many subcell averages. Theorems D.1 and D.2 give its limit
on each fixed pin set. Integration against the nonnegative quadratic weight
is justified by its boundedness and by the confining observation masses;
fixed linear tilts are handled by the exponential moment bounds. Free
component constants are treated by the joint uniform-phase limit before
summing their confining translations.
The full pin interaction also transforms by finite-dimensional quantities. If
W is the added quadratic weight and subscripts denote centered conditioning
on the designated pins, then
D_W(H,p) - D(H,p)
= log E_{H,p} W + log E W - log E_H W - log E_p W . (F.6)
Each expectation has a strictly positive Gaussian limit. This proves, at
fixed t, convergence to the pin interaction for the TILTED continuum
Gaussian precision, not merely convergence of unconditioned smears.
It remains to send t -> inf. In observation-cell units let P_t denote
averages over squares of side 1/t and B the averages over the unit
observation cells. The continuum form obtained in this first limit is
T_t(f) = a^{-1} int |grad f|^2
+ delta sum_{<i,j>} ( (P_t f)_i - (P_t f)_j )^2
+ || B f ||_2^2 . (F.7)
It converges in INVERSE FORMS to
T(f) = a'^{-1} int |grad f|^2 + || B f ||_2^2 , (F.8)
also with any of the specified zero-region pins. Here is the form argument,
including its normalization. If zeta = 1/t, the averaged-gradient term is a
sum of zeta^2 |D_zeta P_t f|^2. Bounded T_t-energy gives weak H^1 and strong
L^2 compactness on each fixed component. The averaged difference quotients
converge weakly to the directional derivatives on its interior, giving the
lower bound with coefficient delta. Conversely, smooth functions give the
matching upper bound by Riemann sums. The averaged-gradient term is uniformly
bounded by a constant times the continuum gradient energy, by the two-cell
Poincare ineuality. Energy approximation therefore extends recovery to all
admissible functions, using corner cutoffs and density with zero
neighbourhoods of the pins, as in [BKT 3.12]. No component is glued through a
point. This proves the claimed inverse-form convergence under (F.5), and with
it the strengthened local limits at coefficient a'. QED (F.1)
Why (F.7) has coefficient a'. The coefficient of int|grad f|^2 in T_t is
a^{-1}, and the block term, at a scale coarser than the continuum limit,
contributes exactly delta to it. So a'^{-1} = a^{-1} + delta, i.e.
a' = (a^{-1}+delta)^{-1}. The fine-scale coefficient a is unchanged; only the
large-scale one moves, and it is the large-scale one that the RG map of
[BKT 8.3] sees.
-------------------------------------------------------------------------------
F.2 A refinement-independent large-field reserve.
-------------------------------------------------------------------------------
The whole point of the strengthened law is that it has a UNIFORM reserve on
large observation data: the block precision (F.2) is fixed and coercive, so
the physical partition function can be compared with the Gaussian one with a
strictly negative coefficient. This is what lets us discard "bad" (large)
data boxes in the entry expansion without ever touching the bare microscopic
chain precision, and without needing b~ to be close to b.
LEMMA F.2 (large-field reserve; cf. [BKT 9.13, 9.20]). There is c_* > 0,
depending on delta, a, s_0, M and the fixed geometry, but NOT on the
microscopic subdivision, and NOT on the physical parameter b~, such that for
every cut topology G and every observation vector y,
e^{E_G(y)/2} Z_G(y; b~) / Z_G(0; b~)
<= e^{(1 - c_*) E_G(y)/2} , i.e.
Z_G(y; b~) / Z_G(0; b~) <= e^{- c_* E_G(y)/2} . (F.9)
We may take c_* <= 1.
Proof (mechanism). Write the physical partition sum relative to its
Gaussian reference through the exact density identity ([BKT, Section 9]),
up to a normalization that cancels between y and 0,
Z_G(y; b~) = int exp( E_G(y - B_s v)/2 ) x (relative-density factor) dN(v) ,
(F.10)
with N the reference Gaussian of coefficient a'. Completing the square in
F.4 gives Z_G^gauss(y)/Z_G^gauss(0) = e^{-E_G(y)/2}, so the Gaussian part
already supplies exactly the stated factor. The relative-density factor is a
product of positive activities; the uniform bound on each activity's
reference cost in the ordinary unit-edge flow norm is Lemma A.3, and the
variational ineuality of [BKT 9.13] bounds the total activity above y by a
constant times E_G(y) with a coefficient we may take below 1/2, because
delta > 0 gives the observation energy a quadratic surplus over the block
form (F.6) that the activities can see. Multiplying, and absorbing the
surplus into c_* in (F.9), gives the claim at the level of the subdivided
quadratic approximants. Passing to the fixed-graph Skellam limit is Lemma
A.2; the constants do not use the microscopic chain precision or the value
of b~. If necessary decrease c_* so that c_* <= 1. QED (F.2)
Reading. (F.9) is the only large-field input taken from the physical
(Bessel) law. Everything else in the entry expansion below is Gaussian
analysis on the reference of coefficient a', which is why a' > 8 pi is the
whole game.
-------------------------------------------------------------------------------
F.3 Entry: exact expansion, patches, and proxies.
-------------------------------------------------------------------------------
We use the nearest-neighbour Gaussian split and the local subset algebra of
[BKT 9.1] (the entry theorem). That theorem is stated for QUADRATIC discrete
heights; it is NOT being applied directly to the physical law. We prove its
needed entry conclusion using Theorem F.1 and Lemma F.2, while retaining its
model-independent Gaussian estimates and finite expansion.
The reference split and the exact expansion. For this observed law the exact
density relative to its Gaussian reference is, up to normalization,
Z_full(y; b~) exp( E_full(y)/2 ) . (F.11)
This is the [BKT, Section 9] observation identity with the new Z. Use the
finite-range nearest-neighbour covariance C_{<m} of [BKT 9.1], followed by
its positive finite-range shells. These covariances, their ranges, and the
reference energies do NOT involve the added precision (F.2).
Covariant isolation. Use the covariant isolation of [BKT 9.2]: tiles of
diameter comparable to R observation cells, with trimmed vertex tiles, a
finite number of COLOURS of a fixed period q, and bounded-size proximity
groups within each colour. The core, the open patch, and the surrounding
guard have widths fixed multiples of R. At an event, delete all remaining
bonds connecting its tiles to other tiles, counting each bond once and
deleting the corresponding A_cell bonds. If o, c are the current open
patches before and after deletion, use the replacement
B_i(y) := C_i exp( -( E_o(y) - E_c(y) )/2 ) ,
C_i := Z_o(0; b) / Z_c(0; b) . (F.12)
The scalar is fixed at b, while the Gaussian SHAPE always has reference a'.
Replace the old weight by this proxy times the cut weight plus their
difference. A selected difference MARKS the event and SKIPS all later events
within a fixed sufficiently large halo. This is an exact finite expansion,
and a later considered patch avoids each earlier marked core and guard, so
its proxy does not depend on the sign chosen inside that difference.
-------------------------------------------------------------------------------
F.4 Entry, continued: cells, good and bad data, and the reserve.
-------------------------------------------------------------------------------
For a positive topology G, the pattern bound produced by the expansion has
the form
e^{C j_1 + C e^{-c R} Ecal(y)} e^{E_G(y)/2}
Z_G(y; b~) / Z_G(0; b~) E_{G,y}[ prod_i delta_i ] , (F.13)
where Ecal is a face-difference cost in the padded halo and, at prescribed
primary guard data gamma_i,
delta_i := | K_o(y,gamma_i) - B_i(y) K_c(y,gamma_i) |
/ ( K_o(y,gamma_i) + B_i(y) K_c(y,gamma_i) ) <= 1 .
All factors exterior to a full guard cancel in this ratio.
Only O(j_1) centered comparisons are charged for j_1 marks. Telescope
separately the positive branch and the no-mark branch on an enlarged union of
their halos: the two paths agree at its artificial boundary, so clipped
proxies there cancel, and final integrals and proxies outside the affected
union cancel as well. All events remaining inside the union number O(j_1).
This is the finite double telescoping of [BKT 9.3]; it applies unchanged to
the present centered sandwich. Finite-patch continuity preserves the bounded
comparisons near b, with the constants C_i fixed as in (F.12). No error
proportional to the unaffected volume is introduced.
Bounded data and large data. We must distinguish the graph used to CLASSIFY
observation data from the positive topologies being compared. Let D^0 be the
observation-cell graph on the pattern's original padded support D, BEFORE any
isolation deletion. Clip this graph by the P-box grid and put
Ecal_box(y) := sum_{ <x,x'> in D^0, both in the box } ( y_x - y_{x'} )^2 .
(F.14)
In particular a face REMAINS in this cost when an isolation step deletes its
physical bonds. Retain a deep mark only if its complete open patch, restored
supports, and guard lie in the same clipped box; the other marks are among
the boundary losses already charged by the seam estimate. Thus every face
across which any retained restoration, Gaussian proxy, or guard comparison
tests a relative translation belongs to this original graph. Clipping adds
no observation cell or exterior field value to D.
Fix the large-field threshold temporarily. On each connected component of
this original clipped graph translate all heights, observations, and guard
data together by one height period, putting one observed value in
[-pi,pi]. Every factor under comparison is invariant under this common
translation, so bounded Ecal_box now gives a COMPACT set of observation
values by following paths in the finite original graph. Independent
translations of components created by a later cut are not used: a
restoration may join those components and detect their relative translation.
At fixed R, P there are finitely many geometries, positive topologies, and
mark choices. Each Gaussian energy and each physical partition function
still uses its OWN actual positive topology, not D^0.
Decompose log[ K_o/(B_i K_c) ] into (a) its centered zero-guard value, which
is O(e^{-R}) after refinement, (b) the variation of its centered
restoration with guard data, which lies between log r_H and 0, and (c) its two
conditional observation Laplace terms minus the Gaussian proxy change. The
conditional terms and r_H converge by Theorem F.1 under either topology and
its fixed observation tilt. Transfer these estimates using the conditional
approximation and r_H <= r_Delta <= 1. The Gaussian limiting r_H is strictly
positive (D.29), so exceptional sets are negligible in both topologies, also
after fixed tilts. On such sets use delta_i <= 1. Since
|(x-1)/(x+1)| <= min(1,|log x|), bounded moments of each delta_i have the
bounds from (E.17). Holder over the at most poly(P) deep marks, and
absorption of polynomial losses, give the GOOD-BOX bound
C exp( - c R j_deep + R^{-12} Ecal_box(y) ) . (F.15)
For example, maximize (1 + Ecal)^{C j} e^{-R^{-12} Ecal}: since j <= poly(P),
a factor [ P^C j^C (1+Ecal)^C ]^j is bounded by e^{O(j log R) + R^{-12} Ecal},
and the O(j log R) loss is absorbed by the exponential rarity. Isolated
selected-tile errors are made smaller than e^{-R} by the same compact
Gaussian convergence.
Positive branches on good boxes. For every ACTUAL positive box topology G,
Theorem F.1 gives
e^{E_G(y)/2} Z_G(y; b) / Z_G(0; b) -> 1 (F.16)
uniformly on the good-data compact set just described, after the common
period recentering. The reference limit is precisely the Gaussian of
coefficient a' on that same topology, and there are only finitely many such
topologies. Choose the refinements so that on every good box
e^{E_G(y)/2} Z_G(y; b~) / Z_G(0; b~)
<= 2 exp( R^{-12} Ecal_box(y) ) . (F.17)
At b~ = b choose the stronger upper bound 3/2 using (F.16); finite-patch
continuity gives the bound 2 after shrinking the eventual parameter interval.
This is not obtained by removing delta_i from (F.15); the displayed allowance
is kept so that the same quadratic-cost accounting serves both estimates.
Bad boxes. On a bad box discard rarity and apply Lemma F.2: the
centered-normalized factor is at most
e^{E_G(y)/2} Z_G(y; b~)/Z_G(0; b~) <= e^{(1-c_*) E_G(y)/2} . (F.18)
This replaces the bare quadratic bad-box estimate and is the only large-field
input needed from the physical height law.
Gaussian averaging with reserve. Here is the energy accounting for the
original-face costs. For any real trial field v on the uncut padded domain,
two-cell Poincare and the observation residual give
sum_{<x,x'> in D^0} (y_x - y_{x'})^2
<= C { a'^{-1} <v, A_D v> + || B_s v - y ||_2^2 } . (F.19)
Indeed insert the two cell averages of v in each difference: the difference
of averages is bounded by the gradient energy on the two adjacent cells, as
in the two-cell ineuality, and the two residuals are charged at most four
times. All faces are internal to D^0, so the two-cell regions are present
before minimization and have bounded overlap, also at a clipped boundary.
Minimizing over v proves the bound by C E_D(y). This argument does not ask a
cut topology to control a deleted face.
For every positive branch of the expansion, restriction of that SAME uncut
trial field to the disjoint cut boxes bounds the sum of their actual
minimized energies E_G by E_D. The bad-box physical contribution is
consequently at most (1-c_*) E_D/2. All remaining data costs -- the
R^{-12} Ecal_box good-box allowance, seam and halo costs, Gaussian links, and
the eps_1 Ecal_box bad-box threshold charges -- use faces of D^0. Their
summed coefficients are at most
C ( R^{-12} + R/P + e^{-R} + eps_1 ) per face . (F.20)
For halos this follows from their bounded multiplicity; for links one sums
their exponentially decaying displacement weights. The same bound applies to
every restoration branch. Choose R large and then eps_1 > 0 small so that
(F.19) makes their total less than [ h_1 - (1-c_*) ] E_D / 2, i.e. so that
the total positive quadratic cost is at most h_1 E_D / 2 with a strict
reserve. The large-field threshold is chosen only AFTER this reserve is
fixed.
The regulator. The reference covariance contracts the observation energy,
exactly as in [BKT 9.15]. At scale u = m L^j the regulator is
V_j(X,psi) = (1/2) sup_{zeta,e}
{ h_* E_{D_u(X)}( B_s alpha (psi+zeta) + e )
- p_1^{-1} ( ||zeta||_{<u}^2 + ||e||_2^2 ) } , (F.21)
where the first norm is the Cameron norm of the reference pieces below u
(singular covariances mean their Gaussian Hilbert spaces). The covariance
contraction and p_1 h_* < 1 make this supremum finite. Its composition,
localization, and terminal bounds are [BKT 9.2, 4.4]; they involve only the
unchanged reference covariance and whole observation cells.
-------------------------------------------------------------------------------
F.5 Analytic norm, one RG step, and the torus variance diagnostic.
-------------------------------------------------------------------------------
(a) Imaginary directions. Bounded-real-data convergence alone does NOT give
the full analytic norm, so we record the imaginary-direction bound
separately. On the extended quadratic lattice, including (F.2), the theta
parity identity [BKT 9.29] gives
| Z_G(y+iz; b~) | / Z_G(y; b~)
<= Z_G(iz; b~) / Z_G(0; b~) . (F.22)
It survives the fixed-graph Skellam limit. The centered imaginary ratio is
its Gaussian prefactor times a positive characteristic function, and centered
pinning increases that characteristic factor.
(b) Three-lines. Choose r_0 > 4 e h_0, then H > 4 r_0, all before R. Apply
the three-lines theorem on the upper and lower strips to
F_pi(psi+z f) exp( - D_0 z^2 |X| ) with D_0 > C_H. The real boundary has the
real pattern bound (F.15)/(F.17) and the real-translation bound; the other
boundary has the rough-strip bound (F.22). On |z| <= r_0 the real-boundary
harmonic weight is at least 3/4, so the exponential item gains survive with a
smaller constant. Removing the damping costs e^{C|X|}, which is absorbed
because
|X| <= C (j_1 + j_2) + C sum_l ( 1 + d_l / R ) . (F.23)
Cauchy's estimate and real polarization bound the order-r contribution to
the analytic point norm by the pattern bound times (h_0/r_0)^r r^r / r!,
whose sum converges. Hence the real pattern bound, with weaker constants,
holds in the FULL analytic point norm. This is the version the RG map
consumes.
(c) The RG step. Apply [BKT 8.3] with the cosine coordinate omitted, which
is permitted precisely because a' > 8 pi. The reference kernel is fixed and
nearest-neighbour; take its starting scale sufficiently large. The modified
regulator (F.21) is allowed by [BKT 9.2]. All map hypotheses now concern the
reference and the small activities supplied by entry ((F.15), (F.17), (b)).
Why the permitted smallness can be chosen BEFORE the large entry scale. The
Gaussian-curve construction [BKT 10.3] gives exact trajectories by
initializing a small multiple d of the block gradient energy and no lattice
factor. Their expansion has a linear gradient singleton and an O(d^2)
remainder, and these estimates are uniform over sufficiently large starting
scales: scaled block gradient energy has a uniform point norm, and its small
exponential is absorbed by the same regulator reserve. Terminal Gaussian
completion is also uniform, because the tail covariance contracts the
kinetic energy and the terminal torus has a fixed number of blocks. On a
fixed complex disk, (Id - d C^{1/2} A C^{1/2})^{-1} is uniformly bounded, and
determinants cancel in the shifted ratios. The map contraction and the
terminal gradient diagnostic therefore prevent a first exit on that disk,
exactly as in the proof of [BKT 10.3]. Its analytic inverse disk is
consequently uniform in the starting scale as well as in the RG step.
(d) The gapped tube. Use the exact Gaussian parameter u for the gradient
coordinate and subtract the corresponding Gaussian remainder, writing the
transverse remainder as B_rem. The balanced linearization is contracting
transversely and the pure Gaussian curve has no drift. The analytic map
estimates give
|| B_rem,+ || <= [ theta + C ( |u| + || B_rem || ) ] || B_rem || , (F.24)
| u_+ - u | <= C ( |u| + || B_rem || ) || B_rem || . (F.25)
These are [BKT 10.10, 10.11]. Choose a small tube so that the first factor is
at most theta' < 1. Small enough entry data stay in that tube, with
|| B_rem,j || <= (theta')^j || B_rem,0 || and summable gradient drift. In
particular all activities remain as small as desired. The separate torus
remainder obeys the same uniform small bound by its contraction with
quadratic forcing and the exact copying of singleton coefficients in
[BKT 8.3].
(e) Consequence on the torus. Take a periodic smooth mean-zero density
f = f_n of the observation-cell form B_s^* f_n (with s fixed and n -> inf).
Since (i) the strengthened periodic law is strongly confined by the block
precision, (ii) its RG step keeps a strictly positive gradient coefficient
because a' > 8 pi, and (iii) the gapped tube (d) is uniform in the starting
scale, its torus variance does not collapse:
liminf_{j -> inf} Var_{b~, tor}[ <h, B_s^* f_{n_j}> ] > 0 . (F.26)
This holds for EVERY b~ in a real neighbourhood I of b found by
finite-patch continuity in (F.17) and the uniformity of (d): all constants
used are either reference constants or fixed at b.
(f) The torus variance diagnostic. Now we show that (F.26) forces
a(b~) > 0 for the ORIGINAL coefficient. Two facts.
First, adding the block precision (F.2) can only DECREASE centered
variances, by Lemma A.2(i). Hence
Var^{orig}_{b~, tor}[ <h, B_s^* f_n> ]
>= Var^{strength}_{b~, tor}[ <h, B_s^* f_n> ] . (F.27)
Second, IF a(b~) = 0 then the original periodic variance must VANISH. To see
this directly, take a smooth periodic gradient test. Split the torus into
macroscopic rectangles, release the bonds between them, and retain the
internal gradient observations. For a fixed macroscopic partition there are
O(n) seam edges with coefficients O(n^{-1}), so the discarded seams have
reference cost O(n^{-1}); the free-cell variances converge to zero by
Theorem B.1. The variance comparison and Lemma A.3 give vanishing variance
for the original periodic test. The same conclusion holds for smooth
mean-zero density tests by using the periodic inverse Laplacian and the
reference-energy approximation. Finally the observation-cell profile
B_s^* f_n approximates the corresponding smooth density in reference cost as
n -> inf, with error O((s/n)^2) since s is fixed, so its variance also tends
to zero.
Combining (F.26), (F.27) and the last paragraph: for b~ in I the original
periodic variance of <h, B_s^* f_n> stays bounded away from zero, whereas if
a(b~) = 0 it would tend to zero. Hence a(b~) > 0 for every b~ in I.
PROPOSITION (openness). If a(b) > 8 pi, there is an open interval I
containing b, contained in (0, inf), such that a(b~) > 0 for every b~ in I.
The interval depends on b but not on any continuity of a.
Shrinking I inside (0, inf) completes the proof. QED (openness)
===============================================================================
PART G. ASSEMBLY AT CRITICALITY
===============================================================================
Summary. Everything is now in place. At b_c two independent facts pin the
coefficient exactly: the published polynomial lower bound rules out a = 0,
and openness rules out a > 8 pi. The gap theorem then gives a = 8 pi, and the
two exponent bounds meet.
PROOF OF THEOREM 1.1.
(1) Apply Proposition C.1 (spin upper) at b = b_c. If a(b_c) = 0 it gives
limsup log C_{b_c}(n)/log n = -inf, while Proposition 1.2 (the published
lower bound C_{b_c}(n) >= 1/(8n)) gives liminf >= -1. Contradiction.
Hence a(b_c) > 0.
(2) With a(b_c) > 0, Theorem B.2 (the gap) says a(b_c) >= 8 pi.
(3) Suppose the ineuality were strict: a(b_c) > 8 pi. Then the openness
proposition in Part F gives a(b~) > 0 for every b~ in an open interval I
around b_c. Since b_c > 0, the interval I contains some 0 < b~ < b_c.
At that b~, Proposition E.1 (spin lower) gives the polynomial lower bound
C_{b~}(n) >= n^{-2 pi/a(b~) - o(1)}, hence m(b~) = 0 (because
0 < C_{b~}(n) <= 1 and -log C_{b~}(n) = O(log n)). But b_c was defined as
the infimum of the b with m(b) = 0, so m(b~) = 0 with b~ < b_c is
impossible. Therefore a(b_c) = 8 pi.
(4) Feed a(b_c) = 8 pi into the two exponent bounds:
Proposition C.1: limsup log C_{b_c}(n)/log n <= -2 pi/(8 pi) = -1/4,
Proposition E.1: liminf log C_{b_c}(n)/log n >= -2 pi/(8 pi) = -1/4.
They meet, so the limit exists and equals -1/4. QED (1.1)
===============================================================================
WHERE EACH FACT IS USED (a map of the proof)
===============================================================================
model and definition of a(b) Section 1; Theorem B.1 (free limits)
duality: spin corr = sector ratio Lemma A.1
Gaussian rules on Bessel weights Lemma A.2, Lemma A.3 (reference cost)
a(b) in {0} u [8pi, inf) Theorem B.2 (soft joining, radial flow)
upper bound -2pi/a Proposition C.1 (annuli, real tilt)
hard pins and conditional laws Theorems D.1, D.2, Lemma D.8
lower bound -2pi/a Proposition E.1 (sectors, pattern gain,
filling), via Prop E.3
openness above 8 pi Part F (strengthened law, entry,
one RG step, torus diagnostic)
assembly Part G
external non-sharp input Proposition 1.2 (van Engelenburg-Lis)
WHAT IS NOT CLAIMED.
* No statement that n^{1/4} C_{b_c}(n) converges: multiplicative
logarithmic corrections are allowed and are absorbed in the o(1).
* No claim that a(b) is continuous or monotone in b; Part G needs only the
gap (Theorem B.2) and openness (Part F), neither of which differentiates or
interpolates a as a function of b.
* No statement about nonderivative lattice observables (energy density,
susceptibility) at b_c; only the two-point function is treated.
WHAT IS IMPORTED. The generic Gaussian analysis of [BKT] (covariance limits
for a real quadratic field, band and pin localization, the analytic RG map,
and the Gaussian-curve trajectories). Those are quoted with explicit numbers
and their hypotheses are checked here for Bessel weights through the
subdivision comparison Lemma A.2 and the reference-cost bound Lemma A.3.
Every XY-specific step -- the duality, the twist energy, the sector
comparison and filling, the pin transfers, and the openness transfer -- is
proved in this document.
ONE-SENTENCE SUMMARY OF THE WHOLE THING. A vortex pair costs 4 pi log n in
gradient energy, so the spin correlation decays like n^{-2 pi/a}; any nonzero
a is at least 8 pi; and at criticality a is pinned to exactly 8 pi, because
a = 0 contradicts the published polynomial lower bound while a > 8 pi
contradicts the definition of b_c through openness. Hence -2 pi/(8 pi) = -1/4.
===============================================================================
END
===============================================================================