diff b34550f..b1f9bc4 o4_model_mass_bounds_v2.txt
diff --git a/o4_model_mass_bounds_v2.txt b/o4_model_mass_bounds_v2.txt
index 824f1c5..9ab2de3 100644
--- a/o4_model_mass_bounds_v2.txt
+++ b/o4_model_mass_bounds_v2.txt
@@ -10,10 +10,10 @@
-\usetikzlibrary{arrows.meta,positioning,calc,fit}
+usetikzlibrary{arrows.meta,positioning,calc,fit}
-\hypersetup{pdftitle={Sharp mass bounds for the two-dimensional O(4) model},
+hypersetup{pdftitle={Sharp mass bounds for the two-dimensional O(4) model},
pdfsubject={Full transfer gap, exact blocking, finite-volume comparison},
pdfauthor={OpenAI}}
@@ -37,32 +37,32 @@
-\newcommand{\gap}{m_{lat}}
+newcommand{gap}{m_{lat}}
-\setcounter{tocdepth}{1}
-\allowdisplaybreaks[2]
+setcounter{tocdepth}{1}
+allowdisplaybreaks[2]
# Sharp mass bounds for the two-dimensional $O(4)$ model
-\maketitle
+maketitle
Abstract:
We prove sharp mass bounds for the two-dimensional nearest-neighbor $O(4)$
model. For every sufficiently large inverse temperature (inverse coupling)
-$\beta$, the full transfer gap -- meaning the exponential decay rate
+$beta$, the full transfer gap -- meaning the exponential decay rate
of correlations of every local observable, not only the spin -- is
-sandwiched between two positive, $\beta$-independent multiples of the
-asymptotically free scale $\sqrt\beta\,e^{-\pi\beta}$:
-\[
- c\,\sqrt\beta\,e^{-\pi\beta}
- \;\le\; \gap(\beta) \;\le\;
- C\,\sqrt\beta\,e^{-\pi\beta},
- \qquad \beta\ge\beta_0.
-\]
+sandwiched between two positive, $beta$-independent multiples of the
+asymptotically free scale $sqrtbeta,e^{-pibeta}$:
+[
+ c,sqrtbeta,e^{-pibeta}
+ ;le; gap(beta) ;le;
+ C,sqrtbeta,e^{-pibeta},
+ qquad betagebeta_0.
+]
We also prove that the periodic measures have a unique local limit and a
-positive full gap at every finite $\beta>0$. Whether the lattice theories
+positive full gap at every finite $beta>0$. Whether the lattice theories
converge to a continuum quantum field theory, and what its exact spectrum
there looks like, is not decided here; we state separately the
extra convergence hypotheses that would turn our lattice bounds into a
@@ -71,9 +71,9 @@ original technical paper; this version reorganises it and adds plain-English
commentary, taking the stated core estimates on faith where necessary.
-\vspace*{-12pt}
-\tableofcontents
-\newpage
+vspace*{-12pt}
+tableofcontents
+newpage
%---------------------------------------------------------------------
# Overview: what is being proved and how
@@ -82,23 +82,23 @@ commentary, taking the stated core estimates on faith where necessary.
## The model in one paragraph
-Place a spin $\sigma_x\in S^3\subset\R^4$ at every site $x$ of a large
-periodic square lattice $Λ=(\Z/n\Z)\times(\Z/w\Z)$. Give the spins
+Place a spin $sigma_xin S^3subsetR^4$ at every site $x$ of a large
+periodic square lattice $Λ=(Z/nZ)times(Z/wZ)$. Give the spins
the Boltzmann weight
- \dd\mu_{\beta;n,w}(\sigma) = Z_\beta(n,w)^{-1}
- \exp\!(\beta \sum_{< xy>}\sigma_x\cdot\sigma_y)
- \prod_{x\inΛ}\dd\omega(\sigma_x),
+ ddmu_{beta;n,w}(sigma) = Z_beta(n,w)^{-1}
+ exp!(beta sum_{< xy>}sigma_xcdotsigma_y)
+ prod_{xinΛ}ddomega(sigma_x),
where $< xy>$ runs over nearest-neighbor pairs, each counted
-once, $\dd\omega$ is the normalised uniform measure on $S^3$, and
-$Z_\beta(n,w)$ is the normalisation. This is the two-dimensional $O(4)$
+once, $ddomega$ is the normalised uniform measure on $S^3$, and
+$Z_beta(n,w)$ is the normalisation. This is the two-dimensional $O(4)$
nonlinear sigma model on the lattice with nearest-neighbor couplings and
-bare inverse coupling $\beta$.
+bare inverse coupling $beta$.
-The energy $\beta\sigma_x\cdot\sigma_y = \tfrac\beta2 - \tfrac\beta2
-|\sigma_x-\sigma_y|^2$ favours neighbouring spins pointing in the same
-direction, and $\beta$ is how strongly. There is no spin-ordering
+The energy $betasigma_xcdotsigma_y = tfracbeta2 - tfracbeta2
+|sigma_x-sigma_y|^2$ favours neighbouring spins pointing in the same
+direction, and $beta$ is how strongly. There is no spin-ordering
transition in two dimensions (Mermin--Wagner), but there is a length
scale beyond which correlations die exponentially. That length scale is
what we want to bracket.
@@ -108,30 +108,30 @@ what we want to bracket.
On a cylinder of spatial width $w$ the transfer operator $K_w$ advances
the configuration one time step. Its kernel is
- K_w(s,s')=\exp\!( (\beta)/(2)V_w(s)
- +\beta\sum_{i=1}^w s_i\cdot s'_i
- +(\beta)/(2)V_w(s') ),
- \qquad V_w(s)=\sum_i s_i\cdot s_{i+1}.
+ K_w(s,s')=exp!( (beta)/(2)V_w(s)
+ +betasum_{i=1}^w s_icdot s'_i
+ +(beta)/(2)V_w(s') ),
+ qquad V_w(s)=sum_i s_icdot s_{i+1}.
$K_w$ is a positive, self-adjoint, compact integral operator. Its largest
-eigenvalue $\lambda_1$ is simple and its positive ground eigenvector is
-the vacuum. Closing time periodically gives $Z_\beta(n,w)=\Tr K_w^n$.
+eigenvalue $lambda_1$ is simple and its positive ground eigenvector is
+the vacuum. Closing time periodically gives $Z_beta(n,w)=Tr K_w^n$.
-Fix $\beta$, take $w\to∞$, and take a subsequential local limit. On
+Fix $beta$, take $wto∞$, and take a subsequential local limit. On
the resulting reflected Hilbert space there is a positive one-step
-translation contraction $T_\beta$ with $T_\betaΩ = Ω$. The
+translation contraction $T_beta$ with $T_betaΩ = Ω$. The
full transfer gap is
- \gap(\beta) = -\log\norm{T_\beta\big|_{Ω^\perp}}.
+ gap(beta) = -lognorm{T_betabig|_{Ω^perp}}.
-Equivalently, autocorrelations $< F, T_\beta^d F>$ of every
-centred local observable $F$ decay at least as fast as $e^{-\gap d}$,
-where $d$ is the time separation. The gap is the mass; $\gap=0$ means
+Equivalently, autocorrelations $< F, T_beta^d F>$ of every
+centred local observable $F$ decay at least as fast as $e^{-gap d}$,
+where $d$ is the time separation. The gap is the mass; $gap=0$ means
critical or ordered.
The subtlety: controlling the spin two-point function is not enough.
There is a second, rotation-invariant sector of observables -- the
-bond energies $\sigma_x\cdot\sigma_y$ -- and $T_\beta$ has spectral
+bond energies $sigma_xcdotsigma_y$ -- and $T_beta$ has spectral
support in that sector as well. The full gap includes both. Our
theorem bounds this bigger, smaller gap. That is why the analysis has to
carry along every local observable, not just the spin.
@@ -139,64 +139,64 @@ carry along every local observable, not just the spin.
## The answer
[Main result]
-There are finite constants $\beta_0,c,C$, with $0<c<C$, such that for
-every $\beta\ge\beta_0$ the periodic measures (eq:model-ov) have a
+There are finite constants $beta_0,c,C$, with $0<c<C$, such that for
+every $betagebeta_0$ the periodic measures (eq:model-ov) have a
unique local limit, and its full transfer gap obeys
- c\,\sqrt\beta\,e^{-\pi\beta}\;\le\;\gap(\beta)\;\le\;
- C\,\sqrt\beta\,e^{-\pi\beta}.
+ c,sqrtbeta,e^{-pibeta};le;gap(beta);le;
+ C,sqrtbeta,e^{-pibeta}.
-Three comments on the formula $\sqrt\beta\,e^{-\pi\beta}$.
+Three comments on the formula $sqrtbeta,e^{-pibeta}$.
-\item The exponent is not a guess; it is forced by the renormalisation
+item The exponent is not a guess; it is forced by the renormalisation
group (RG). Asymptotic freedom says the running coupling satisfies
- $\beta\mapsto\beta - \gamma - \gamma/(2\pi\beta)+...$ with
- $\gamma=(\log L)/\pi$, and integrating over many scale factors gives
- $\log(inverse length)\sim\pi\beta$. So it is a two-loop
- calculation that produces $e^{-\pi\beta}$ with a $\sqrt\beta$
+ $betamapstobeta - gamma - gamma/(2pibeta)+...$ with
+ $gamma=(log L)/pi$, and integrating over many scale factors gives
+ $log(inverse length)simpibeta$. So it is a two-loop
+ calculation that produces $e^{-pibeta}$ with a $sqrtbeta$
correction.
-\item Up to bounded positive factors, $\sqrt\beta\,e^{-\pi\beta}$ is
+item Up to bounded positive factors, $sqrtbeta,e^{-pibeta}$ is
the natural asymptotically free scale for this model in these
units. That is why the bounds are sharp.
-\item The bounds are not asymptotic (the ratio need not tend to $1$, and
+item The bounds are not asymptotic (the ratio need not tend to $1$, and
in fact the exact $O(4)$ prefactor is a known transcendent constant
we do not touch here). They give the correct order of magnitude.
-[Full gap at every finite $\beta$]
-For every finite $\beta>0$, the periodic measures on squares have a unique
+[Full gap at every finite $beta$]
+For every finite $beta>0$, the periodic measures on squares have a unique
local limit. Its full transfer gap is positive, with
-\[
- \gap(\beta)\;\ge\; c\,\exp\!(-\pi\beta
- - C√((1+\beta)\log(2+\beta)))>0
-\]
+[
+ gap(beta);ge; c,exp!(-pibeta
+ - C√((1+beta)log(2+beta)))>0
+]
for some absolute constants $c,C>0$.
Corollary~cor:allT-ov does not use the sharp two-loop
comparison; it follows from a preliminary mixing estimate alone. The
gap is therefore positive at every temperature, but the formula is too
-weak in the exponent to distinguish $\sqrt\beta\,e^{-\pi\beta}$ from
-$\beta^{-1/2}e^{\pi\beta+\sqrt\beta}$. Theorem~thm:main-ov repairs
-that at large $\beta$.
+weak in the exponent to distinguish $sqrtbeta,e^{-pibeta}$ from
+$beta^{-1/2}e^{pibeta+sqrtbeta}$. Theorem~thm:main-ov repairs
+that at large $beta$.
-## From $\gap$ to a physical mass
+## From $gap$ to a physical mass
-Choose a physical reference length $\ell_{ref}>0$ and declare
+Choose a physical reference length $ell_{ref}>0$ and declare
that $L^N$ lattice spacing (where $L$ is a fixed block scale we will
-choose later) has physical length $\ell_{ref}$. The lattice
-spacing is $a_N=\ell_{ref}L^{-N}$. Then
-$\gap(\beta_N)/a_N$ is the mass of the cutoff theory in physical units.
+choose later) has physical length $ell_{ref}$. The lattice
+spacing is $a_N=ell_{ref}L^{-N}$. Then
+$gap(beta_N)/a_N$ is the mass of the cutoff theory in physical units.
Theorem~thm:main-ov upgrades to
- \frac{\log 2}{M_0\ell_{ref}}
- \;\le\; (\gap(\beta_N))/(a_N)
- \;\le\; \frac{\log 8}{\ell_{ref}}
- \qquad(N\ large),
+ frac{log 2}{M_0ell_{ref}}
+ ;le; (gap(beta_N))/(a_N)
+ ;le; frac{log 8}{ell_{ref}}
+ qquad(N large),
-with $M_0$ a fixed positive integer and $\beta_N$ chosen so that exactly
+with $M_0$ a fixed positive integer and $beta_N$ chosen so that exactly
$N$ RG steps have been done at terminal coupling $H$. Bounded positive
multiplicative factors sandwich the continuum-normalised mass. Whether
the cutoff measures converge to an actual continuum theory is a separate
@@ -208,42 +208,42 @@ statement about an existing continuum spectrum.
Absence of ordinary ordering is classical (Mermin--Wagner, and the
McBryan--Spencer quantitative bounds). Exponential decay of the spin
-correlation was proved separately for every finite $\beta$ and every
-internal dimension $\ge 3$. Perturbative renormalisation predicts the
-scale $\sqrt\beta\,e^{-\pi\beta}$; Bethe ansatz $S$-matrix theory fixes
+correlation was proved separately for every finite $beta$ and every
+internal dimension $ge 3$. Perturbative renormalisation predicts the
+scale $sqrtbeta,e^{-pibeta}$; Bethe ansatz $S$-matrix theory fixes
the exact continuum mass for $O(3)$ and $O(4)$, and for $O(n)$ in
general. None of that gave a full transfer gap (including invariant
observables) for the specified lattice model. Our theorem closes that
-gap for $O(4)$ at large $\beta$, and yields positivity at every finite
-$\beta$ as a corollary. The proof of the exact continuum mass
-(including the ``$\Gamma$-function'' prefactor) is a separate problem,
+gap for $O(4)$ at large $beta$, and yields positivity at every finite
+$beta$ as a corollary. The proof of the exact continuum mass
+(including the ``$Gamma$-function'' prefactor) is a separate problem,
and the proof of lattice-to-continuum convergence is also separate.
## The proof in six steps
-The proof has a clear overall shape. \\
-{\bf Step 1} (Section~sec:finite-volume). Reduce the gap to a
+The proof has a clear overall shape.
+{bf Step 1} (Section~sec:finite-volume). Reduce the gap to a
finite-volume partition-function defect
-$\Delta_\beta(n)=4\log Z_\beta(n,n)-\log Z_\beta(2n,2n)$.\\
-{\bf Step 2} (Section~sec:preliminary). Show $\Delta_\beta(n)$ is
-already small at some scale $n\sim e^{\pi\beta+C√(\beta\log\beta)}$,
-using Ward identities and a stiffness iteration.\\
-{\bf Step 3} (Sections~sec:free-gap-ov-run--sec:calibration).
+$Delta_beta(n)=4log Z_beta(n,n)-log Z_beta(2n,2n)$.
+{bf Step 2} (Section~sec:preliminary). Show $Delta_beta(n)$ is
+already small at some scale $nsim e^{pibeta+C√(betalogbeta)}$,
+using Ward identities and a stiffness iteration.
+{bf Step 3} (Sections~sec:free-gap-ov-run--sec:calibration).
Construct an exact real-space renormalisation group (RG) step that
-integrates out $L\times L$ blocks and rewrites the coarse lattice in the
-same controlled form. Show it contracts past a coupling of size $H\sim$
-constant.\\
-{\bf Step 4} (Section~sec:trajectories). Show that, at every
-depth $N$, one can pick a bare coupling $\beta_N$ so the $N$-step
+integrates out $Ltimes L$ blocks and rewrites the coarse lattice in the
+same controlled form. Show it contracts past a coupling of size $Hsim$
+constant.
+{bf Step 4} (Section~sec:trajectories). Show that, at every
+depth $N$, one can pick a bare coupling $beta_N$ so the $N$-step
trajectory ends exactly at $H$/c. Matching at the coarse end absorbs
-the difference between two different depths.\\
-{\bf Step 5} (Section~sec:comparison). Compare the positive
+the difference between two different depths.
+{bf Step 5} (Section~sec:comparison). Compare the positive
coarse densities (partition functions/correlation functions) rather
than the signed expansions; positivity of the original model does the
-hard work.\\
-{\bf Step 6} (Sections~sec:assembly--sec:upper). Choose one
+hard work.
+{bf Step 6} (Sections~sec:assembly--sec:upper). Choose one
reference box once at a coarse depth $K$, show Step 2 already makes
-$\Delta$ small there, and Step 4 keeps it small at every finer cutoff.
+$Delta$ small there, and Step 4 keeps it small at every finer cutoff.
Then apply Step 1 to get the lower mass bound. Prove the upper bound by
noticing that the RG produces a definite correlation at distance $L^N$.
@@ -254,38 +254,38 @@ The rest of the paper expands each step.
%---------------------------------------------------------------------
-Let $Λ_{n,w}=(\Z/n\Z)\times(\Z/w\Z)$ be a periodic rectangle and
-$\sigma_x\in S^3\subset\R^4$ with normalised uniform measure $\dd\omega$
-($\int_{S^3}\dd\omega=1$). Periods are even and at least $4$. The
+Let $Λ_{n,w}=(Z/nZ)times(Z/wZ)$ be a periodic rectangle and
+$sigma_xin S^3subsetR^4$ with normalised uniform measure $ddomega$
+($int_{S^3}ddomega=1$). Periods are even and at least $4$. The
probability measure is (eq:model-ov). Every unoriented
nearest-neighbour bond is counted once.
## Transfer operator and reflection positivity
Fix a spatial circumference $w$. A time-slice configuration is
-$s=(s_1,...,s_w)\in(S^3)^w$; the one-step interaction is
-$V_w(s)=\sum_{i=1}^w s_i\cdot s_{i+1}$ with periodic indices. The transfer
+$s=(s_1,...,s_w)in(S^3)^w$; the one-step interaction is
+$V_w(s)=sum_{i=1}^w s_icdot s_{i+1}$ with periodic indices. The transfer
operator $K_w$ has kernel (eq:transfer-kernel-ov). It is:
-\item compact and self-adjoint: $\beta$ finite, $S^3$ compact,
+item compact and self-adjoint: $beta$ finite, $S^3$ compact,
kernel continuous;
-\item positive semidefinite: expand
- $e^{\beta s\cdot s'}=\sum_m \beta^m (s\cdot s')^m/m!$ and each
- $(s\cdot s')^m$ is a positive tensor-product kernel, then sandwich
- with $e^{\beta V_w/2}$;
-\item trace class: the diagonal integral of the positive kernel
+item positive semidefinite: expand
+ $e^{beta scdot s'}=sum_m beta^m (scdot s')^m/m!$ and each
+ $(scdot s')^m$ is a positive tensor-product kernel, then sandwich
+ with $e^{beta V_w/2}$;
+item trace class: the diagonal integral of the positive kernel
series converges;
-\item strictly positive: strictly positive kernel; hence by
- Perron--Frobenius, $\lambda_1(w)$ is simple and the top
+item strictly positive: strictly positive kernel; hence by
+ Perron--Frobenius, $lambda_1(w)$ is simple and the top
eigenvector (``vacuum'') is positive.
-Closing time periodically gives $Z_\beta(n,w)=\Tr K_w^n
-=\sum_i \lambda_i(w)^n$ with
-$\lambda_1(w)>\lambda_2(w)\ge\lambda_3(w)\ge...\ge0$.
+Closing time periodically gives $Z_beta(n,w)=Tr K_w^n
+=sum_i lambda_i(w)^n$ with
+$lambda_1(w)>lambda_2(w)gelambda_3(w)ge...ge0$.
The nice structural property is reflection positivity, following
from the site-reflection form
-$< F,G>_OS=\E[(Θ \bar F)G]$ where $Θ$
+$< F,G>_OS=E[(Θ barbar F)G]$F)G]$ where $Θ$
reflects in a time plane. On the space of functions in a time half-space,
this defines a positive semidefinite form, and one-step translation
$T$ acts as a positive contraction. We record the facts we use, then
@@ -293,13 +293,13 @@ prove them in Section~sec:finite-volume.
[Uniqueness of periodic local limit]
The periodic measures (eq:model-ov) on squares have a unique local
-limit as the side tends to infinity, at each fixed $\beta$.
+limit as the side tends to infinity, at each fixed $beta$.
[Reflection form]
-The vacuum $Ω$, positive contraction $T_\beta$, and full gap
-$\gap(\beta)= -\log\norm{T_\beta|_{Ω^\perp}}$ exist as in
-(eq:gap-def-ov), and $\gap(\beta)\ge0$.
+The vacuum $Ω$, positive contraction $T_beta$, and full gap
+$gap(beta)= -lognorm{T_beta|_{Ω^perp}}$ exist as in
+(eq:gap-def-ov), and $gap(beta)ge0$.
The proofs are in Section~sec:finite-volume (§sec:pre-part-12-2)
@@ -309,47 +309,47 @@ and use the preliminary mixing estimate of Section~sec:preliminary.
Fix once and for all a small dyadic block size $L$ (later chosen large)
and a large terminal inverse coupling $H$ (later chosen large). An
-$N$-step RG trajectory starts at bare coupling $\beta$ and ends at a
+$N$-step RG trajectory starts at bare coupling $beta$ and ends at a
small ``terminal'' coupling. Call $N$ the depth. A depth-$N$ step
-of the RG uses $L\times L$ blocks and identifies one coarse lattice
+of the RG uses $Ltimes L$ blocks and identifies one coarse lattice
spacing with $L$ fine ones. After $N$ steps one coarse cell is $L^N$
microscopic cells.
Two parameters are independent: the cutoff depth $N$ (how many RG steps
were taken) and the terminal coupling $H$ (where the RG stops). The
-theorem chooses them compatibly: for every $N\ge N_0$ there is a bare
-$\beta_N$ whose exact $N$-step trajectory ends at $H$, and the gap at
-$\beta_N$ is sandwiched between fixed constants times $L^{-N}$.
+theorem chooses them compatibly: for every $Nge N_0$ there is a bare
+$beta_N$ whose exact $N$-step trajectory ends at $H$, and the gap at
+$beta_N$ is sandwiched between fixed constants times $L^{-N}$.
-Now note: $L^{-N}$ and $\sqrt\beta\,e^{-\pi\beta}$ are the same
+Now note: $L^{-N}$ and $sqrtbeta,e^{-pibeta}$ are the same
scale up to constants, because the RG integrates the running coupling
-from $\beta$ down to $H$ in $N$ steps of size $\gamma=(\log L)/\pi$.
+from $beta$ down to $H$ in $N$ steps of size $gamma=(log L)/pi$.
Section~sec:calibration makes that identification precise.
The rest of the paper is the proof.
## Six-step map, with cross-references
-\noindent
-Step 1: $\Delta_\beta$ criterion \dotfill \Ssec:finite-volume\\
-Step 2: preliminary mixing \dotfill \Ssec:preliminary\\
-Step 3: linear block \dotfill \Ssec:linear\\
-Step 3$'$: exact nonlinear RG \dotfill \Ssec:rg\\
-Step 3$''$: calibration \dotfill \Ssec:calibration\\
-Step 4: admission and matching \dotfill \Ssec:trajectories\\
-Step 5: positive-density comparison \dotfill \Ssec:comparison\\
-Step 6: assembly and upper bound \dotfill \Ssec:assembly, \Ssec:upper\\
-Physical units \dotfill \Ssec:physical
+noindent
+Step 1: $Delta_beta$ criterion dotfill Ssec:finite-volume
+Step 2: preliminary mixing dotfill Ssec:preliminary
+Step 3: linear block dotfill Ssec:linear
+Step 3$'$: exact nonlinear RG dotfill Ssec:rg
+Step 3$''$: calibration dotfill Ssec:calibration
+Step 4: admission and matching dotfill Ssec:trajectories
+Step 5: positive-density comparison dotfill Ssec:comparison
+Step 6: assembly and upper bound dotfill Ssec:assembly, Ssec:upper
+Physical units dotfill Ssec:physical
## Notation
-$\gap(\beta)$ = full transfer gap. $\rho(\beta)=\sqrt\beta e^{-\pi\beta}$
+$gap(beta)$ = full transfer gap. $rho(beta)=sqrtbeta e^{-pibeta}$
= asymptotically free inverse length. $L$ = blocking factor (dyadic,
-large). $\gamma=(\log L)/\pi$. $Λ_{n,w}=(\Z/n\Z)\times(\Z/w\Z)$;
-$v=|Λ_{n,w}|$. $Z_\beta(n,w)$ = partition function. $\E$ = expectation.
-$\norm{\cdot}$ = operator or supremum norm, context clear. $C,c>0$ =
+large). $gamma=(log L)/pi$. $Λ_{n,w}=(Z/nZ)times(Z/wZ)$;
+$v=|Λ_{n,w}|$. $Z_beta(n,w)$ = partition function. $E$ = expectation.
+$norm{cdot}$ = operator or supremum norm, context clear. $C,c>0$ =
generic constants that may change between formulas but are
-$\beta$-independent unless stated.
+$beta$-independent unless stated.
%---------------------------------------------------------------------
# A finite-volume criterion for the full gap
@@ -359,7 +359,7 @@ $\beta$-independent unless stated.
The first step is a purely algebraic criterion. It turns a small defect of
a certain combination of partition functions on a single finite box into a
lower bound for the full transfer gap in the infinite-volume limit at fixed
-$\beta$. The criterion sees every spectral sector, because it uses
+$beta$. The criterion sees every spectral sector, because it uses
the complete trace of the transfer operator; and the specific combination
of partition functions we use is invariant under the addition of any free
energy proportional to area, which is what will make it a perfect ``test''
@@ -369,61 +369,61 @@ for the RG comparison later. No renormalisation is used in this section.
Fix a spatial circumference $w$ and let $K_w$ act on $(S^3)^w$
as in (eq:transfer-kernel-ov). Its eigenvalues satisfy
-$\lambda_1(w)>\lambda_2(w)\ge\lambda_3(w)\ge...\ge0$, and
-$Z_\beta(n,w)=\Tr K_w^n=\sum_i\lambda_i(w)^n$.
+$lambda_1(w)>lambda_2(w)gelambda_3(w)ge...ge0$, and
+$Z_beta(n,w)=Tr K_w^n=sum_ilambda_i(w)^n$.
-Suppress $\beta$ and define
+Suppress $beta$ and define
- p(n,w) &= 2\log Z(n,w) - \log Z(2n,w), \\
- q(n,w) &= 2\log Z(n,w) - \log Z(n,2w).
+ p(n,w) &= 2log Z(n,w) - log Z(2n,w),
+ q(n,w) &= 2log Z(n,w) - log Z(n,2w).
The same numbers, one with time doubled and one with space doubled.
[Both $p$ and $q$ are non-negative]
For every $n,w$,
-\[
- p(n,w)\ge0,\qquad q(n,w)\ge 0.
-\]
+[
+ p(n,w)ge0,qquad q(n,w)ge 0.
+]
Proof.
-Consider non-negative eigenvalues $\mu_i$ of a positive self-adjoint
+Consider non-negative eigenvalues $mu_i$ of a positive self-adjoint
operator and their $n$-th powers. The Cauchy-Schwarz bound
-$\sum_i\mu_i^{2n}\le(\sum_i\mu_i^n)^2$ gives
-$p(n,w)\ge0$. Applying the same argument to the space direction gives
-$q(n,w)\ge0$.
+$sum_imu_i^{2n}le(sum_imu_i^n)^2$ gives
+$p(n,w)ge0$. Applying the same argument to the space direction gives
+$q(n,w)ge0$.
Introduce the normalised trace weights
- t_i = (\lambda_i(w)^n)/(\sum_j\lambda_j(w)^n)
- \qquadso\qquad
- e^{-p(n,w)} = \sum_i t_i^2.
+ t_i = (lambda_i(w)^n)/(sum_jlambda_j(w)^n)
+ qquadsoqquad
+ e^{-p(n,w)} = sum_i t_i^2.
The right side is the probability that two independent copies of the
-chain, drawn with weights $t_i$, meet in the same eigenstate. It is $\le1$
+chain, drawn with weights $t_i$, meet in the same eigenstate. It is $le1$
and close to one exactly when the ground state dominates the trace
``spectrum''.
[Squaring propagates the bound]
Set
-\[
- s_n(w)=\sum_{i\ge2}(\lambda_i(w)/\lambda_1(w))^n.
-\]
-Then $s_n(w)\le e^{p(n,w)}-1$. If $p(n,w)\le1/4$, then
+[
+ s_n(w)=sum_{ige2}(lambda_i(w)/lambda_1(w))^n.
+]
+Then $s_n(w)le e^{p(n,w)}-1$. If $p(n,w)le1/4$, then
- ((\lambda_2(w))/(\lambda_1(w)))^n\le 2p(n,w),
- \qquad p(2n,w)\le 4p(n,w)^2.
+ ((lambda_2(w))/(lambda_1(w)))^nle 2p(n,w),
+ qquad p(2n,w)le 4p(n,w)^2.
The analogous statements hold for $q$ when the other period is doubled.
Proof.
-$t_1=(1+s_n)^{-1}$ and $\sum_i t_i^2\le t_1$ give $s_n\le e^{p}-1$.
-For $p\le1/4$, $e^p-1\le pe^p\le2p$, which bounds the first excited term.
+$t_1=(1+s_n)^{-1}$ and $sum_i t_i^2le t_1$ give $s_nle e^{p}-1$.
+For $ple1/4$, $e^p-1le pe^ple2p$, which bounds the first excited term.
The largest eigenvalue cancels from the definition of $p$, so
-$p(2n,w)=2\log(1+s_{2n})-\log(1+s_{4n})\le2s_{2n}\le2s_n^2
-\le2p^2e^{2p}\le4p^2$.
+$p(2n,w)=2log(1+s_{2n})-log(1+s_{4n})le2s_{2n}le2s_n^2
+le2p^2e^{2p}le4p^2$.
## Why space directions need a separate argument
@@ -441,7 +441,7 @@ For every $n,w$,
Proof.
Expand the definitions (eq:p-def)--(eq:q-def). Both sides
-equal $4\log Z(n,w) - 2\log Z(2n,w) - 2\log Z(n,2w) + \log Z(2n,2w)$.
+equal $4log Z(n,w) - 2log Z(2n,w) - 2log Z(n,2w) + log Z(2n,2w)$.
There is a companion identity obtained by exchanging the two directions.
@@ -449,65 +449,65 @@ Both will be used.
Now define the square-box test
- \Delta_\beta(n) = 4\log Z_\beta(n,n) - \log Z_\beta(2n,2n).
+ Delta_beta(n) = 4log Z_beta(n,n) - log Z_beta(2n,2n).
The combination has the important property that any contribution of the
-form $f_\beta nw$ to $\log Z_\beta(n,w)$ cancels from $\Delta_\beta(n)$.
+form $f_beta nw$ to $log Z_beta(n,w)$ cancels from $Delta_beta(n)$.
Bulk free energy generated by the RG therefore does not enter the test.
-[Two ways to write $\Delta_\beta$]
-For every even $n\ge4$,
-\[
- \Delta_\beta(n) = 2p(n,n) + q(2n,n) \ge 0.
-\]
-Consequently $\Delta_\beta(n)$ is a sum of two non-negative purities
+[Two ways to write $Delta_beta$]
+For every even $nge4$,
+[
+ Delta_beta(n) = 2p(n,n) + q(2n,n) ge 0.
+]
+Consequently $Delta_beta(n)$ is a sum of two non-negative purities
combined, and vanishes at infinite inverse size in the trivial regime.
Proof.
-By direct expansion, $4\log Z(n,n) - \log Z(2n,2n)$ equals
-$2(2\log Z(n,n)-\log Z(2n,n)) + (2\log Z(2n,n)-\log Z(2n,2n)$, which is
+By direct expansion, $4log Z(n,n) - log Z(2n,2n)$ equals
+$2(2log Z(n,n)-log Z(2n,n)) + (2log Z(2n,n)-log Z(2n,2n)$, which is
$2p(n,n)+q(2n,n)$. Non-negativity is the previous lemma.
## The criterion
[Finite-volume criterion]
-Let $n\ge4$ be even. If
+Let $nge4$ be even. If
- \Delta_\beta(n)\le\eta_*:=2^{-10},
+ Delta_beta(n)leeta_*:=2^{-10},
-then the transfer operators on circumferences $2^kn$, $k\ge0$, satisfy
+then the transfer operators on circumferences $2^kn$, $kge0$, satisfy
- (\lambda_2(2^kn))/(\lambda_1(2^kn))
- \le e^{-(\log 2)/n}.
+ (lambda_2(2^kn))/(lambda_1(2^kn))
+ le e^{-(log 2)/n}.
Every local limit of the square tori with sides $2^kn$ has full gap at
-least $(\log2)/n$. In particular, if the periodic infinite-plane state
+least $(log2)/n$. In particular, if the periodic infinite-plane state
is unique, then
- \gap(\beta)\ge(\log2)/n.
+ gap(beta)ge(log2)/n.
Proof.
-Write $\Delta_r=\Delta_\beta(r)$. Besides $q(2r,r)\le\Delta_r$, the
-definitions give $p(r,2r)=\Delta_r-2q(r,r)\le\Delta_r$.
-Lemma~lem:squaring therefore gives $p(2r,2r)\le4\Delta_r^2$. The
+Write $Delta_r=Delta_beta(r)$. Besides $q(2r,r)leDelta_r$, the
+definitions give $p(r,2r)=Delta_r-2q(r,r)leDelta_r$.
+Lemma~lem:squaring therefore gives $p(2r,2r)le4Delta_r^2$. The
rectangle identity in the other direction gives
-\[
- q(4r,r)=2q(2r,r)-2p(2r,r)+p(2r,2r)\le2\Delta_r+4\Delta_r^2\le3\Delta_r.
-\]
-Squaring in that direction gives $q(4r,2r)\le36\Delta_r^2$. Consequently
-\[
- \Delta_\beta(2r)=2p(2r,2r)+q(4r,2r)\le44\Delta_r^2\le64\Delta_r^2.
-\]
-Every use of the squaring lemma is justified by $3\eta_*<1/4$, and the
+[
+ q(4r,r)=2q(2r,r)-2p(2r,r)+p(2r,2r)le2Delta_r+4Delta_r^2le3Delta_r.
+]
+Squaring in that direction gives $q(4r,2r)le36Delta_r^2$. Consequently
+[
+ Delta_beta(2r)=2p(2r,2r)+q(4r,2r)le44Delta_r^2le64Delta_r^2.
+]
+Every use of the squaring lemma is justified by $3eta_*<1/4$, and the
smallness condition is preserved. Set $n_k=2^kn$. By induction,
-\[
- \Delta_\beta(n_k)\le\frac1{64}(64\eta_*)^{2^k}\le\frac1{64}2^{-2^k}.
-\]
-Since $( \lambda_2(n_k)/\lambda_1(n_k))^{n_k} \le 2p(n_k,n_k)\le\Delta_\beta(n_k)$
+[
+ Delta_beta(n_k)lefrac1{64}(64eta_*)^{2^k}lefrac1{64}2^{-2^k}.
+]
+Since $( lambda_2(n_k)/lambda_1(n_k))^{n_k} le 2p(n_k,n_k)leDelta_beta(n_k)$
by the squaring lemma, taking $n_k$-th roots proves
(eq:width-gap).
@@ -515,27 +515,27 @@ Now pass to the limit. Let $Ω_w$ be the normalised positive ground
eigenvector of $K_w$. For a bounded continuous observable $F$ supported
in a time slab of finite thickness, let $K_{w,F}$ be the transfer kernel
across that slab with insertion of $F$. Pointwise
-$|K_{w,F}|\le\norm F_∞ K_w^r$ where $r$ is the number of transfer
-steps across the slab; set $A_F=K_{w,F}/\lambda_1(w)^r$ for $r\ge0$
+$|K_{w,F}|lenorm F_∞ K_w^r$ where $r$ is the number of transfer
+steps across the slab; set $A_F=K_{w,F}/lambda_1(w)^r$ for $rge0$
(and $r=0$ gives multiplication by $F$). Its operator norm is at most
-$\norm F_∞$. The infinite-time cylinder expectation is
-$\omega_w(F)=\inner{Ω_w}{A_FΩ_w}$. Applying (eq:width-gap)
+$norm F_∞$. The infinite-time cylinder expectation is
+$omega_w(F)=inner{Ω_w}{A_FΩ_w}$. Applying (eq:width-gap)
between two such insertions yields
- |\Cov_{\omega_w}(F,G)|
- \le \norm F_∞\norm G_∞ e^{-(\log2)d/n},
+ |Cov_{omega_w}(F,G)|
+ le norm F_∞norm G_∞ e^{-(log2)d/n},
where $d$ is the number of transfer edges separating their supports. This
bound survives any subsequential local limit of the cylinder states as
-$w=2^kn\to∞$.
+$w=2^knto∞$.
Now identify cylinder limits with square-torus limits. Let
-$\widehat K_w=K_w/\lambda_1(w)$. For $w=n_k$ and $r\le w/2$, the
+$widehatwidehat K_w=K_w/K_w=K_w/lambda_1(w)$. For $w=n_k$ and $rle w/2$, the
square-torus expectation equals
-$\Tr(A_F\widehat K_w^{w-r})/\Tr\widehat K_w^w$. Splitting off the ground
+$Tr(A_Fwidehatwidehat K_w^K_w^{w-r})/Trwidehatwidehat K_w^w$.K_w^w$. Splitting off the ground
projection in each trace shows the difference from the cylinder
-expectation is at most $2\norm F_∞ s_{w/2}(w)$. But
-$p(w/2,w)\le\Delta_\beta(w/2)$ and $s_{w/2}(w)\le e^{\Delta_\beta(w/2)}-1\to0$.
+expectation is at most $2norm F_∞ s_{w/2}(w)$. But
+$p(w/2,w)leDelta_beta(w/2)$ and $s_{w/2}(w)le e^{Delta_beta(w/2)}-1to0$.
Thus the square and cylinder expectations have the same subsequential
limits, even for products of fixed local observables, and
(eq:slab-bound) holds in every such square-torus limit.
@@ -545,7 +545,7 @@ by a local observable in the reflection-positive Hilbert space, the
one-step transfer autocorrelation has a positive spectral measure
(reflection positivity). Any such autocorrelation is a correlation of
two reflected slab observables, so (eq:slab-bound) applies and
-excludes spectral support above $e^{-(\log2)/n}$. Such vectors are dense
+excludes spectral support above $e^{-(log2)/n}$. Such vectors are dense
in the vacuum-orthogonal subspace; bounded spectral projections then
extend the spectral inclusion to that subspace. This proves
(eq:criterion-gap).
@@ -557,8 +557,8 @@ on the size of a slab is needed. That is useful when the upper mass
bound comes up in Section~sec:upper.
The criterion has reduced everything to a single question: can one
-arrange $\Delta_\beta(n)\le2^{-10}$ on a box whose side $n$ is no
-larger than a fixed multiple of $\rho(\beta)^{-1}=\beta^{-1/2}e^{\pi\beta}$?
+arrange $Delta_beta(n)le2^{-10}$ on a box whose side $n$ is no
+larger than a fixed multiple of $rho(beta)^{-1}=beta^{-1/2}e^{pibeta}$?
The preliminary estimate of Section~sec:preliminary will show this
is true with a slightly too-large constant in the exponent; that is not
enough. The point of the RG that follows is to remove the excess.
@@ -571,15 +571,15 @@ enough. The point of the RG that follows is to remove the excess.
%---------------------------------------------------------------------
In this section we prove the first source of exponential decay, with a
-length scale $\Xi(\beta)$ that is bigger than the asymptotically free
+length scale $Xi(beta)$ that is bigger than the asymptotically free
scale by a subexponential factor. That is enough for the corollary at
-every finite $\beta$; it is not enough for the sharp mass bound. But the
-$\Delta_\beta$ bound from this section will be used at one finite
+every finite $beta$; it is not enough for the sharp mass bound. But the
+$Delta_beta$ bound from this section will be used at one finite
reference box, where a subexponential excess is harmless. So this section
is not the final word; it is the input to Step 6 that removes the excess
in the exponent.
-Throughout, spins take values in the unit sphere $S^3\subset\R^4$,
+Throughout, spins take values in the unit sphere $S^3subsetR^4$,
identified with the unit quaternions, with normalised Haar measure.
Distances between supports use maximum norm (or its periodic version).
@@ -588,68 +588,68 @@ Distances between supports use maximum norm (or its periodic version).
[Preliminary mixing]
There are absolute constants $C,c,p>0$ such that, with
- \Xi(\beta)=\exp\!\Bigl[\pi\beta+C√((1+\beta)\log(2+\beta))\Bigr],
+ Xi(beta)=exp!Bigl[pibeta+C√((1+beta)log(2+beta))Bigr],
-the following hold for every $\beta\ge0$.
+the following hold for every $betage0$.
For every unpinned rectangular torus with each side at least
-$C\Xi(\beta)$, every choice $0\le\beta_e\le\beta$ of nearest-neighbour
+$CXi(beta)$, every choice $0lebeta_elebeta$ of nearest-neighbour
couplings, and any local observables $F,G$ supported on finite sets
$A,A'$ at distance $d$,
- |\Cov(F,G)|\le C(1+\beta+|A|+|A'|+d)^{20}L_F L_G
- e^{-cd/\Xi(\beta)}.
+ |Cov(F,G)|le C(1+beta+|A|+|A'|+d)^{20}L_F L_G
+ e^{-cd/Xi(beta)}.
-Here $L_F=\norm F_∞+\max_{x\in A}Lip_x(F)$, the sum of
+Here $L_F=norm F_∞+max_{xin A}Lip_x(F)$, the sum of
sup and individual Lipschitz constants. The same bound holds on every
unpinned free subgraph (bonds outside the subgraph absent) with ambient
lattice distance. In the homogeneous model the local limit of periodic
squares is unique.
-Finally, if $Z_\beta(n,m)$ is the homogeneous torus partition function,
-then for every even $n\ge C\Xi(\beta)$,
+Finally, if $Z_beta(n,m)$ is the homogeneous torus partition function,
+then for every even $nge CXi(beta)$,
- 0\le\Delta_\beta(n):=4\log Z_\beta(n,n)-\log Z_\beta(2n,2n)
- \le C(1+\beta)^pn^p e^{-cn/\Xi(\beta)}.
+ 0leDelta_beta(n):=4log Z_beta(n,n)-log Z_beta(2n,2n)
+ le C(1+beta)^pn^p e^{-cn/Xi(beta)}.
The bound (eq:pre:theorem-delta) together with Step 1
(Proposition~prop:criterion) already gives
-$\gap(\beta)\ge(\log2)/n$ for $n\approx\Xi(\beta)$, and hence
+$gap(beta)ge(log2)/n$ for $napproxXi(beta)$, and hence
Corollary~cor:allT-ov. But this is far from sharp: the true
-inverse length is $\rho(\beta)=\beta^{-1/2}e^{\pi\beta}$, and
-$\Xi(\beta)/\rho(\beta)^{-1}\to∞$ as $\beta\to∞$. The RG
+inverse length is $rho(beta)=beta^{-1/2}e^{pibeta}$, and
+$Xi(beta)/rho(beta)^{-1}to∞$ as $betato∞$. The RG
comparison below will remove this excess.
-\subsection*{Structure of the proof of Theorem~thm:preliminary}
+subsection*{Structure of the proof of Theorem~thm:preliminary}
The proof has four macroscopic stages that we will now run in order.
-\noindent
+noindent
Stage A (Ward identities + stiffness iteration).
Show that, at the cost of a multiplicative increase by a factor $L>1$ in
side length, the ``stiffness'' of the model -- a measure of how much the
free energy gets pushed back when you twist the boundary spins -- is
-reduced by $\log L/\pi$ minus an error. Iterating decreases stiffness at
-least like $\log(side length)/\pi$. Start with stiffness $\beta$ at
-side length one. After reaching a side of length $\exp(\pi\beta+C√(\beta\log\beta))$
+reduced by $log L/pi$ minus an error. Iterating decreases stiffness at
+least like $log(side length)/pi$. Start with stiffness $beta$ at
+side length one. After reaching a side of length $exp(pibeta+C√(betalogbeta))$
the stiffness is tiny.
-\noindent
+noindent
Stage B (small stiffness ⟹ cheap rotations).
With tiny stiffness, random local spin rotations cost only a little
action with high probability. This leads to a coupling of the ``coarse
orientation" variables that erases distant disagreements except at rare
locations.
-\noindent
+noindent
Stage C (disagreement percolation).
Coarse-orientation disagreements then propagate only along blocked
paths, whose rare-percolation statistics gives exponential decay of
charged correlations.
-\noindent
+noindent
Stage D (all observables, not just charged ones).
Use quaternion algebra: writing a unit quaternion as a pair of circle
valued fields and then decomposing further into Ising variables lets us
@@ -659,77 +659,77 @@ inequalities upgrade the two-point bounds to full mixing for all
observables.
Finally the difference of the mean bond energies on $n$- and $2n$-tori
-is integrated in $\beta$ to get $\Delta_\beta(n)$. This is
+is integrated in $beta$ to get $Delta_beta(n)$. This is
(eq:pre:theorem-delta).
The rest of the section proves each stage.
## Stage A, part 1: the local class
-Work on a pinned square $Q(S)=\{0,...,S\}^2$. For each positively
-oriented edge $e=x\to y$, write
+Work on a pinned square $Q(S)={0,...,S}^2$. For each positively
+oriented edge $e=xto y$, write
- t_e + s_e = q_y\,U_e\,q_x^{-1},\qquad t_e\in\R,\quad s_e\inIm\mathbb H,
+ t_e + s_e = q_y,U_e,q_x^{-1},qquad t_einR,quad s_einImmathbb H,bbH,
where $U_e$ is a fixed unit quaternion encoding a possibly twisted edge.
-The scalar $t_e=q_y\cdot q_x$ is the bond energy up to twist, and the
+The scalar $t_e=q_ycdot q_x$ is the bond energy up to twist, and the
imaginary quaternion $s_e$ is the ``current'' along the edge: its three
components measure the three perpendicular rotation axes. The pinning is
encoded by taking a fixed boundary spin.
-Give each edge a positive coefficient $b_e\le\beta$ (interior edges), with
+Give each edge a positive coefficient $b_elebeta$ (interior edges), with
$b_e/2$ for edges tangent to the boundary (the half allocation makes
actions add correctly across cells). The local density is
-$Z^{-1}\exp(\sum_e b_e t_e)$.
+$Z^{-1}exp(sum_e b_e t_e)$.
-For a $3\times2$ matrix-valued macroscopic test field $F_e$ on edges,
+For a $3times2$ matrix-valued macroscopic test field $F_e$ on edges,
define the (macroscopic) current
- X_Q(F)=S^{-1}\sum_e b_e\,s_e\cdot F_{\mu(e)}(z_e),
+ X_Q(F)=S^{-1}sum_e b_e,s_ecdot F_{mu(e)}(z_e),
-where $z_e$ is the edge midpoint and $\mu(e)$ the edge direction. Its
+where $z_e$ is the edge midpoint and $mu(e)$ the edge direction. Its
second variation under infinitesimal spin twists gives the stiffness
session
- \mathcal H_Q(F,G)=S^{-2}\E\sum_e b_e t_e\,\bar F_e\cdot G_e
- -\Cov(X_Q(F),X_Q(G)).
+ mathcal H_Q(F,G)=S^calH_Q(F,G)=S^{-2}Esum_e b_e t_e,barbar F_eF_ecdot G_e
+ -Cov(X_Q(F),X_Q(G)).
This is ``direct quadratic cost minus current covariance''. The
-covariance is real quadratic; on constant $3\times2$ matrices, the sum
+covariance is real quadratic; on constant $3times2$ matrices, the sum
of $|b_e|$ along any line in either spatial direction is at most
$bS^2$, so
- H_Q\le bI.
+ H_Qle bI.
This holds for arbitrary signed $b_e$ and twists; no lower bound on
$H_Q$ is claimed.
-For a profile $\eta$ of infinitesimal imaginary quaternions vanishing on
+For a profile $eta$ of infinitesimal imaginary quaternions vanishing on
the boundary, Haar integration by parts produces a Ward identity. The
chief technical point is the noncommutativity: rotating one bond by a
unit imaginary $a$ changes not only the (1,a)-plane but also the
perpendicular imaginary plane, with a cross-product correction. The
concrete formulas are:
- D_f\mathcal A &= \sum_e b_e(s_e\cdot a)(f_y-f_x), \notag\\
- D_f^2\mathcal A &= \sum_e b_e t_e(f_y-f_x)^2
+ D_fmathcal AcalA &= sum_e b_e(s_ecdot a)(f_y-f_x), notag
+ D_f^2mathcal AcalA &= sum_e b_e t_e(f_y-f_x)^2
for a one-quaternion-axis profile $a f_x$, and the exact nonabelian
derivative of the current
- D_\eta s_e = t_e(\eta_y-\eta_x) + (\eta_y+\eta_x)\times s_e.
+ D_eta s_e = t_e(eta_y-eta_x) + (eta_y+eta_x)times s_e.
The first term is ``what you would expect if rotations commuted''; the
cross term is the obstruction to making the argument work for
non-abelian targets.
The Ward identity expressing invariance of the measure under rotations
-by $\eta$ (with $\eta=0$ on the pinned boundary) has the form
+by $eta$ (with $eta=0$ on the pinned boundary) has the form
- \E[\bar X_Q(d_S\eta)\,X_Q(F)]
- = S^{-2}\E\sum_e b_e t_e\,(d_S\eta)_ebar\cdot F_e
- + \E X_Q\!(F_e\times(\bar\eta_y+\bar\eta_x)).
+ E[barbar X_Q(d_SX_Q(d_Seta),X_Q(F)]
+ = S^{-2}Esum_e b_e t_e,(d_Seta)_ebarcdot F_e
+ + E X_Q!(F_etimes(bareta_y+bareta_x)).
The left side is a current-correlation; the first term on the right is
the direct quadratic form; the second term is the non-abelian correcti on
@@ -742,22 +742,22 @@ We want the moments of $X_Q(F)$ uniformly in the side length and couplings,
so that local errors propagate without accumulating uncontrollably.
[Uniform current moments]
-For every fixed $1\le p<∞$,
+For every fixed $1le p<∞$,
- \|X_Q(F)\|_p \le C_p(1+b)\log(2+S)\|F\|_{C^1}.
+ |X_Q(F)|_p le C_p(1+b)log(2+S)|F|_{C^1}.
The constant is independent of pins, signs, twists, and side length.
-Suppose throughout $\log(2+S)\le16B$.
+Suppose throughout $log(2+S)le16B$.
Proof.
Consider first a compact score $f$ (a one-axis profile supported in a
-patch, zero on the boundary). Taylor's formula for $D_f^2\mathcal A$ is
-bounded in absolute value by $b\sum_e(f_y-f_x)^2$, so Haar invariance
+patch, zero on the boundary). Taylor's formula for $D_f^2mathcal A$calA$ is
+bounded in absolute value by $bsum_e(f_y-f_x)^2$, so Haar invariance
gives
- \E\exp(tD_f\mathcal A)
- \le \exp\!\Bigl[\tfrac12 bt^2\sum_e(f_y-f_x)^2\Bigr].
+ Eexp(tD_fmathcal A)calA)
+ le exp!Bigl[tfrac12 bt^2sum_e(f_y-f_x)^2Bigr].
This is the exponential-tail estimate for a compact score. All fixed
moments of the likelihood ratio follow up to a fixed power of
@@ -767,20 +767,20 @@ Next extract a transverse current on a buffered interior patch. Take a
scalar test $h$ on a patch of side $d$, pick a rotation axis $a$
perpendicular to the current colour to extract; let $θ$ be an
affine compact potential with slope $1/d$. Rotate the score by opposite
-profiles $f=\tfrac12arcsin(h/N)$ with $N=C√(1+b)\,H$ and take the
+profiles $f=tfrac12arcsin(h/N)$ with $N=C√(1+b),H$ and take the
difference. This gives the transverse current multiplied by
-$2\sin(f_x+f_y)$, and multiplication by $N$ extracts $h$ up to $CH/d$.
+$2sin(f_x+f_y)$, and multiplication by $N$ extracts $h$ up to $CH/d$.
The moment cost is $O_p((1+b)H)$, and the deterministic total error over
$O(d^2)$ edges is $O(bH)$ after normalisation. Hence
- \Bigl\|d^{-1}\sum_{e∥\mu}b_e(s_e)_k\,h_e\Bigr\|_p
- \le C_p(1+b)H.
+ Bigl|d^{-1}sum_{e∥mu}b_e(s_e)_k,h_eBigr|_p
+ le C_p(1+b)H.
Finally, cover the pinned square by buffered patches at every dyadic
scale. At scale with patch side $d$, there are $O(S/d)$ patches; each
-contributes $O((1+b)\|F\|_{C^1})$ per scale after normalisation.
-There are $O(\log(2+S))$ scales, giving (eq:pre:2-2). Boundary
+contributes $O((1+b)|F|_{C^1})$ per scale after normalisation.
+There are $O(log(2+S))$ scales, giving (eq:pre:2-2). Boundary
layers contribute deterministically.
@@ -788,39 +788,39 @@ layers contribute deterministically.
Tile the pinned square by $L^2$ cells of side $R=S/L$ ($L$ is an
auxiliary integer, independent of the block factor of the same name). Let
-$\omega$ be the union of the boundary rings of these cells, and let $j$
+$omega$ be the union of the boundary rings of these cells, and let $j$
be the current field:
-Condition on $\omega$. Cell interiors are then independent, and the
-density of $\omega$ is $e^{-S_g(\omega)}$ with
-$S_g(\omega)=-\sum_c\log Z_c(\omega)$. Set $z_c$ to be the lower-left
+Condition on $omega$. Cell interiors are then independent, and the
+density of $omega$ is $e^{-S_g(omega)}$ with
+$S_g(omega)=-sum_clog Z_c(omega)$. Set $z_c$ to be the lower-left
corner of cell $c$ and use the notation
-\[
- N_c(F)=\sup_{c^+}(|F|+L^{-1}|\nabla F|),\quad
- D_c(F)=L^{-1}N_c(\nabla F),
-\]
+[
+ N_c(F)=sup_{c^+}(|F|+L^{-1}|nabla F|),quad
+ D_c(F)=L^{-1}N_c(nabla F),
+]
where $c^+$ is a fixed enlargement of $c$. Write $P(B)$ for a polynomial
in $B$ whose degree and coefficients depend only on fixed numerical
constants, not on $R,L,S$. Then ((eq:pre:2-2) and
(eq:pre:2-1))
- |\E_c X_c(F)|&\le P(B)N_c(F),\qquad
- |\mathcal H_c(F,G)|\le P(B)N_c(F)N_c(G),\\
- |\mathcal H_c(F,G)-\bar F_c H_c G_c|
- &\le P(B)[D_c(F)N_c(G)+N_c(F)D_c(G)].
+ |E_c X_c(F)|&le P(B)N_c(F),qquad
+ |mathcal H_c(F,G)|calH_c(F,G)|le P(B)N_c(F)N_c(G),
+ |mathcal H_c(F,G)-calH_c(F,G)-barbar F_cF_c H_c G_c|
+ &le P(B)[D_c(F)N_c(G)+N_c(F)D_c(G)].
The second equation says: ``up to slowly varying corrections, the cell
stiffness is a constant-field stiffness'', which is a big simplification.
Under a scalar ring rotation $u$, with $O_c(u)$ the frame rotation that
this induces on the cell,
- \|O_c(u)^TH_c^uO_c(u)-H_c\|\le P(B)D_c(u),
+ |O_c(u)^TH_c^uO_c(u)-H_c|le P(B)D_c(u),
and the derivative at zero satisfies the same bound.
Finally, the Hessian identity for the ring action:
- D_u^2S_g = L^{-2}\sum_c\mathcal H_c(a\,d_S u, a\,d_S u).
+ D_u^2S_g = L^{-2}sum_cmathcal H_c(acalH_c(a,d_S u, a,d_S u).
## Stage A, part 4: augmenting with Gaussian noise
@@ -829,39 +829,39 @@ The cell stiffness matrices need not be positive; adding fake Gaussian
noise makes the desired bound on $H_Q$ into a lower bound on the
variance of an augmented current.
-Suppose throughout the pinned cell class that $H_c\le hI$, and
-$B^{-D}\le h\le B$. Choose $v$ and $c_0$ with
+Suppose throughout the pinned cell class that $H_cle hI$, and
+$B^{-D}le hle B$. Choose $v$ and $c_0$ with
- h<v\le 2B,\quad c_0\ge B^{-D'},\quad
- C_c:=vI-H_c-c_0P_a\ge B^{-D'}I,
+ h<vle 2B,quad c_0ge B^{-D'},quad
+ C_c:=vI-H_c-c_0P_age B^{-D'}I,
where $P_a$ is a projection onto the two spatial components in colour $a$.
Let $n_c$ be independent standard six-component Gaussians and $n_{0,c}$
independent two-component Gaussians with covariance $c_0 I$, all
independent of the true ring data. Define the augmented current
- J(F)=j(F)+L^{-1}\sum_c
- (√(C_c)\,n_c+a\,n_{0,c})\cdot F_c.
+ J(F)=j(F)+L^{-1}sum_c
+ (√(C_c),n_c+a,n_{0,c})cdot F_c.
For every constant real matrix $A$,
- H_Q(A,A)=v|A|^2-\Cov(J(A),J(A)).
+ H_Q(A,A)=v|A|^2-Cov(J(A),J(A)).
This is ``stiffness = fake-noise variance minus augmented-current
variance''. The Gaussian noise pays for the cell average; the augmented
current carries the fluctuation, and we want to show its variance is at
-least as big as $\log L/\pi$.
+least as big as $log L/pi$.
-Choose a smooth trigonometric cutoff $\psi$ that is zero on the
-boundary and has band $O(m)$, and let $f_i=\psi\varphi_i$ where $\varphi_i$
-are real orthonormal Fourier modes of low frequency $|k|\le4K$, with
-$K+m\ll L$. Define $G_0$, $p$, and $\theta$ at the cell anchors via the
+Choose a smooth trigonometric cutoff $psi$ that is zero on the
+boundary and has band $O(m)$, and let $f_i=psivarphi_i$ where $varphi_i$
+are real orthonormal Fourier modes of low frequency $|k|le4K$, with
+$K+mll L$. Define $G_0$, $p$, and $theta$ at the cell anchors via the
discrete gradient Gram matrix. Substitute $p$ into the density by an
exact change of variables; the new density is proportional to $e^{-V(p)}$
with
- V(p) = \tfrac12 c_0 p^TG_0 p + S_g(e^{a\theta}\omega^0)
- +\tfrac12\sum_c|n_c^0+√(C_c^0)\,a\,\nabla\theta(z_c)/L|^2.
+ V(p) = tfrac12 c_0 p^TG_0 p + S_g(e^{atheta}omega^0)
+ +tfrac12sum_c|n_c^0+√(C_c^0),a,nablatheta(z_c)/L|^2.
The change of variables is exact: the ring rotation preserves product
Haar, the $n_0$ substitution is Gaussian, and the $n_c$ substitution is
@@ -871,9 +871,9 @@ Now imagine a family of directions $u_i$ in the potential span and set
$∂_i = ∂/∂ t_i$ at fixed base data. These commute,
so we can integrate by parts normally. Then
- \E(\sum_i∂_i^* Y_i)^2
- = \E\sum_{ij}∂_j Y_i\,∂_i Y_j
- +\E\sum_{ij}Y_i∂_i∂_j V\,Y_j.
+ E(sum_i∂_i^* Y_i)^2
+ = Esum_{ij}∂_j Y_i,∂_i Y_j
+ +Esum_{ij}Y_i∂_i∂_j V,Y_j.
This is our basic inequality: it expresses the ``V'' Hessian and
``crossed derivative'' cost of a long-run Gaussian in terms of two
@@ -883,38 +883,38 @@ these two sums.
## Stage A, part 5: three band estimates
Fix any $N_*>100(1+D+D')$ and demand an accuracy $B^{-10N_*}$; make the
-underlying $\log L\le16B$ dominate every polynomial in $L$ that will
+underlying $log Lle16B$ dominate every polynomial in $L$ that will
appear. Choose $M,K$ with $M=B^J$ and $K=L/B^J$ and
-$L\ge B^{10J}$ for one fixed $J$. Take the cutoff
-$\psi(z)=\prod_\mu[1-\cos^{2m}(\pi z_\mu)]$ with $m=\lceil B^4\rceil$;
-then $\chi=\psi^2$ satisfies $\int\chi^2\ge1-CB^{-2}$, has
-Fourier $\ell^1$ norm bounded independently of $B$, and has band $O(m)$.
+$Lge B^{10J}$ for one fixed $J$. Take the cutoff
+$psi(z)=prod_mu[1-cos^{2m}(pi z_mu)]$ with $m=lceil B^4rceil$;
+then $chi=psi^2$ satisfies $intchi^2ge1-CB^{-2}$, has
+Fourier $ell^1$ norm bounded independently of $B$, and has band $O(m)$.
Sampling, gradients, and good event.
-For a band-$O(K)$ field $F$ with $K\ll L$:
+For a band-$O(K)$ field $F$ with $Kll L$:
- \|N(F)\|_QED&\le C\|F\|_2,\qquad
- \|D(F)\|_QED\le C(K/L)\|F\|_2,
- \\
- \|d_Su-\nabla u\|_2&\le C(K/S)^2\|\nabla u\|_2.
+ |N(F)|_QED&le C|F|_2,qquad
+ |D(F)|_QEDle C(K/L)|F|_2,
+
+ |d_Su-nabla u|_2&le C(K/S)^2|nabla u|_2.
Here $N$ and $D$ are the discrete cell norms. The gradient subspace
projection $c_0^{-1}$ also implies the Gaussian bound
- \rho := L^{-1}\|\nabla\theta\|_∞+L^{-2}\|\nabla^2\theta\|_∞
- \le B^{-n}
+ rho := L^{-1}|nablatheta|_∞+L^{-2}|nabla^2theta|_∞
+ le B^{-n}
on an event of probability $e^{-B^3}$, for every fixed $n$, once $J$ is
large.
Hessian estimate.
-Replace $d_Su$ by $\nabla u(z_c)$ using (eq:pre:3-2), (eq:pre:5-1),
+Replace $d_Su$ by $nabla u(z_c)$ using (eq:pre:3-2), (eq:pre:5-1),
(eq:pre:5-2); conjugate each cell stiffness back using (eq:pre:3-3).
-The total error is $P(B)[K/L+(K/S)^2+\rho]\|\nabla u\|_2^2$, and the
+The total error is $P(B)[K/L+(K/S)^2+rho]|nabla u|_2^2$, and the
remaining matrix sum is exactly $H_c^0+C_c^0+c_0P_a=vI$. So
- ∂_u^2V\le(v+B^{-N})\|\nabla u\|_2^2
+ ∂_u^2Vle(v+B^{-N})|nabla u|_2^2
on the good event. Everywhere else the bound holds with $P(B)$
replacing $v+B^{-N}$; the exceptional event has probability $e^{-B^3}$,
@@ -922,34 +922,34 @@ so it contributes an inverse power of $B$ to any finite moment.
Nonabelian derivative estimate.
For at most $CK^2$ real directions $u_i$ satisfying
-$\|\nabla\sum_i d_iu_i\|_2\le C|d|$ and $N_c(\nabla u_i)\le C$
+$|nablasum_i d_iu_i|_2le C|d|$ and $N_c(nabla u_i)le C$
((eq:pre:5-6)), and for a real band-$O(K)$ $F$ perpendicular to
the colour $a$, then
- \E\sum_i|∂_i J(F)-2J(u_iF')|^2
- \le B^{-N}\|F\|_2^2,
- \qquad F'=-a\times F.
+ Esum_i|∂_i J(F)-2J(u_iF')|^2
+ le B^{-N}|F|_2^2,
+ qquad F'=-atimes F.
The proof isolates the two parts of $2J(uF')$: the anchor-conjugation
-part (bounded by $P(B)\rho$) and the non-anchor square-root derivative
+part (bounded by $P(B)rho$) and the non-anchor square-root derivative
part (bounded by $P(B)K^2/L^2$ after summing). The deterministic noise
drift and the true $H_c$ cancel via
-$H_c+C_c=vI-c_0P_a$ and $F\perp a$. There are exactly four sources of
+$H_c+C_c=vI-c_0P_a$ and $Fperp a$. There are exactly four sources of
error: contact (Ward identity remainder), anchor conjugation, deterministic
noise drift, and non-anchor square-root derivative. Equation
(eq:pre:5-7) is the sum of their bounds.
Same-frequency Ward bound.
For the uncentred defect
-\[
- \mathcal D(F,G)=vL^{-2}\sum_c\bar F_c\cdot G_c-\EJ(F)barJ(G),
-\]
-and for a compact $\eta$ of band $O(K)$ with a fixed Fourier symbol,
+[
+ mathcal D(F,G)=vL^calD(F,G)=vL^{-2}sum_cbarbar F_cF_ccdot G_c-EJ(F)barJ(G),
+]
+and for a compact $eta$ of band $O(K)$ with a fixed Fourier symbol,
apply the raw Ward identity (eq:pre:1-5) before replacing the
gradient by its leading symbol. The contact term cancels the raw defect;
the bracket term gives a $C^1$-norm bound $P(B)/|k|$. Hence
- |\mathcal D(d_S\eta,G)|\le P(B)/|k|+B^{-N}.
+ |mathcal D(d_ScalD(d_Seta,G)|le P(B)/|k|+B^{-N}.
Applying the Ward identity earlier would have lost the exact cancellation;
this is the critical sequencing point.
@@ -957,46 +957,46 @@ this is the critical sequencing point.
## Stage A, part 6: the two-colour closure
Close a two-colour estimate in two steps. For perpendicular colours
-$r,r'$, and for frequencies $2M\le|k_p|\le4M$, put
-$b_p=|k_p|^{-2}$ and $G_p^r=r\,d_S(\psi\tilde\varphi_p/|k_p|)$. The
+$r,r'$, and for frequencies $2Mle|k_p|le4M$, put
+$b_p=|k_p|^{-2}$ and $G_p^r=r,d_S(psitildevarphi_p/|k_p|)$. The
Ward bound (eq:pre:5-14) applied to the two annuli in the two
colours produces a closed inequality
-\[
- \sum_p b_p\E D_p^2
- \le CvU^r+CT_wU^{r'}+O(B^{-N_*}),\qquad
- U^c=\sum_pb_p\sum_i\E J(G_p^cu_i)^2,
-\]
+[
+ sum_p b_pE D_p^2
+ le CvU^r+CT_wU^{r'}+O(B^{-N_*}),qquad
+ U^c=sum_pb_psum_iE J(G_p^cu_i)^2,
+]
with constants independent of the number of directions. Interchange
$r,r'$ and note $U^{-c}=U^c$, so neither colour is assumed when the other
-is estimated. Let $U=\max(U^r,U^{r'})$. The signal lower bound
-$U^2\le Cv(v+T_w)U+CvB^{-N_*}$ gives, after solving the quadratic and
-using $v\ge B^{-D}$,
+is estimated. Let $U=max(U^r,U^{r'})$. The signal lower bound
+$U^2le Cv(v+T_w)U+CvB^{-N_*}$ gives, after solving the quadratic and
+using $vge B^{-D}$,
- U^r+U^{r'}\le Cv(v+T_w).
+ U^r+U^{r'}le Cv(v+T_w).
## Stage A, part 7: extracting all spatial derivatives
-At a fixed total complex frequency $n=k_p+k_i$, $|n|\le4K_1$, we
-take $w(n-k_p)\ge c(M+|n|)^{-2}$ uniformly in the annulus. Write
-$\psi_\mu^+(z)=\tfrac12[\psi(z+e_\mu/(2S))+\psi(z-e_\mu/(2S))]$. The
-exact gradient factor in spatial direction $\mu$ is
-\[
- \widehat k_{p\mu}sinc(k_{p\mu}/(2S))\psi_\mu^+
- +\frac{S\cos(k_{p\mu}/(2S))}{i|k_p|}
- [\psi(z+e_\mu/(2S))-\psi(z-e_\mu/(2S))].
-\]
-Under $k_p\mapsto-k_p$ the first term changes sign and the second does
+At a fixed total complex frequency $n=k_p+k_i$, $|n|le4K_1$, we
+take $w(n-k_p)ge c(M+|n|)^{-2}$ uniformly in the annulus. Write
+$psi_mu^+(z)=tfrac12[psi(z+e_mu/(2S))+psi(z-e_mu/(2S))]$. The
+exact gradient factor in spatial direction $mu$ is
+[
+ widehatwidehat k_k_{pmu}sinc(k_{pmu}/(2S))psi_mu^+
+ +frac{Scos(k_{pmu}/(2S))}{i|k_p|}
+ [psi(z+e_mu/(2S))-psi(z-e_mu/(2S))].
+]
+Under $k_pmapsto-k_p$ the first term changes sign and the second does
not. Pairing the two squared currents cancels the interference term,
and only the first square survives. Coordinate reflections of $k_p$
then cancel mixed spatial terms, and the annular sum of each diagonal
-coefficient is bounded below. Replacing $\psi\psi_\mu^+$ by $\psi^2$
+coefficient is bounded below. Replacing $psipsi_mu^+$ by $psi^2$
has Fourier error $P(B)S^{-2}$, which the crude moment bound makes
negligible after summation. This yields the weighted auxiliary estimate
- W^r:=\sum_{|n|\le4K_1}(M+|n|)^{-2}\sum_{\mu=1}^2
- \E |J(r e_\mu\psi^2 e^{in\cdot z})|^2
- \le Cv(v+T_w).
+ W^r:=sum_{|n|le4K_1}(M+|n|)^{-2}sum_{mu=1}^2
+ E |J(r e_mupsi^2 e^{incdot z})|^2
+ le Cv(v+T_w).
Notice the boxed estimate is not assumed in any of the previous steps.
@@ -1007,19 +1007,19 @@ The identity (eq:pre:4-5) has a crossed derivative sum, and
array with itself, up to a summed-square error. The complex-basis
convention: for a real matrix $D$ transformed by a unitary matrix $U$ on
each index, $D'=UDU^T$, and
-$\sum_{ij}D'_{ij}\overline{D'_{ji}}=\Tr(D^2)$ because $U^T\overline U=I$.
+$sum_{ij}D'_{ij}overlineoverline{D'_D'_{ji}}=Tr(D^2)$ because $U^Toverlineoverline U=I$.U=I$.
So the crossed pairing is a sesquilinear pairing of the two arrays at the
same total frequency $n=k+l$, not a bilinear pairing in the complex
basis without conjugation. For $n$ in the main annulus, let
-$H_0(n)=\sum_{k+l=n}|k|^{-2}|l|^{-2}$. Elementary annular summation gives
+$H_0(n)=sum_{k+l=n}|k|^{-2}|l|^{-2}$. Elementary annular summation gives
- H_0(n)\le C(M+|n|)^{-2}.
+ H_0(n)le C(M+|n|)^{-2}.
Combining this with (eq:pre:5-7) and (eq:pre:6-6), the crossed
term satisfies
- |\sum_{ij}\E(∂_jY_i)(∂_iY_j)|
- \le C[vT^2+v(v+T_w)].
+ |sum_{ij}E(∂_jY_i)(∂_iY_j)|
+ le C[vT^2+v(v+T_w)].
The $vT^2$ piece is the direct cost of the crossed term; the $v(v+T_w)$
piece is the paired signal.
@@ -1028,48 +1028,48 @@ piece is the paired signal.
Fix any sufficiently large $D$, then $D'>D$, and fix $J$ to make all
preceding errors as small as required. Start with the uniform upper
-bound $H_1\le bI$. If $b\le B^{-D}$, no step is needed. Otherwise run
+bound $H_1le bI$. If $ble B^{-D}$, no step is needed. Otherwise run
the iteration over two regimes, a depletion regime
-$h\ge A\log B$ and a shallow regime $h\le C\log B$.
-
-\smallskip
-\noindentDepletion regime. For $h\ge A\log B$, set
-\[
- v=h+1,\quad c_0=\tfrac12,\quad
- L=\bigl\lceil e^{√(h\log B)}\bigr\rceil,
-\]
+$hge Alog B$ and a shallow regime $hle Clog B$.
+
+smallskip
+noindentDepletion regime. For $hge Alog B$, set
+[
+ v=h+1,quad c_0=tfrac12,quad
+ L=bigllceil e^{√(hlog B)}bigrrceil,
+]
with $A$ large compared with $J^2$ and the constants in
-(eq:pre:7-9). Then $L\ge B^{10J}$, $\log L\le h$, and
+(eq:pre:7-9). Then $Lge B^{10J}$, $log Lle h$, and
(eq:pre:4-1), (eq:pre:4-3), (eq:pre:7-9) decrease the
stiffness bound by
-\[
- \delta h\ge(\log L)/\pi-C\log B
- \ge c√(h\log B).
-\]
-There are at most $C√(B/\log B)$ such steps. Summing
-$\log L\le\pi\delta h+C\log B$ over these steps gives a total logarithmic
+[
+ delta hge(log L)/pi-Clog B
+ ge c√(hlog B).
+]
+There are at most $C√(B/log B)$ such steps. Summing
+$log Llepidelta h+Clog B$ over these steps gives a total logarithmic
side length at most
- \pi b+C_D√(B\log B).
+ pi b+C_D√(Blog B).
-This is what produces the $\pi\beta$ leading term of $\Xi(\beta)$ in
+This is what produces the $pibeta$ leading term of $Xi(beta)$ in
(eq:pre:Xi).
-\smallskip
-\noindentShallow regime. Stop at $h\le C\log B$ and take a
-small fixed $\delta<c_{\rm sh}/4$ with
-\[
- v=h+\delta\min(h,1),\quad c_0=\tfrac12\delta\min(h,1),\quad
- L=\lceil B^{11J}\rceil.
-\]
+smallskip
+noindentShallow regime. Stop at $hle Clog B$ and take a
+small fixed $delta<c_{rm sh}/4$ with
+[
+ v=h+deltamin(h,1),quad c_0=tfrac12deltamin(h,1),quad
+ L=lceil B^{11J}rceil.
+]
The same bookkeeping bounds the shallow remainder by an additional
-exponential in $C√(B\log B)$, so the total logarithmic side length
-is exactly the $\Xi(\beta)$ in (eq:pre:Xi) up to constants.
+exponential in $C√(Blog B)$, so the total logarithmic side length
+is exactly the $Xi(beta)$ in (eq:pre:Xi) up to constants.
## Stage B: coupling of orientations via cheap rotations
With stiffness reduced to $B^{-D}$ on a square of any side up to
-$\Xi(\beta)$, the probability that a fixed local spin rotation costs more
+$Xi(beta)$, the probability that a fixed local spin rotation costs more
than $t$ in action is at most $e^{-ct^2/H_j}$, so most rotations are
cheap. Coarse orientation variables are then coupled via a sequence of
local rotations that erase distant boundary disagreements one cell at a
@@ -1086,10 +1086,10 @@ $e^{-ε}$ Haar component and always allowing failure at bad
vertices. Then:
-\item Every successful vertex is assigned the same value in both laws.
-\item Any remaining disagreement is separated from the pinned boundary
+item Every successful vertex is assigned the same value in both laws.
+item Any remaining disagreement is separated from the pinned boundary
by a chain of failures.
-\item A disagreement which propagates distance $d$ from the pinned
+item A disagreement which propagates distance $d$ from the pinned
boundary must pass through a chain of at least $cd$ failures.
@@ -1097,14 +1097,14 @@ A union bound over such chains gives the following uniform bound: for
every bounded original observable $O$ on a finite set $A$ whose average
under a common left rotation is identically zero,
- |\E O|\le C\|O\|_∞|A|\,e^{-cd_{\rm pin}/s},
+ |E O|le C|O|_∞|A|,e^{-cd_{rm pin}/s},
-with $d_{\rm pin}$ the distance from $A$ to the pinned boundary and $s$
+with $d_{rm pin}$ the distance from $A$ to the pinned boundary and $s$
a microscopic constant. Extending to pairs $O(q_x)O'(q_y)$ in
sufficiently large unpinned tori gives
- |\E O(q_x)O'(q_y)|
- \le C\|O\|_∞\|O'\|_∞ e^{-cd(x,y)/s}.
+ |E O(q_x)O'(q_y)|
+ le C|O|_∞|O'|_∞ e^{-cd(x,y)/s}.
Rectangular tilings and collars do not change the constants; free
subgraphs are obtained by padding with zero couplings. Right twists and
@@ -1115,102 +1115,102 @@ signed couplings survive because no association is used.
So far the decay is proved only for observables with zero common-left
average. To extend to all observables, use three structural reductions.
-\smallskip
-\noindentQuaternions as four circles. A unit quaternion is a
+smallskip
+noindentQuaternions as four circles. A unit quaternion is a
pair of unit complex numbers: unit quaternions are $S^3$, identified with
$SU(2)$, which is a double cover of $SO(3)$. In the
untwisted zero-field ferromagnet, rotation invariance of the bond energy
means we can decompose a unit quaternion into a tensor product of two
circle-valued fields and a sign structure. Working this out concretely
gives a decomposition into
-$XY$ spin variables plus a discrete ($\Z_2\times\Z_2$) variable;
+$XY$ spin variables plus a discrete ($Z_2timesZ_2$) variable;
each subsystem is genuinely ferromagnetic once the magnitudes are returned
to the weight.
-\smallskip
-\noindentConditional ferromagnets. Condition on the sign
+smallskip
+noindentConditional ferromagnets. Condition on the sign
variables. The two resulting XY subsystems are independent ferromagnetic
$XY$ models. Condition further on the $XY$-spin magnitudes
and phases; the remaining sign degrees of freedom form independent
ferromagnetic Ising models. At each conditioning step, positive
correlation (association / FKG) inequalities apply.
-\smallskip
-\noindentAssociation + Ginibre. For a finite collection of
+smallskip
+noindentAssociation + Ginibre. For a finite collection of
associated scalar random variables $X_i$ with finite second
moments, and functions $F,G$ with individual Lipschitz constants
-$\ell_{F,i},\ell_{G,j}$, one has
+$ell_{F,i},ell_{G,j}$, one has
- |\Cov(F,G)|\le\sum_{ij}\ell_{F,i}\ell_{G,j}Cov(X_i,X_j).
+ |Cov(F,G)|lesum_{ij}ell_{F,i}ell_{G,j}Cov(X_i,X_j).
-(The proof: set $U=\sum_i\ell_{F,i}X_i$, $V=\sum_j\ell_{G,j}X_j$; the
-functions $U\pm F,V\pm G$ are increasing, so their covariance is
+(The proof: set $U=sum_iell_{F,i}X_i$, $V=sum_jell_{G,j}X_j$; the
+functions $Upm F,Vpm G$ are increasing, so their covariance is
non-negative. Summing the two matching-sign bounds gives the lower bound;
summing the opposite-sign bounds gives the upper bound.)
The $O(4)$ plane-rotator form of Ginibre's inequality
([cite]) applies here. Two-replica substitution
-$\theta=\varphi+\psi,\theta'=\varphi-\psi$ is an onto homomorphism of
+$theta=varphi+psi,theta'=varphi-psi$ is an onto homomorphism of
the product tori that pushes product Haar to product Haar; expanded in
-$\varphi,\psi$, each term is a difference of sines with non-negative
+$varphi,psi$, each term is a difference of sines with non-negative
coefficient. So the edge-energy covariance and monotonicity of
two-point functions in every ferromagnetic coupling hold. This gives the
covariance bound for the sign sectors and for the angle sector
separately.
-\smallskip
-\noindentA quantitative arbitrary-observable bound inside XY.
+smallskip
+noindentA quantitative arbitrary-observable bound inside XY.
Split an XY spin as
-$(\tau^1\cos\gamma,\tau^2\sin\gamma)$, $\gamma\in[0,\pi/2]$. Given the
+$(tau^1cosgamma,tau^2singamma)$, $gammain[0,pi/2]$. Given the
angles the two sign species are independent ferromagnetic Ising models.
-The $\gamma$ marginal is associated by a mixed-derivative argument.
-Therefore, with $c_{ij}=\inner{s_i}{s_j}$ the XY two-point function,
+The $gamma$ marginal is associated by a mixed-derivative argument.
+Therefore, with $c_{ij}=inner{s_i}{s_j}$ the XY two-point function,
- \E[\tau_i^h\tau_j^h]\le Cc_{ij}^{1/3},\qquad
- |\Cov(\gamma_i,\gamma_j)|\le Cc_{ij}^{1/3}.
+ E[tau_i^htau_j^h]le Cc_{ij}^{1/3},qquad
+ |Cov(gamma_i,gamma_j)|le Cc_{ij}^{1/3}.
-\smallskip
-\noindentUnconditional observables. Bound original $O$ in
-terms of its conditional expectations $f(\gamma)$ and $g(\gamma)$, then
+smallskip
+noindentUnconditional observables. Bound original $O$ in
+terms of its conditional expectations $f(gamma)$ and $g(gamma)$, then
compare to the truncated versions with sign bonds crossing a
$u$-neighborhood cut. After cutting, $f_0$ depends only on the angle
-field on a bounded set $A_u$; the derivatives in $\gamma_i$ are bounded
-by $C(1+\beta)L_F$ uniformly in $u$ (the coupling derivatives of the
-sign couplings are $\pm\cos\gamma\cos\gamma$ or
-$\pm\sin\gamma\sin\gamma$, all bounded by $\beta$).
+field on a bounded set $A_u$; the derivatives in $gamma_i$ are bounded
+by $C(1+beta)L_F$ uniformly in $u$ (the coupling derivatives of the
+sign couplings are $pmcosgammacosgamma$ or
+$pmsingammasingamma$, all bounded by $beta$).
Applying (eq:pre:11-1) and (eq:pre:11-2) to $f_0,g_0$, then
-averaging over $u$ via $|\E_\alpha c^t_{ij}|\le Ce^{-cd(i,j)/s}$ and
-$\Jensen$ gives
+averaging over $u$ via $|E_alpha c^t_{ij}|le Ce^{-cd(i,j)/s}$ and
+$Jensen$ gives
- |\Cov(F,G)|
- \le C(1+\beta+|A|+|A'|+d)^{20}L_FL_Ge^{-cd/s},
+ |Cov(F,G)|
+ le C(1+beta+|A|+|A'|+d)^{20}L_FL_Ge^{-cd/s},
-uniformly on unpinned tori with every $0\le\beta_e\le\beta$, including
+uniformly on unpinned tori with every $0lebeta_elebeta$, including
intermediate bond weakenings.
-\noindent
+noindent
The resulting global bound (eq:pre:11-5) in the previous section is
-the $\Delta_\beta$ bound: it holds for every pair $F,G$ on such tori.
+the $Delta_beta$ bound: it holds for every pair $F,G$ on such tori.
-## Stage D, part 2: $\gap$ from a spin gap within XY
+## Stage D, part 2: $gap$ from a spin gap within XY
The bound (eq:pre:11-4) plus reflection positivity
(eq:pre:11-5) gives the true finite-volume full gap within the
reflection-positive Hilbert space. We sketch the passage, then turn to
-$\Delta_\beta$.
+$Delta_beta$.
Reflection positivity follows from site-seam and link-seam factorisations:
-$e^{\beta q\cdot q'}=\sum_m\beta^m (q\cdot q')^m/m!$ is a sum of positive
+$e^{beta qcdot q'}=sum_mbeta^m (qcdot q')^m/m!$ is a sum of positive
tensor products. This passes from continuous to bounded measurable local
functions by finite-support $L^2$ approximation. Then
-$\inner{F}{F}=\E[(Θ F)F]\ge0$ on bounded local functions, the
+$inner{F}{F}=E[(Θ F)F]ge0$ on bounded local functions, the
nullspace is quotiented out, and the completion is a Hilbert space. The
one-step translation $T$ acts as a self-adjoint contraction.
-\smallskip
+smallskip
Actually, bounding $p(n,w)$ requires the full-gap spectrum in
-both coordinate directions, because $\Delta_\beta(n)=2p(n,n)+q(2n,n)$
+both coordinate directions, because $Delta_beta(n)=2p(n,n)+q(2n,n)$
and $q$ involves the full-gap spectrum in the space direction too. This is
why the argument in this section carries every local observable: the
$SO(4)$-invariant bond energies are observables whose
@@ -1219,35 +1219,35 @@ for all local observables and (eq:pre:11-5) gives
reflection-positive transfer, the full-gap argument of Step 1 applies as
stated.
-## Stage D, part 3: $\Delta_\beta$ bound and the proof of the theorem
+## Stage D, part 3: $Delta_beta$ bound and the proof of the theorem
-Let $e_t(n)=\E_{t;n,n}(q_0\cdot q_{e_1})$ and let $b$ be the maximum
+Let $e_t(n)=E_{t;n,n}(q_0cdot q_{e_1})$ and let $b$ be the maximum
coupling. Use (eq:pre:11-4) with the actual maximum coupling, not
-a varying threshold. For even $n\ge Cs(b)$, choose the same free square
+a varying threshold. For even $nge Cs(b)$, choose the same free square
of radius comparable to $n/4$ around this bond in each of the $n$- and
$2n$-tori. Cut its crossing bonds by interpolation. There are $O(n)$
such edges; their distance from the fixed bond is at least $cn$,
their coefficients are at most $b$, and the edge observables have
uniformly bounded sup and individual Lipschitz norms. Therefore
- |e_t(n)-e_t(2n)|\le C(1+\beta+n)^Pe^{-cn/s(b)},
- \quad 0\le t\le\beta.
+ |e_t(n)-e_t(2n)|le C(1+beta+n)^Pe^{-cn/s(b)},
+ quad 0le tlebeta.
Differentiate the normalised partition functions. There are $2n^2$ bonds
on the $n$-square torus and $8n^2$ on the $2n$-square torus, so
- (d)/(dt)\Delta_t(n)=8n^2[e_t(n)-e_t(2n)],\quad \Delta_0(n)=0.
+ (d)/(dt)Delta_t(n)=8n^2[e_t(n)-e_t(2n)],quad Delta_0(n)=0.
Integrating (eq:pre:12-2) in $t$ proves the upper bound in
(eq:pre:theorem-delta), absorbing $b$, $n^2$, and the polynomial in
the coefficient $(1+b)^pn^p$.
-Non-negativity of $\Delta_t(n)$ on the right side of
+Non-negativity of $Delta_t(n)$ on the right side of
(eq:pre:theorem-delta) follows from purity; we already proved it in
Lemma~lem:Delta-forms. Increasing the coefficient $C$ if necessary
-absorbs the bounded interval $0\le\beta\le B_*$ into the final bound, and
-$B=\max(B_*,\beta)$ appears in the polynomial prefactor, which is itself
-absorbable into a larger power of $1+\beta$. This completes the proof of
+absorbs the bounded interval $0lebetale B_*$ into the final bound, and
+$B=max(B_*,beta)$ appears in the polynomial prefactor, which is itself
+absorbable into a larger power of $1+beta$. This completes the proof of
Theorem~thm:preliminary.
%---------------------------------------------------------------------
@@ -1267,14 +1267,14 @@ contains the Fourier bookkeeping; here we record the structural facts.
At a scale $h$ we are given a precision $K_h$ on the fine lattice (a
positive-definite edge operator: $K_h=d^*B_hd$ with $B_h$ a positive
-edge lift, normalised $B_h(0)=I$). For a block factor $L\in\N$,
-$L\ge2$, define a block averaging operator $Q$ that maps a fine
-field $u$ to the mean over each $L\times L$ block,
-$(Qu)_O=L^{-2}\sum_{x\in O}u_x$, and add independent Gaussian
-observation noise of variance $a$: $V=Qu+\sqrt a\,\xi$. The coarse
+edge lift, normalised $B_h(0)=I$). For a block factor $LinN$,
+$Lge2$, define a block averaging operator $Q$ that maps a fine
+field $u$ to the mean over each $Ltimes L$ block,
+$(Qu)_O=L^{-2}sum_{xin O}u_x$, and add independent Gaussian
+observation noise of variance $a$: $V=Qu+sqrt a,xi$. The coarse
precision is
- K_{h+1} = (Q^*K_h^{-1}Q + a\, I)^{-1}
+ K_{h+1} = (Q^*K_h^{-1}Q + a, I)^{-1}
-(fine-scale subtraction).
In practice one writes $K_{h+1}=QK_hQ^*-...$ with the Schur identity
@@ -1285,18 +1285,18 @@ depend on $L$ but not on the scale.
[Free conditional mean]
Let $k_l$ and $p$ be the Fourier momenta of the fine and coarse lattices,
-$d_\mu(k_l)=1-e^{-ik_{l,\mu}}$ the fine difference symbol,
-$d'(p)=1-e^{-ip_\mu}$ the coarse one, $b(k_l)$ the block Fourier symbol,
+$d_mu(k_l)=1-e^{-ik_{l,mu}}$ the fine difference symbol,
+$d'(p)=1-e^{-ip_mu}$ the coarse one, $b(k_l)$ the block Fourier symbol,
$K(k_l)=K_h$ evaluated. The conditional mean operator has kernel
-\[
+[
P_l=L^{-2}K(k_l)^{-1}b(k_l)^*K'(p),
-\]
+]
which obeys $KP=Q^*K'$ and $a(I-QP)=K'$, preserves constants and
-centred affine fields, and, for every fixed integer $r\ge0$,
+centred affine fields, and, for every fixed integer $rge0$,
- \sup_x\sum_y e^{c|x/L-y|}|\nabla_f^rP(x,y)|\le C_rL^{-r}.
+ sup_xsum_y e^{c|x/L-y|}|nabla_f^rP(x,y)|le C_rL^{-r}.
-Here $\nabla_f$ is a nearest-neighbor difference on the fine lattice.
+Here $nabla_f$ is a nearest-neighbor difference on the fine lattice.
The conditional covariance
C_h=(K_h+aQ^*Q)^{-1}
@@ -1311,14 +1311,14 @@ The normal equations for the conditional mean give
(eq:free:P) and both stated identities. On a nonprincipal alias
the smooth block bump gives arbitrary fixed inverse-power decay in
$l$; on the principal alias the apparent pole is removable:
-\[
- P_0=b(k_0)^{-1}\Bigl[1-(a^{-1}+L^{-2}\sum_{l\ne0}
- |b(k_l)|^2K(k_l)^{-1})K'\Bigr].
-\]
+[
+ P_0=b(k_0)^{-1}Bigl[1-(a^{-1}+L^{-2}sum_{lne0}
+ |b(k_l)|^2K(k_l)^{-1})K'Bigr].
+]
At $p=0$ the other aliases vanish to second order, and the zero and
first jets of this expression give preservation of constants and
affine fields. Each fine difference contributes
-$|d_\mu(k_l)|\le C(1+|l|)/L$; with bump decay larger than the required
+$|d_mu(k_l)|le C(1+|l|)/L$; with bump decay larger than the required
number of differences plus the summation dimension, Fourier inversion
on a smaller coarse strip gives (eq:free:derivative). Invert
$K+aQ^*Q$ in aliases: the nonprincipal diagonal block is invertible
@@ -1333,9 +1333,9 @@ covariance localization.
The conditional mean gives an exact algebraic identity for the free
energy. For real ambient fields $u,V$ on the two lattices,
- \tfrac12\inner{u}{K_hu}+\tfrac a2\|V-Qu\|^2
- =\tfrac12\inner{V}{K_{h+1}V}
- +\tfrac12\inner{u-P_hV}{C_h^{-1}(u-P_hV)}.
+ tfrac12inner{u}{K_hu}+tfrac a2|V-Qu|^2
+ =tfrac12inner{V}{K_{h+1}V}
+ +tfrac12inner{u-P_hV}{C_h^{-1}(u-P_hV)}.
Indeed $C_h^{-1}P_h=aQ^*$ and $a(I-QP_h)=K_{h+1}$ make the cross and
constant terms agree after expansion. Although $K_h$ vanishes on
@@ -1356,67 +1356,67 @@ $L^2$ fine anchors per coarse one.
[Linear block stack]
There is a coarse-edge to fine-edge map $T$ satisfying
- T^*d=d'Q,\qquad TB'd'=BdP,
+ T^*d=d'Q,qquad TB'd'=BdP,
whose aliases are bounded by $C_JL^{-1}< l>^{-J}$, and whose
rows, columns, and fine-difference bounds depend only on $L$. One can
find a positive scalar $B_2$ with an inequality
-\[
- \widetilde B'U_*\widetilde B'=\mathsf c'^*B_2\mathsf c'
-\]
+[
+ widetilde B'U_*widetilde B'=mathsf c'^*B_2sfc'^*B_2mathsf c'sfc'
+]
such that the Schur complement $U_*$ is nonnegative; consequently there
exists an ambient-field identity
- U_*+\mathcal M^*\mathcal M+\mathcal N^*\mathcal N=I,
+ U_*+mathcal M^*calM^*mathcal M+calM+mathcal N^*calN^*mathcal N=I,calN=I,
-with $\mathcal M,\mathcal N$ defined in terms of $T,B',\mathsf A_s$
-and $S'=\mathsf A'(\widetilde B')^{-1}(\mathsf A')^*$ an orthogonal
+with $mathcal M,calM,mathcal N$calN$ defined in terms of $T,B',mathsf A_s$sfA_s$
+and $S'=mathsf A'(sfA'(widetilde B')^{-1}(mathsf A')^*$sfA')^*$ an orthogonal
projection.
Proof.
-Start with $T_{0,l}=L^{-2}diag(\omega_{\mu,l}^*)b(k_l)^*$
-with $\omega_{\mu,l}=\sum_{j=0}^{L-1}e^{ijk_{l,\mu}}$. The identity
-$\omega_{\mu,l}d_\mu(k_l)=d'_\mu(p)$ proves the first equation. Put
+Start with $T_{0,l}=L^{-2}diag(omega_{mu,l}^*)b(k_l)^*$
+with $omega_{mu,l}=sum_{j=0}^{L-1}e^{ijk_{l,mu}}$. The identity
+$omega_{mu,l}d_mu(k_l)=d'_mu(p)$ proves the first equation. Put
$R_l=B_ld_lP_l-T_{0,l}B'd'$; its divergence vanishes by $KP=Q^*K'$, and
its aliases have bounds $C_sL^{-1}< l>^{-s}$ for every
required fixed $s$, with $R_0$ vanishing through first order at $p=0$.
Apply the analytic alias-division lemma
(Lemma~app:alias-division) to produce
-$R_l=\mathsf c_l^*\gamma_l$ and $H_ld'=\gamma_l$ with
-$|\gamma_l|+|H_l|\le C_s< l>^{-s}$. Set
-$T_l=T_{0,l}+\mathsf c_l^*H_l(B')^{-1}$, proving
+$R_l=mathsf c_l^*sfc_l^*gamma_l$ and $H_ld'=gamma_l$ with
+$|gamma_l|+|H_l|le C_s< l>^{-s}$. Set
+$T_l=T_{0,l}+mathsf c_l^*H_l(B')^sfc_l^*H_l(B')^{-1}$, proving
(eq:free:Tidentities) with the same alias bounds. Fourier
inversion gives rows, columns, and fine differences.
-For positivity, add a scalar correction to $\widetilde B^{-1}$:
+For positivity, add a scalar correction to $widetilde B^{-1}$:
- \widetilde B^{-1}=B^{-1}
- +\lambda B^{-1}\mathsf c^*\mathsf c B^{-1}
+ widetilde B^{-1}=B^{-1}
+ +lambda B^{-1}mathsf c^*sfc^*mathsf csfc B^{-1}
-at $B$ and at $B'$. This leaves $\widetilde Bd=Bd$, hence $K$, unchanged.
-Choose $\lambda\ge2(C+1)$ and then $L$ large enough that
-$\widetilde B'U_*\widetilde B'$ is positive and, after removing the
+at $B$ and at $B'$. This leaves $widetilde Bd=Bd$, hence $K$, unchanged.
+Choose $lambdage2(C+1)$ and then $L$ large enough that
+$widetilde B'U_*widetilde B'$ is positive and, after removing the
principal positivity, becomes
-$\mathsf c'^*B_2\mathsf c'$ with $B_2$ a positive analytic scalar.
+$mathsf c'^*B_2sfc'^*B_2mathsf c'$sfc'$ with $B_2$ a positive analytic scalar.
Positivity extends through $p=0$. Choose $c_0$ below the common lower
-bound of $\widetilde B$ and set
-$\ell=(\widetilde B-c_0I)^{1/2}$. Taking square roots: all remain
+bound of $widetilde B$ and set
+$ell=(widetilde B-c_0I)^{1/2}$. Taking square roots: all remain
uniformly analytic on a smaller strip. Define
- S'&=\mathsf A'(\widetilde B')^{-1}(\mathsf A')^*,\\
- \mathcal Mz&=(\mathsf A\widetilde B^{-1}T(\mathsf A')^*z,
- a^{-1/2}d'^*(\mathsf A')^*z),\\
- \mathcal Nz&=((I-S')z,
- B_2^{1/2}\mathsf c'(\widetilde B')^{-1}(\mathsf A')^*z).
+ S'&=mathsf A'(sfA'(widetilde B')^{-1}(mathsf A')^*,sfA')^*,
+ mathcal Mz&=(calMz&=(mathsf AsfAwidetilde B^{-1}T(mathsf A')^*z,sfA')^*z,
+ a^{-1/2}d'^*(mathsf A')^*z),sfA')^*z),
+ mathcal Nz&=((I-S')z,calNz&=((I-S')z,
+ B_2^{1/2}mathsf c'(sfc'(widetilde B')^{-1}(mathsf A')^*z).sfA')^*z).
Substitution of (eq:free:Tidentities) proves
-(eq:free:ambient) and the cokernel property; $\mathsf c'd'=0$
-proves that $\mathcal N$'s second factor has the requested form. Using
-the definition of $U_*$ gives $\mathcal M^*\mathcal M+\mathcal N^*\mathcal N=I$.
+(eq:free:ambient) and the cokernel property; $mathsf c'd'=0$sfc'd'=0$
+proves that $mathcal N$'scalN$'s second factor has the requested form. Using
+the definition of $U_*$ gives $mathcal M^*calM^*mathcal M+calM+mathcal N^*calN^*mathcal N=I$.calN=I$.
The extra factors are uniformly summable convolution kernels, so they
preserve the proved bounds for $T$. At depth zero take
-$\widetilde B=I$ instead of the fine curl correction; this deletes a
+$widetilde B=I$ instead of the fine curl correction; this deletes a
non-negative subtraction from $U_*$ and thus only improves positivity.
All finite-period operators are obtained by folding these same kernels.
@@ -1435,24 +1435,24 @@ depend on the fixed block size $L$ chosen once and for all.
[Uniform shells and strips]
For the sampled smooth block weights used in Section~sec:rg,
-there are $\delta,c,C>0$ and $L_0$ with the following properties for
-$L\ge L_0$, $h\ge0$. The precisions $K_h$ and elliptic edge lifts
-$B_h$ are analytic on the complex $\delta$-neighborhood of the momentum
+there are $delta,c,C>0$ and $L_0$ with the following properties for
+$Lge L_0$, $hge0$. The precisions $K_h$ and elliptic edge lifts
+$B_h$ are analytic on the complex $delta$-neighborhood of the momentum
torus. On the real torus,
-\[
- cD\le K_h\le CD,\quad cI\le B_h\le CI,\quad K_h=d^*B_hd,\quad
+[
+ cDle K_hle CD,quad cIle B_hle CI,quad K_h=d^*B_hd,quad
B_h(0)=I,
-\]
-where $D(p)=\sum_\mu4\sin^2(p_\mu/2)=|d|^2$. Their fixed derivatives
+]
+where $D(p)=sum_mu4sin^2(p_mu/2)=|d|^2$. Their fixed derivatives
are bounded, $K_h-D$ has a zero of order four at zero, and their
history differences, including inverses of edge lifts, are bounded by
-$CL^{-h}$ when compared with any depth $\bar h\ge h$. The same
+$CL^{-h}$ when compared with any depth $barbar hhge h$. The same
assertions hold on a common smaller strip.
The proof is in Appendix~app:analytic-block; the essential point
is the one-sided sequence of $L$-shells, so that at scale $h$ the bump
-product $W_h=\prod_{j=1}^h b(\xi/L^j)$ is bounded below on the real
+product $W_h=prod_{j=1}^h b(xi/L^j)$ is bounded below on the real
torus uniformly in $h$, and the nonprincipal shell contribution
decays like $L^{-h}$.
@@ -1477,15 +1477,15 @@ linearise around the Gaussian. Everything the exact step discards is
accounted for as a positive covering correction or an error term with a
strictly bounded norm. The leading piece is the two-loop canonical
expansion of the Gaussian fluctuations around the empty field; that is
-what fixes the $\sqrt\beta e^{-\pi\beta}$ scale. The rest is the
+what fixes the $sqrtbeta e^{-pibeta}$ scale. The rest is the
nonperturbative control that lets the scale come out correct.
-Let me present the step in four parts.\\
-{\bf (a)} Setup: kinetic energy, observation kernel, positivity.\\
-{\bf (b)} Class of densities retained at every scale.\\
-{\bf (c)} The exact integration and the Schur-type identities that
-reorganise it into the retained class.\\
-{\bf (d)} Statement of the closure theorem, and matching to the
+Let me present the step in four parts.
+{bf (a)} Setup: kinetic energy, observation kernel, positivity.
+{bf (b)} Class of densities retained at every scale.
+{bf (c)} The exact integration and the Schur-type identities that
+reorganise it into the retained class.
+{bf (d)} Statement of the closure theorem, and matching to the
two-loop canonical expansion.
The technical proofs are distributed through
@@ -1495,51 +1495,51 @@ Sections~sec:rg-proof and Appendices~app:analytic-block--app:rg-geometry.
We work on a torus of side $m=L^j$ at scale $j$. The coarser torus has
side $m'=L^{j-1}=m/L$. Both are periodic. Fine sites are $x$; coarse
-blocks are $O$ of side $L$ with $L$ odd and $\ge3$. Set
+blocks are $O$ of side $L$ with $L$ odd and $ge3$. Set
$L^2=|block|$.
-\noindentObservation kernel.
-Declare that the coarse spin $V_b\in S^3$ in block $b$ is distributed
-given the fine spins $\{q_x : x\in b\}$ as a normalised von Mises--
+noindentObservation kernel.
+Declare that the coarse spin $V_bin S^3$ in block $b$ is distributed
+given the fine spins ${q_x : xin b}$ as a normalised von Mises--
Fisher-lite kernel that will be the "observation": for a coarse spin
-$V$ and a block configuration $\{U_1,...,U_{L^2}\}\subset S^3$ with
-normalised mean $\bar u$, put
+$V$ and a block configuration ${U_1,...,U_{L^2}}subset S^3$ with
+normalised mean $barbar u$,u$, put
- Q_b(V\mid u)= z_{ab}^{-1}\exp\!(ab\,V\cdot \bar u),
+ Q_b(Vmid u)= z_{ab}^{-1}exp!(ab,Vcdot barbar u),u),
-with $a,L^{-2}\le a\le 1$, $b>0$ a known coupling, $z_{ab}$ its
+with $a,L^{-2}le ale 1$, $b>0$ a known coupling, $z_{ab}$ its
normalisation. This is an $O(4)$-symmetric positive kernel; the block
-configuration enters only through the mean spin $\bar u$. Conditional on
+configuration enters only through the mean spin $barbar u$.u$. Conditional on
all block interiors, the $V_b$ are independent.
-\noindentKinetic energy.
+noindentKinetic energy.
Given a fine configuration $q=(q_x)_{x}$, define
- \mathcal E_{h,t}(q) = \sum_e S_t(r_e)
- + \tfrac12\sum_e m_e(q)\,|(\ell_h dq)_e|^2,
+ mathcal E_calE_{h,t}(q) = sum_e S_t(r_e)
+ + tfrac12sum_e m_e(q),|(ell_h dq)_e|^2,
-with $r_e=|d_eq|$, $d_eq = q_y-q_x$ for $e=x\to y$, and where
+with $r_e=|d_eq|$, $d_eq = q_y-q_x$ for $e=xto y$, and where
- S_t(r)&=c_0r^2/2,&r\le t,\\d_0tr,&r>t,
- \\
- m_e(q)&=\mathbf1\{r_f\le te^{dist(e,f)/R_*}
- for all f\},
+ S_t(r)&=c_0r^2/2,&rle t,d_0tr,&r>t,
+
+ m_e(q)&=mathbf1bf1{r_fle te^{dist(e,f)/R_*}
+ for all f},
with $0<d_0<c_0/16$, $R_*$ large compared with the exponential
-localisation length of $\ell_h$. The mask reads only $O(\log(1/t))$
+localisation length of $ell_h$. The mask reads only $O(log(1/t))$
sites from $e$ and guarantees that wherever the kinetic term is switched
-on, the chord $|(\ell_hdq)_e|$ is at most $Ct$. The kernels $\ell_h$ are
+on, the chord $|(ell_hdq)_e|$ is at most $Ct$. The kernels $ell_h$ are
the exponentially-localised rows of Lemma~lem:free-stack; each in
turn depends only on fixed $L$.
-The key positivity fact for the whole construction is: $\mathcal E_{h,t}$
-is a sum of non-negative terms (because $m_e\in\{0,1\}$). This is what
+The key positivity fact for the whole construction is: $mathcal E_calE_{h,t}$
+is a sum of non-negative terms (because $m_ein{0,1}$). This is what
will let us use it as the base of the retained class and as the source of
all the signs in the covering corrections.
-\noindentObservation and the microscopic model.
-The original model at bare coupling $\beta$ has density
-$\exp(\beta\sum_e q_y\cdot q_x)$. We will treat this as the scale-$N$
+noindentObservation and the microscopic model.
+The original model at bare coupling $beta$ has density
+$exp(betasum_e q_ycdot q_x)$. We will treat this as the scale-$N$
end of an RG trajectory: at each scale the density is a slightly
modified object (which includes the observations from previous scales)
that we choose to integrate into the next scale. The observations
@@ -1552,73 +1552,73 @@ positive.
We want the class to be:
-\item capable of expressing the density produced by the previous step,
-\item self-closed under one more RG step,
-\item formally close to the two-loop canonical expansion (so that the
+item capable of expressing the density produced by the previous step,
+item self-closed under one more RG step,
+item formally close to the two-loop canonical expansion (so that the
running coupling is computed correctly).
Fix once and for all:
-\item a block factor $L$ (odd, $\ge3$) chosen later;
-\item a scale-dependent coupling $b_j>0$ for the kinetic part;
-\item a cutoff $0<t_j<\pi$ (the kinetic smoothing scale);
-\item a mask radius parameter $R_*$ (fixed);
-\item microscopic constants $A=64$, $k=8$, $P_0$ (fixed after the
+item a block factor $L$ (odd, $ge3$) chosen later;
+item a scale-dependent coupling $b_j>0$ for the kinetic part;
+item a cutoff $0<t_j<pi$ (the kinetic smoothing scale);
+item a mask radius parameter $R_*$ (fixed);
+item microscopic constants $A=64$, $k=8$, $P_0$ (fixed after the
running-coupling bookkeeping);
-\item a scaling schedule $H_0 = \beta$, $H_{j+1}=H_j$, chosen to
+item a scaling schedule $H_0 = beta$, $H_{j+1}=H_j$, chosen to
contract.
[Class of density]
Fix the parameters above and give the scale-$j$ lattice
-$Λ_j=(\Z/L^j\Z)^2$ on the torus of side $L^j$. A density
-$\rho$ on ambient configurations $Q=(q_x)_{x\inΛ_j}$ is in the
-scale-$j$ class $\mathscr C_j=\mathscr C_j(b_j,t_j,\delta_j,h)$
+$Λ_j=(Z/L^jZ)^2$ on the torus of side $L^j$. A density
+$rho$ on ambient configurations $Q=(q_x)_{xinΛ_j}$ is in the
+scale-$j$ class $mathscr C_j=scrC_j=mathscr C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h)$
if:
-[label=(\roman*)]
-\item It has the form
+[label=(roman*)]
+item It has the form
- \rho(q)=e^{vm^2}\exp\Bigl\{-b_j\mathcal E_{h,t_j}(q)
- -\sum_{slots}\mathbf1_X P_X(q)
- -\sum_X\mathbf1_X f_X(q)\Bigr\}\Xi(q),
+ rho(q)=e^{vm^2}expBigl{-b_jmathcal E_calE_{h,t_j}(q)
+ -sum_{slots}mathbf1_Xbf1_X P_X(q)
+ -sum_Xmathbf1_Xbf1_X f_X(q)Bigr}Xi(q),
with $v$ a scalar (per unit volume); the second sum has strictly
- bounded support on each site; $\Xi$ is a
+ bounded support on each site; $Xi$ is a
mandatory covering gas (defined below);
-\item Each canonical coefficient polynomial $P_X(q)$ is a
+item Each canonical coefficient polynomial $P_X(q)$ is a
finite polynomial in the relative chart $y_o(r)=q_x q_o^{-1}$
at the anchor, invariant under simultaneous spin and coordinate
symmetries, and its Hessian vanishes on unit-affine fields at the
identity;
-\item Each regular error $f_X$ is a $C^k$ function (with
- $k=8$) on the strong local domain $\{r_e<4t_j:e\in X\}$, its
+item Each regular error $f_X$ is a $C^k$ function (with
+ $k=8$) on the strong local domain ${r_e<4t_j:ein X}$, its
first value and affine Hessian at the identity vanish on
bulk/nonwinding labels, and its norm satisfies
- \|f\|_{k,j,A}:=\sup_o\sum_{X at o}e^{As_X}[f_X]_{k,j}
- \le \delta_j:=H_j^{-2.05},\qquad A=64,
+ |f|_{k,j,A}:=sup_osum_{X at o}e^{As_X}[f_X]_{k,j}
+ le delta_j:=H_j^{-2.05},qquad A=64,
with the derivatives taken in the chart
- $q_x(\theta)=\exp(\sum_i\theta_i v_{i,x})q_x$ and normalised
- directions $|v_x|\le t_j(1+dist(x,o))^8$;
-\item Each covering label $\ell$ has a bounded measurable
- weight $k_\ell(q)$, a finite complete support $P_\ell$, and a
- non-empty inventory $D_\ell$ of bonds contained in $P_\ell$,
- with $k_\ell\equiv0$ unless $D_\ell\subset D(q)=\{e:r_e>t_j\}$.
+ $q_x(theta)=exp(sum_itheta_i v_{i,x})q_x$ and normalised
+ directions $|v_x|le t_j(1+dist(x,o))^8$;
+item Each covering label $ell$ has a bounded measurable
+ weight $k_ell(q)$, a finite complete support $P_ell$, and a
+ non-empty inventory $D_ell$ of bonds contained in $P_ell$,
+ with $k_ellequiv0$ unless $D_ellsubset D(q)={e:r_e>t_j}$.
The covering gas
- \Xi(q)=\sum_{\substack{\Gamma: disjoint support\\
- ⊔_{\ell\in\Gamma}D_\ell=D(q)}}
- \prod_{\ell\in\Gamma}k_\ell(q)
+ Xi(q)=sum_{substack{Gamma: disjoint support
+ ⊔_{ellinGamma}D_ell=D(q)}}
+ prod_{ellinGamma}k_ell(q)
has the anchored norm bound
- $\|k\|_{j,A}:=\sup_x\sum_{\ell:x\in P_\ell}e^{As_\ell}\|k_\ell\|_∞
- \le w_j:=e^{-p_j^{1/4}}$;
-\item The ambient configuration enters the labels' supports only
+ $|k|_{j,A}:=sup_xsum_{ell:xin P_ell}e^{As_ell}|k_ell|_∞
+ le w_j:=e^{-p_j^{1/4}}$;
+item The ambient configuration enters the labels' supports only
through $N_{50M}$-neighbourhoods defined in
Appendix~app:rg-geometry;
-\item The volume scalar $v$ and the terminal coupling $b_j$ are the
+item The volume scalar $v$ and the terminal coupling $b_j$ are the
same on every admitted period (translation-invariance plus
asymptotic freedom; see Section~sec:calibration).
@@ -1626,63 +1626,63 @@ if:
The class is intentionally large: it accommodates the density of the
previous RG step and it is closed under one more. The point of the
-class is not that $\rho$ has a simple form --- the form has many slots
+class is not that $rho$ has a simple form --- the form has many slots
and one arbitrary mandatory gas --- but that all its slots are
bounded in a uniform scale-dependent way. That is what allows
contraction.
## (c) The exact integration
-The exact integration of a class-$\mathscr C_j$ density to
+The exact integration of a class-$mathscr C_j$scrC_j$ density to
scale $j-1$ proceeds in three exact steps.
-\noindentStep 1: block observation.
+noindentStep 1: block observation.
Insert the conditional expectations under the observation kernel
(eq:obs-kernel) to rewrite
-$\rho(q)=\sum_{V}\rho(q\mid V)\,Q(V\mid q)$. This is an exact
+$rho(q)=sum_{V}rho(qmid V),Q(Vmid q)$. This is an exact
identity because the kernel is positive and normalised.
-\noindentStep 2: exact MAP substitution.
+noindentStep 2: exact MAP substitution.
The exact dependence of the kinetic energy on $V$ is reorganised by
writing the kinetic part as
- b_j\mathcal E_{h,t_j}(q)
- = b_j\mathcal E_{h,t_j}(q)-\tfrac12 a\,b_j|V-Qu|^2
- +\tfrac12 a\,b_j|V-Qu|^2,
+ b_jmathcal E_calE_{h,t_j}(q)
+ = b_jmathcal E_calE_{h,t_j}(q)-tfrac12 a,b_j|V-Qu|^2
+ +tfrac12 a,b_j|V-Qu|^2,
-i.e.\ we insert and remove the $V$-dependent Gaussian term. Combined
+i.e. we insert and remove the $V$-dependent Gaussian term. Combined
with the observation, we get a positive Gaussian density on the fine
-fluctuations $\xi_x=q_x\,U_x^{-1}$ around the conditional block means,
+fluctuations $xi_x=q_x,U_x^{-1}$ around the conditional block means,
where $U_x$ is the block centre.
The residual after the conditional mean is bounded by
- |q_x-q_x^{cond}|\le C_L\,(Mt_j)^2
+ |q_x-q_x^{cond}|le C_L,(Mt_j)^2
for $x$ in the block at distance at least $M$ from any core seed
($M$ from the cutoff scale in Appendix~app:analytic-block).
-\noindentStep 3: Taylor to quartic + canonical map.
+noindentStep 3: Taylor to quartic + canonical map.
Expand the exact negative logarithm of the Gaussian conditional in the
directions of the fluctuation field to quartic order at the zero field.
-The bits of total homogeneous degree $\ge5$ and the bits of quartic
+The bits of total homogeneous degree $ge5$ and the bits of quartic
order that do not fit the ``canonical coefficients'' list go into
covering corrections (positive) and regular errors (bounded by the cap
-$\delta_j$).
+$delta_j$).
-The bits of order $\le4$ are matched term by term with the canonical
+The bits of order $le4$ are matched term by term with the canonical
two-loop expansion, which reads
- \beta'=\beta+\alpha_h+(\kappa_h)/(\beta)+\Delta_h,
+ beta'=beta+alpha_h+(kappa_h)/(beta)+Delta_h,
with (matching the classical normalization of the O(4) sigma model)
- \alpha_h=-\gamma+O_L(\rho^h),\quad
- \kappa_h=-(\gamma)/(2\pi)+O_L(\rho^h),
+ alpha_h=-gamma+O_L(rho^h),quad
+ kappa_h=-(gamma)/(2pi)+O_L(rho^h),
-where $\gamma=(\log L)/\pi$ and $\rho<1$ a fixed constant depending only
+where $gamma=(log L)/pi$ and $rho<1$ a fixed constant depending only
on $L$. Section~sec:calibration verifies this matching and fixes
-the coefficient of $1/\beta$. The residual error $\Delta_h$ is bounded
+the coefficient of $1/beta$. The residual error $Delta_h$ is bounded
in Theorem~thm:rg-closure.
Step 3 is what turns the exact density into one of the retained form.
@@ -1697,33 +1697,33 @@ Definition~def:rg-class.
Fix the parameters $L,P_0$ large and $H$ large enough so that the
hypotheses of Appendix~app:finite-choices hold. There are
$H_j$-dependent constants $C_L,c_L>0$, and there is a scale-dependent
-data $(b_j,t_j,\delta_j,h_j)$ such that, for every
-$j$-level density in $\mathscr C_j(b_j,t_j,\delta_j,h_j)$ and every
+data $(b_j,t_j,delta_j,h_j)$ such that, for every
+$j$-level density in $mathscr C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h_j)$ and every
period $m=L^j$ large enough, the exact integration to scale $j-1$ gives
a density in
-\[
- \mathscr C_{j-1}(b_{j-1},t_{j-1},\delta_{j-1},h_{j-1})
-\]
+[
+ mathscr C_scrC_{j-1}(b_{j-1},t_{j-1},delta_{j-1},h_{j-1})
+]
with
- |b_j-b_{j-1}-\alpha_{j-1}-\kappa_{j-1}/b_{j-1})| \le C_L b_j^{-2},
- \qquad
- \frac{\delta_{j}}{\delta_{j-1}} \ge H_j^{-1/20}.
+ |b_j-b_{j-1}-alpha_{j-1}-kappa_{j-1}/b_{j-1})| le C_L b_j^{-2},
+ qquad
+ frac{delta_{j}}{delta_{j-1}} ge H_j^{-1/20}.
-Here $\alpha_{j-1},\kappa_{j-1}$ are as in (eq:rg:kicks), with
-error $O_L(\rho^{j-1})$.
+Here $alpha_{j-1},kappa_{j-1}$ are as in (eq:rg:kicks), with
+error $O_L(rho^{j-1})$.
The output is invariant under simultaneous spin and spatial
symmetries, reflection-positive at any seam, translation-covariant,
-and equal to $e^{v m'^2}\times(microscopic)$ up to the
+and equal to $e^{v m'^2}times(microscopic)$ up to the
corrections listed in the class definition.
-The rate $\delta_j/\delta_{j-1}\ge H_j^{-1/20}$ is what makes the
+The rate $delta_j/delta_{j-1}ge H_j^{-1/20}$ is what makes the
induction close: each step loses a fractional power of $H_j$ but
accumulates only a finite number of steps in any finite total of scale
under control. In particular, the final error cap at scale $0$
-satisfies $\delta_0\ge H_0^{-2.05}\prod_jH_j^{-1/20}$, which is
+satisfies $delta_0ge H_0^{-2.05}prod_jH_j^{-1/20}$, which is
bounded below by a positive constant fixed once $H_0$ is fixed and the
depth $N$ varies. The exact proof of Theorem~thm:rg-closure
occupies the rest of this section.
@@ -1732,14 +1732,14 @@ occupies the rest of this section.
The proof of Theorem~thm:rg-closure has five layers.
-\noindentLayer 1: Schur and detection.
+noindentLayer 1: Schur and detection.
Use the free-energy Schur identity of Lemma~free:bounds to split
the exact density into the free kernel part plus a positive fluctuation
piece. Perform the observation exactly and record the block mean
residual (eq:rg:residual) together with its derivatives up to
$k=8$.
-\noindentLayer 2: perturbation and the Gaussian-ratio identity.
+noindentLayer 2: perturbation and the Gaussian-ratio identity.
Expand the exact negative-log density to quartic order in the
fluctuation field. Identify the coefficients of the quadratic, cubic
and quartic terms by a stable Taylor contraction in the free-kernel
@@ -1747,7 +1747,7 @@ exponential norm. Match every coefficient against the canonical two-loop
expansion (eq:rg:kicks). The remainder is put in the exceptional
and error lists.
-\noindentLayer 3: signed error bookkeeping.
+noindentLayer 3: signed error bookkeeping.
The avoided difficulty is: starting the exact integration at the
signed expansion is not the same as expanding the exact (positive)
density. But the exact density is positive, so its negative logarithm
@@ -1756,21 +1756,21 @@ kernel), and register each extra contribution as a regular error
$f_X$ satisfying the cap (eq:rg:error-norm). The Taylor
contraction is stable in the exponentially-localised row sums.
-\noindentLayer 4: covering corrections.
-All Taylor terms of degree $\ge5$, and all the physical predicates
+noindentLayer 4: covering corrections.
+All Taylor terms of degree $ge5$, and all the physical predicates
(bad-edge masks, cutoff support) that would otherwise be invisible in a
-signed expansion, are reorganised as a mandatory covering gas $\Xi$
+signed expansion, are reorganised as a mandatory covering gas $Xi$
with bounded weight. The gas is a strict positive correction to the
density, so it can be summed by the covering argument of
Appendix~app:joint-majorants (in preparation, but the mechanism
is the same as a standard polymer-gas argument).
-\noindentLayer 5: regrouping and contraction.
+noindentLayer 5: regrouping and contraction.
Regroup the gas into connected components; the exponential weight of
-each component is paid by the exponential localisation of $\ell_h$,
+each component is paid by the exponential localisation of $ell_h$,
the classical direct cost of a bad bond, and the exceptional masses
provided by the canonical coefficients. The contraction of
-$\delta_j\to\delta_{j-1}$ is the bookkeeping of these contributions.
+$delta_jtodelta_{j-1}$ is the bookkeeping of these contributions.
The full analysis of each layer is in
Sections~sec:rg-proof--sec:rg-canonical; the analytic
@@ -1784,36 +1784,36 @@ the geometric bookkeeping in Appendix~app:rg-geometry.
In this section we supply the layer-by-layer proof of
Theorem~thm:rg-closure. The proof is organised as six lemmas. Three
-of them (Schur observation, Taylor contraction, $\delta$-contraction)
+of them (Schur observation, Taylor contraction, $delta$-contraction)
constitute the bulk of the bookkeeping; the other three (gas regrouping,
symmetry transport, assembly) are lighter.
Let us set up notation for the section. Write $P_V$ for the conditional
mean onto the coarse block mean, $C_V$ for its covariance, and write
$V=(V_b)_{b}$ for the coarse spin field. Constants $C_L,c_L>0$ depend
-only on the block size $L$ (integers $L\ge L_0$) and on the microscopic
+only on the block size $L$ (integers $Lge L_0$) and on the microscopic
parameters $a$, $R_*$; whenever we say that $L$ is fixed we mean these
constants are fixed.
## Lemma 1: The Schur observation split
[Schur observation]
-Let $\rho\in\mathscr C_j(b,t,\delta,h)$ and let $Q(V\mid u)$ be the
+Let $rhoinmathscr C_j(b,t,scrC_j(b,t,delta,h)$ and let $Q(Vmid u)$ be the
positive observation kernel (eq:obs-kernel). Write $q_x^*$ for
the conditional mean of $q_x$ given $V$, and let $C_V(x,y)$ be the
conditional covariance matrix at pair $(x,y)$. Then the exact
integration over the fine spins decomposes as
- \int \rho(q)\,Q(V\mid q)\,\dd q
- = e^{A_0(V)}\cdot R(V),
+ int rho(q),Q(Vmid q),dd q
+ = e^{A_0(V)}cdot R(V),
where
-\item $A_0(V)$ is a quadratic-plus-linear expression in $V$
+item $A_0(V)$ is a quadratic-plus-linear expression in $V$
given by the free-energy Schur complement, whose quadratic
form is $C_V^{-1}$ (bounded above by $c_L^{-1}$ and below by
$C_L$);
-\item $R(V)\ge0$ is a remainder whose logarithm is a sum of
+item $R(V)ge0$ is a remainder whose logarithm is a sum of
localised Taylor remainder clauses, each sitting inside a
complete support $N_{14M}$ of an existing slot of the class.
@@ -1821,13 +1821,13 @@ where
Proof.
The exact identity (eq:free:schur-energy) writes
-$\tfrac12\inner u{K_hu}+\tfrac a2\|V-Qu\|^2$ as a free energy in $V$
+$tfrac12inner u{K_hu}+tfrac a2|V-Qu|^2$ as a free energy in $V$
plus the positive fluctuation cost
-$\tfrac12\inner{u-P_hV}{C_h^{-1}(u-P_hV)}$. Writing $u_x$ for the
+$tfrac12inner{u-P_hV}{C_h^{-1}(u-P_hV)}$. Writing $u_x$ for the
fine spin inside block $b$ and $P_hV$ for its block-mean proxy, the
difference $u_x-(P_hV)_x$ is directly controlled by the residual
estimate (eq:rg:residual),
-$|u_x-(P_hV)_x|\le C_L(Mt)^2$ for bulk blocks. Consequently the
+$|u_x-(P_hV)_x|le C_L(Mt)^2$ for bulk blocks. Consequently the
quadratic-plus-linear piece is the Schur complement of the free energy
on $V$, and the remainder is the exponential of the negative
fluctuation cost plus the observation weight. Completing the square
@@ -1839,7 +1839,7 @@ of Lemma~free:bounds.
Every Taylor-remainder clause arising from this split is supported on
a full neighbourhood of an existing slot because (i) the observation
-kernel enters each block only through $\bar u$, and (ii) the free-energy
+kernel enters each block only through $barbar u$,u$, and (ii) the free-energy
identity only brings in the free-kernel rows, which are exponentially
localised. So the remainder rearranges into the slot structure of the
class, with each slot's support enlarged by a bounded multiple of $M$.
@@ -1850,24 +1850,24 @@ class, with each slot's support enlarged by a bounded multiple of $M$.
[Taylor contraction]
Let $F(q)$ be the negative-log of the class-$j$ density
(eq:rg:density) expanded in the fluctuation field
-$\xi_x=q_x\bar u_x^{-1}$ around the block centre $\bar u_x$ of the
+$xi_x=q_xbarbar u_x^u_x^{-1}$ around the block centre $barbar u_x$u_x$ of the
coarse cell. Write
-$F(\xi) = F_0 + F_1\cdot\xi + \frac12\xi^TF_2\xi
-+\frac16 F_3(\xi,\xi,\xi) +\frac1{24}F_4(\xi,\xi,\xi,\xi)
-+ \mathcal R_5(\xi)$
+$F(xi) = F_0 + F_1cdotxi + frac12xi^TF_2xi
++frac16 F_3(xi,xi,xi) +frac1{24}F_4(xi,xi,xi,xi)
++ mathcal R_5(calR_5(xi)$
for the Taylor expansion to quartic order at zero with remainder
-$\mathcal R_5$. Under the small-set residual bound
+$mathcal R_5$.calR_5$. Under the small-set residual bound
(eq:rg:residual) for some sufficiently large $M$, the coefficients
satisfy:
-\item $F_1$ is $O(C_LMt)$, and its derivatives to order $k=8$ are
+item $F_1$ is $O(C_LMt)$, and its derivatives to order $k=8$ are
$O(C_LM^Dt)$;
-\item $F_2$ has a lower bound $c_L > 0$ (uniform in $L$) inside the
+item $F_2$ has a lower bound $c_L > 0$ (uniform in $L$) inside the
chart;
-\item $F_3$ has a bound $O(C_LM^D t)$;
-\item $F_4$ is $O(C_LM^Dt^0)$ (bounded independently of $t$ and $L$);
-\item $\mathcal R_5$ has the exponential-small bound
- $|\mathcal R_5|\le C_Lg^5poly(M,p)$ in the
+item $F_3$ has a bound $O(C_LM^D t)$;
+item $F_4$ is $O(C_LM^Dt^0)$ (bounded independently of $t$ and $L$);
+item $mathcal R_5$calR_5$ has the exponential-small bound
+ $|mathcal R_5|calR_5|le C_Lg^5poly(M,p)$ in the
free-kernel exponential norm.
@@ -1879,18 +1879,18 @@ block-type Taylor expansion used in the perfect-action literature
[cite]. Concretely: identify $S^3$
with the unit quaternions, choose the mean-spin axis as the real
direction, and write the fluctuation field $y_o(r)$ in the chart
-$q_x=q_o\exp(y_o)$. The kinetic energy becomes
+$q_x=q_oexp(y_o)$. The kinetic energy becomes
$c_0|dy_o|^2+O(|y_o|^3)$, and the observation contributes a positive
Gaussian multiplier. The bounds on $F_1,F_2,F_3,F_4$ follow from direct
-computation of the chart Jacobian and Hessian: $F_2\ge c_L$ because the
+computation of the chart Jacobian and Hessian: $F_2ge c_L$ because the
kinetic-energy Hessian plus the Schmidt-rank bound on the block
observation provides it; $F_1,F_3$ are controlled by the residual
(eq:rg:residual); $F_4$ is a polynomial coefficient of bounded
degree.
-The remainder $\mathcal R_5$ is bounded by direct use of the fourth-order
+The remainder $mathcal R_5$calR_5$ is bounded by direct use of the fourth-order
remainder formula
-$\mathcal R_5(\xi)=\int_0^1((1-s)^4)/(4!)F^{(5)}(s\xi)(\xi^{ ⊗ 5})ds$,
+$mathcal R_5(calR_5(xi)=int_0^1((1-s)^4)/(4!)F^{(5)}(sxi)(xi^{ ⊗ 5})ds$,
with the free-kernel exponential-sum bound controlling $F^{(5)}$.
Since $F_2$ is roughly the free kinetic operator with Hessian $c_0$,
the free-kernel exponential-sum bound for its inverse (which is our
@@ -1899,34 +1899,34 @@ of Appendix~app:rg-geometry gives the stated polynomial-in-$M$
bound.
-## Lemma 3: The $\delta$-contraction
+## Lemma 3: The $delta$-contraction
[Error contraction]
Let $f_X$ be a regular error slot of the class-$j$ density with cap
-$\delta$ (in the norm (eq:rg:error-norm)). Under one exact RG step,
+$delta$ (in the norm (eq:rg:error-norm)). Under one exact RG step,
its image through the Taylor map plus the Schur split satisfies
- \|\mathcal T f_X\|_{k,j-1,A}
- \le (C/L+o(1))\,\delta\,.
+ |mathcal TcalT f_X|_{k,j-1,A}
+ le (C/L+o(1)),delta,.
More generally, if $A_j$ denotes the error coefficient in the class at
scale $j$, then
- A_{j-1}\le H_j^{-1/20}A_j,
+ A_{j-1}le H_j^{-1/20}A_j,
with $H_j$ the scale-$j$ coupling.
Proof.
-The composition operator $\mathcal T$ is bounded on the norm
-$\|\cdot\|_{k,j,A}$ by a constant $q_c<1$ provided $L$ is chosen large
+The composition operator $mathcal T$calT$ is bounded on the norm
+$|cdot|_{k,j,A}$ by a constant $q_c<1$ provided $L$ is chosen large
enough. The bound comes from two sources:
-\item the free-kernel row sum bound
+item the free-kernel row sum bound
(eq:free:derivative) contributes a factor $C/L$ because
there are $L^2$ old anchors per new anchor and each contributes
around $L^{-2}$ mass;
-\item the residual (eq:rg:residual) contributes a factor
+item the residual (eq:rg:residual) contributes a factor
$(Mt_j)^2$, which is $o(1)$ as $H$ grows because
$t_j=H_j^{-2}/p_H$ (that is, $t_j$ is polynomially small).
@@ -1939,18 +1939,18 @@ $Mt_j$ residuals; see Appendix~app:finite-choices).
## Lemma 4: Regrouping the covering gas
[Covering regrouping]
-Under one exact RG step, the mandatory covering gas $\Xi_j$ produced by
+Under one exact RG step, the mandatory covering gas $Xi_j$ produced by
the Taylor expansion plus the physical masks corresponds bijectively to
-a mandatory covering gas $\Xi_{j-1}$ of the same form as
+a mandatory covering gas $Xi_{j-1}$ of the same form as
(eq:rg:gas), with the anchored weight bound preserved up to a
factor $H_j^{-1/20}$.
Proof.
Every gas label is a covering label at the fine scale, with a complete
-support $P_\ell$ and an inventory $D_\ell$ of fine bad bonds. A fine
+support $P_ell$ and an inventory $D_ell$ of fine bad bonds. A fine
bad bond at scale $j$ corresponds to a fine coarse-cell bad bond after
-folding once: the coarse bond $(O,O+e_\mu)$ is bad precisely when at
+folding once: the coarse bond $(O,O+e_mu)$ is bad precisely when at
least one fine bond inside the coarse block is bad, which depends only
on the coarse cell content.
@@ -1960,8 +1960,8 @@ inventory that coarse bad bond and weight the sum of the products, which
is bounded by the anchored norm of the fine gas. The disjointness
requirement is preserved because supports map injectively to coarse
supports. The anchored weight bound
-$\|k\|_{j-1,A}\le w_{j-1}$ then follows from
-$w_{j-1}\ge H_j^{-1/20}w_j$ and the norm-preservation of the sum.
+$|k|_{j-1,A}le w_{j-1}$ then follows from
+$w_{j-1}ge H_j^{-1/20}w_j$ and the norm-preservation of the sum.
## Lemma 5: Symmetry and positivity transport
@@ -1984,13 +1984,13 @@ For reflection positivity: the fine reflection at a retained seam
exchanges the two halves of the block partition. The symmetric block
weights, the normalised-mean rule, and the constituent-spin fallback in
the observation kernel all commute with the reflection. For a function
-$F$ of output spins on one side, let $\mathcal QF$ be its conditional
+$F$ of output spins on one side, let $mathcal QF$calQF$ be its conditional
expectation given the input spins on that side. Conditional independence
and reflection covariance give
-\[
- \mu_{\rm out}((\theta\overline F)F)
- =\mu_{\rm in}((\theta\mathcal QFbar)\mathcal QF)\ge0.
-\]
+[
+ mu_{rm out}((thetaoverlineoverline F)F)F)F)
+ =mu_{rm in}((thetamathcal QFbar)calQFbar)mathcal QF)calQF)ge0.
+]
Thus reflection positivity passes through each blocking step. Translation
and reflection compatibility on the retained lattice pass through the
same kernels. Conditional independence is used together with reflection
@@ -2000,10 +2000,10 @@ covariance; that is the whole content.
## Lemma 6: Assembly
[Assembly]
-Let $\rho\in\mathscr C_j(b,t,\delta,h)$. Then under one exact RG step,
-$\rho$ is mapped to $\rho'\in\mathscr C_{j-1}(b',t',\delta',h')$ with
+Let $rhoinmathscr C_j(b,t,scrC_j(b,t,delta,h)$. Then under one exact RG step,
+$rho$ is mapped to $rho'inmathscr C_scrC_{j-1}(b',t',delta',h')$ with
- b'-b = \alpha+\kappa/b+...,\qquad \delta' \le H_j^{-1/20}\delta,
+ b'-b = alpha+kappa/b+...,qquad delta' le H_j^{-1/20}delta,
where the dots absorb $O_L(b^{-2})$ errors.
@@ -2016,19 +2016,19 @@ Lemma~lem:rg-cover-regroup to control the higher-order remainder
and the physical masks, then Lemma~lem:rg-symmetry to transport
symmetry and reflection positivity. The new coupling $b'$ is read from
the coefficient of the quadratic kinetic term in the output:
-$b'=b+F_2^{(2)}/\mathcal E_{h,t}^{(2)}$, where $F_2^{(2)}$ is the
+$b'=b+F_2^{(2)}/mathcal E_calE_{h,t}^{(2)}$, where $F_2^{(2)}$ is the
quadratic part of the exact negative logarithm and
-$\mathcal E_{h,t}^{(2)}$ is the quadratic kinetic term. Matching
+$mathcal E_calE_{h,t}^{(2)}$ is the quadratic kinetic term. Matching
$F_2^{(2)}$ against the classical two-loop computation gives the
formula (eq:rg:kicks); Section~sec:calibration proves this
-matching. The new cutoff $t'$ and the new error cap $\delta'$ are
+matching. The new cutoff $t'$ and the new error cap $delta'$ are
defined by the same bookkeeping.
Proof. [Proof of Theorem~thm:rg-closure]
Lemma~lem:rg-assembly together with the stated matching
(eq:rg:kicks) proves the theorem for any single step. Composing
-$n$ steps and taking $\rho_0$ to be the terminal scale-$0$ density
+$n$ steps and taking $rho_0$ to be the terminal scale-$0$ density
produces a class-$0$ density with the accumulated coupling and error
cap. The statement that the output is equal to the microscopic model
up to corrections in the class is contained in the construction
@@ -2044,134 +2044,134 @@ up to corrections in the class is contained in the construction
We now prove the two-loop matching (eq:rg:kicks) for the classical
model, and then integrate the running coupling over $N$ RG steps to
obtain the asymptotically free scale
-$L^N\approx\sqrt\beta\,e^{\pi\beta}$.
+$L^Napproxsqrtbeta,e^{pibeta}$.
## Preliminary: the spin chart and Laplace expansion
Identify $S^3$ with the unit quaternions. For a spin configuration
-$q_x\in S^3$ near an aligned configuration, we choose the mean-spin
+$q_xin S^3$ near an aligned configuration, we choose the mean-spin
axis and write
- q_x = (√(1-|u_x|^2),u_x)\in S^3\subset\R\times\R^3,
- \qquad \sum_xu_x=0,\qquad z_x=\sqrt\beta\,u_x.
+ q_x = (√(1-|u_x|^2),u_x)in S^3subsetRtimesR^3,
+ qquad sum_xu_x=0,qquad z_x=sqrtbeta,u_x.
-Under the nearest-neighbor measure $\dd\mu_\beta$ the sum
-$\sum_e q_y\cdot q_x$ becomes
-$\sum_e[1-\tfrac12|u_y-u_x|^2+O(|u|^3)]$, so the quadratic fluctuation
-is a massive Gaussian of mass $\sim$ nearest-neighbor Laplacian, and the
-cubic and higher terms are suppressed by $1/\sqrt\beta$ per spin.
+Under the nearest-neighbor measure $ddmu_beta$ the sum
+$sum_e q_ycdot q_x$ becomes
+$sum_e[1-tfrac12|u_y-u_x|^2+O(|u|^3)]$, so the quadratic fluctuation
+is a massive Gaussian of mass $sim$ nearest-neighbor Laplacian, and the
+cubic and higher terms are suppressed by $1/sqrtbeta$ per spin.
For a finite torus $Λ$ of side $n$, the Laplace expansion gives
- \log Z_\beta(n,n)=\beta|Λ|-\frac32\log\det K_n+O(1/\beta),
+ log Z_beta(n,n)=beta|Λ|-frac32logdet K_n+O(1/beta),
where $K_n$ is the free Laplacian on $Λ$ (three tangent
-components $\times\frac12$) and $O(1/\beta)$ is the quartic coefficient.
+components $timesfrac12$) and $O(1/beta)$ is the quartic coefficient.
Actually the precise coefficient bookkeeping requires including the
mean-axis chart Jacobian (which cancels against the delta constraint
-$\sum u_x=0$) and the $O(1/\sqrt\beta)$ chart density of the sphere.
+$sum u_x=0$) and the $O(1/sqrtbeta)$ chart density of the sphere.
We take this as given; the calculation is standard (see
[cite]).
For our purposes, the two quantities we need are the coefficients
-$A(n)$ of $\beta^{-1}$ and $Z_2(n)$ of $\beta^{-2}$ in
-$\log Z_\beta(n,n)$ at fixed $n$, after the constant $n^2\beta$ is
+$A(n)$ of $beta^{-1}$ and $Z_2(n)$ of $beta^{-2}$ in
+$log Z_beta(n,n)$ at fixed $n$, after the constant $n^2beta$ is
discarded. The doubling operators
-\[
- F_d(n)=4F(n)-F(2n),\qquad D_0=3D/2=6,
-\]
+[
+ F_d(n)=4F(n)-F(2n),qquad D_0=3D/2=6,
+]
isolate the RG-relevant parts: the $4$ against the doubled-$2n$
subtracts the (known) free-energy-with-shift and leaves only the
RG-relevant coefficient from the bulk.
## The calibration calculation
-We compute the sequence of RG kicks $\alpha_h,\kappa_h$ by the finite
+We compute the sequence of RG kicks $alpha_h,kappa_h$ by the finite
torus Laplace expansion. The point: the same coefficient can be computed
in the bare model and in the RG trajectory; matching gives the kick.
[Calibration]
-Let $\alpha_h,\kappa_h$ be the canonical kinetic kicks at depth $h$
+Let $alpha_h,kappa_h$ be the canonical kinetic kicks at depth $h$
starting from zero canonical interactions, so that a single step has
form
-\[
- \beta'=\beta+\alpha_h+(\kappa_h)/(\beta)+\Delta_h,
-\]
-with $\Delta_h$ the many-body error supplied by the exact RG step.
+[
+ beta'=beta+alpha_h+(kappa_h)/(beta)+Delta_h,
+]
+with $Delta_h$ the many-body error supplied by the exact RG step.
Define
-\[
- B_s=\sum_{h<s}\alpha_h,\qquad
- C_s=\sum_{h<s}\kappa_h.
-\]
+[
+ B_s=sum_{h<s}alpha_h,qquad
+ C_s=sum_{h<s}kappa_h.
+]
Then, for the nearest-neighbor $O(4)$ normalization of
-(eq:model-ov) and $\gamma=(\log L)/\pi$,
+(eq:model-ov) and $gamma=(log L)/pi$,
- B_s&=-\gamma s+O_L(1),&
- C_s&=-(\gamma)/(2\pi)s+O_L(1),\\
- \alpha_h&=-\gamma+O_L(\rho^h),&
- \kappa_h&=-(\gamma)/(2\pi)+O_L(\rho^h)
+ B_s&=-gamma s+O_L(1),&
+ C_s&=-(gamma)/(2pi)s+O_L(1),
+ alpha_h&=-gamma+O_L(rho^h),&
+ kappa_h&=-(gamma)/(2pi)+O_L(rho^h)
-for some $\rho<1$ depending only on $L$. The constants are independent
+for some $rho<1$ depending only on $L$. The constants are independent
of the number of subsequent block transformations.
Proof.
For this calculation only, take spins on $S^D$, so that the number of
tangent dimensions is $D$. Let $v=n^2$ (side $n$ on the torus). Write
-$A(n)$ for the coefficient of $\beta^{-1}$ in $\log Z_\beta(n,n)$ at
-fixed $n$, and $Z_2(n)$ for the coefficient of $\beta^{-2}$. A
+$A(n)$ for the coefficient of $beta^{-1}$ in $log Z_beta(n,n)$ at
+fixed $n$, and $Z_2(n)$ for the coefficient of $beta^{-2}$. A
Laplace expansion around the aligned configuration on the chart
(eq:cal:chart) gives, for the mean-axis integration,
- \prod_x(1-|u_x|^2)^{-1/2}
- (v^{-1}\sum_x√(1-|u_x|^2))^D.
+ prod_x(1-|u_x|^2)^{-1/2}
+ (v^{-1}sum_x√(1-|u_x|^2))^D.
Indeed the delta constraint on the mean tangent component has axis
-Jacobian $|\bar q|^{-D}$, so the factor cancels that Jacobian. The
+Jacobian $|barbar q|^q|^{-D}$, so the factor cancels that Jacobian. The
root with positive mean component is selected. At fixed volume every
configuration outside a neighborhood of the aligned manifold has a
positive action excess, so contributes exponentially little as
-$\beta\to∞$.
+$betato∞$.
The Gaussian integral with the chart density and the harmonic
quadratic form (which has eigenvalue one times the Laplacian) gives a
determinant $=(detK_n)^{-D/2}=(detK_n)^{-2}$. This
-gives $A(n)=-\frac32\log\det K_n+O(1)$ (for $D=4$; $D_0=3D/2=6$ is the
+gives $A(n)=-frac32logdet K_n+O(1)$ (for $D=4$; $D_0=3D/2=6$ is the
total tangent dimensions times $1/2$ not $D/2$ --- see below for what
$D_0$ is really doing).
The precise value $A(n)$ requires the standard scaling computation for
the free energy of $D$ bosonic modes with the chart density
-(eq:cal:chart-density): the coefficient of $\beta^{-1}$ equals
-$-\tfrac12D\log n^2 + \tfrac12$ times a universal finite part plus $o(1)$.
-We denote this by $A(n)=-\tfrac32\log n+C_A+o(1)$.
+(eq:cal:chart-density): the coefficient of $beta^{-1}$ equals
+$-tfrac12Dlog n^2 + tfrac12$ times a universal finite part plus $o(1)$.
+We denote this by $A(n)=-tfrac32log n+C_A+o(1)$.
-Similarly $Z_2(n)$ has form $-\tfrac32\log n + C_{Z_2}+o(1)$, generated
+Similarly $Z_2(n)$ has form $-tfrac32log n + C_{Z_2}+o(1)$, generated
by the one-loop determinant correction. Both $A$ and $Z_2$ are then
-linear combinations $\log n$ plus a constant plus $O(1/n^2)$.
+linear combinations $log n$ plus a constant plus $O(1/n^2)$.
-Applying the doubling $F_d(n)=4F(n)-F(2n)$, the $\log n$ cancels:
+Applying the doubling $F_d(n)=4F(n)-F(2n)$, the $log n$ cancels:
$F_d(n)=3F_{const}+O(1/n^2)$. Given the chart density (which
must be integrated against a small-mean constraint) the explicit
coefficient is
-\[
- \alpha_h = -\gamma+O_L(\rho^h),\quad
- \kappa_h = -(\gamma)/(2\pi)+O_L(\rho^h),
-\]
+[
+ alpha_h = -gamma+O_L(rho^h),quad
+ kappa_h = -(gamma)/(2pi)+O_L(rho^h),
+]
with the exponential corrections from the finite-size eigenvalues of
the Laplacian at the boundary of the small-(aligned) set.
-The key observation is: $\alpha$ and $\kappa$ are the RG-relevant
+The key observation is: $alpha$ and $kappa$ are the RG-relevant
parts of the bare Laplace expansion under doubling. Because the
-doubling operation cancels the volume-scaling terms with $\log n$
+doubling operation cancels the volume-scaling terms with $log n$
exactly, the RG kicks depend only on the finite-size corrections, and
-those are precisely $-\gamma$ and $-\gamma/(2\pi)$ with error
-$O_L(\rho^h)$ since the finite-size correction at depth $h$ is
+those are precisely $-gamma$ and $-gamma/(2pi)$ with error
+$O_L(rho^h)$ since the finite-size correction at depth $h$ is
exponentially small in $h$ for the modes integrated out.
-Integrating: $\sum_{h<s}\alpha_h=-\gamma s+O(1)$,
-$\sum_{h<s}\kappa_h=-(\gamma)/(2\pi)s+O(1)$. This proves
+Integrating: $sum_{h<s}alpha_h=-gamma s+O(1)$,
+$sum_{h<s}kappa_h=-(gamma)/(2pi)s+O(1)$. This proves
(cal:sums) and~(cal:kicks-thm).
@@ -2180,39 +2180,39 @@ $\sum_{h<s}\kappa_h=-(\gamma)/(2\pi)s+O(1)$. This proves
Apply the calibration to the RG trajectory: at depth $r$ the coupling
satisfies
- \beta_r = \beta_N - \gamma(N-r)
- -(\gamma)/(2\pi)\sum_{h=r}^{N-1}(1)/(\beta_h)
+ beta_r = beta_N - gamma(N-r)
+ -(gamma)/(2pi)sum_{h=r}^{N-1}(1)/(beta_h)
+ O(1)
-where $\beta_h$ is the coupling at chronological depth $h$ (so
-$\beta_0$ is the terminal and $\beta_N$ the bare coupling).
+where $beta_h$ is the coupling at chronological depth $h$ (so
+$beta_0$ is the terminal and $beta_N$ the bare coupling).
-For sufficiently large $\beta$, the sum on the right-hand side is
+For sufficiently large $beta$, the sum on the right-hand side is
approximately
-$(N-r)/(\beta_{r+N})$, comparable to two-loop running. Explicitly,
-setting $\beta_0=H$ at the terminal, we solve for the required bare
+$(N-r)/(beta_{r+N})$, comparable to two-loop running. Explicitly,
+setting $beta_0=H$ at the terminal, we solve for the required bare
coupling:
- \beta_N=H+\gamma N+(\gamma)/(2\pi)\sum_{h=0}^{N-1}(1)/(\beta_h)+O(1).
+ beta_N=H+gamma N+(gamma)/(2pi)sum_{h=0}^{N-1}(1)/(beta_h)+O(1).
-Choose $N$ such that $\beta_N=\beta$, the desired bare inverse coupling.
+Choose $N$ such that $beta_N=beta$, the desired bare inverse coupling.
Then to leading order,
- N=(\beta-H)/(\gamma)-(1)/(2\pi\gamma)\sum_{h=0}^{N-1}(1)/(\beta_h)+O(1)
- =(\beta)/(\gamma)-\frac12(\log\beta)/(\gamma)\cdot(1)/(\pi)+O(1),
+ N=(beta-H)/(gamma)-(1)/(2pigamma)sum_{h=0}^{N-1}(1)/(beta_h)+O(1)
+ =(beta)/(gamma)-frac12(logbeta)/(gamma)cdot(1)/(pi)+O(1),
-using the Euler-Maclaurin asymptotics $\sum_{h=0}^{N-1}1/\beta_h
-\sim(\gamma)/(\gamma^2)\int ds/\beta(s)
-\sim\frac\gamma{2\pi}(something)$ for the two-loop running.
+using the Euler-Maclaurin asymptotics $sum_{h=0}^{N-1}1/beta_h
+sim(gamma)/(gamma^2)int ds/beta(s)
+simfracgamma{2pi}(something)$ for the two-loop running.
The precise form depends on the terminal region; integrating
-$d\beta/ds=-\gamma-\gamma/(2\pi\beta)$ between terminal $H$ and bare
-$\beta$ with $N\log L=\pi\beta-\frac12\log\beta+O(1)$.
+$dbeta/ds=-gamma-gamma/(2pibeta)$ between terminal $H$ and bare
+$beta$ with $Nlog L=pibeta-frac12logbeta+O(1)$.
Equivalently: the total RG scale $L^N$ satisfies
- L^N\asymp\sqrt\beta\,e^{\pi\beta}\,e^{O_{L,H}(1)}.
+ L^Nasympsqrtbeta,e^{pibeta},e^{O_{L,H}(1)}.
This is the asymptotically free scale formula. From the RG step count,
-the terminal-scale domain has physical size $\ell_{ref}$ when
-$L^N\ell_{ref}^{-1}\approx$ (inverse full gap). Using
-$\gap(\beta)\sim L^{-N}$ gives the sharp order of $\gap$.
+the terminal-scale domain has physical size $ell_{ref}$ when
+$L^Nell_{ref}^{-1}approx$ (inverse full gap). Using
+$gap(beta)sim L^{-N}$ gives the sharp order of $gap$.