2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe chessboard density κ_cb, simultaneous flips, two uniformities · Kiichi, Shiori and Rin

The chessboard and simultaneous flips

The notation is as in 13-01 §01. The sector bias $\theta_D(\Phi)=|\mathbb E[\sigma_D\mid\Phi]|$ of a container $D\times T^2_L$ is a function of the frame $\Phi$, and if mutually disjoint containers $D_j$ in the plane lie inside the loop $C$, then

(Mack–Petkova Cited). The right-hand side is the mean of a product of factors sharing the same frame. If the factors were independent it would split into $\prod_j\mathbb E\theta_{D_j}$, but the frame field is connected between containers, so it does not. If we use the Hölder inequality, $\mathbb E\prod_{j=1}^n\theta_j\le\prod_j(\mathbb E\theta_j^n)^{1/n}$, each factor on the right is the $n$-th root of the mean of the $n$-th power of $\theta$; this is not a product of typical values $\theta$, and a power proportional to the area is lost (in the nested form only $\mathbb E\theta_{2R}^4\le\mathbb E\theta_R^4$ comes out).

So 13-02 §02 set frame decorrelation,

as an assumption Assumption. (2) played two roles: the power of the nesting (steps 1 and 2 of Theorem 1 of 13-02), and the bridge from the bias to the Wilson loop (Proposition 2 of 13-01). This chapter moves the role of splitting the product, on a slab with $L$ fixed, to a single translation-invariant quantity.

Setting. Consider the z ensemble with the twist $n_{12}=1$ fixed ((4) of 13-04) on the lattice $(\mathbb Z/L)^2\times(\mathbb Z/M)^2$. Divide the plane (3,4) into square blocks $Q_t$ of side $B$ ($t\in\mathcal T$, including the boundary lattice points), and let $M$ be a multiple of $2B$. Inside each block put one container $D_t\times T^2_L$, at positions carried into one another by reflections.

Theorem 1 (the chessboard estimate) Cited Suppose the measure is reflection positive for reflections in planar directions 3 and 4 through all the lattice planes on the boundaries of the blocks. For each $t$ let $f_t$ be a bounded real function depending only on the variables of the block $Q_t$ (boundary included), and let $f_t^{(s)}$ be the function obtained by carrying $f_t$ to the block $Q_s$ by a composition of reflections. Then

This is the standard argument of Fröhlich–Israel–Lieb–Simon, obtained by using the reflection Cauchy–Schwarz form repeatedly (13-04 §03; Lean's ReflPos (ReflPos.refl_cs) is the algebra after splitting into those factors). (The original paper is Fröhlich–Israel–Lieb–Simon (1978). In this book we checked from the abstract of M. Biskup's lecture notes, arXiv:math-ph/0610025, that this is the standard form of obtaining the chessboard estimate from reflection positivity.)

In the z ensemble, reflection positivity is given by Theorem 3 of 13-04. Theorem 3 speaks of reflections through $x_3=0$ and $x_3=M/2$, but the measure is invariant under translations in direction 3, so the same holds for reflections through any lattice plane $x_3=c$. Direction 4 is the same.

Next we check that the bias is a function of the block. Given the frame, the interiors of the containers $D_t\times T^2_L$ are conditionally independent, and the conditional distribution depends only on the variables of the shell just outside the container (the Markov property). Since the container lies inside the block, the shell lies in the block, and $\theta_t:=\theta_{D_t}$ is a function of the variables of $Q_t$. Also, a reflection preserves the measure and carries the sector $\sigma_D=\prod_{p\in D\times\{y_0\}}z_p$ to the sector of the reflected container, so the function obtained by carrying $\theta_t$ to the block $Q_s$ by reflection is $\theta_s$.

Corollary 2 (the chessboard with $L$ fixed) Proved here Fix $L$ and $M$. In the setting above, put

Then for every subset $S\subset\mathcal T$, $\mathbb E\prod_{t\in S}\theta_t\le\kappa_{\rm cb}^{\,|S|}$. In particular, if $n(C)$ is the number of blocks whose container lies wholly inside the loop $C$,

Proof In (3) put $f_t=\theta_t$ if $t\in S$ and $f_t=1$ if $t\notin S$. Since $\theta_t\ge0$, the left-hand side is $\mathbb E\prod_{t\in S}\theta_t$. On the right, the factors with $t\in S$ are $\bigl(\mathbb E\prod_{s\in\mathcal T}\theta_s\bigr)^{1/|\mathcal T|}=\kappa_{\rm cb}$, because $\theta_t^{(s)}=\theta_s$ as seen above, and the factors with $t\notin S$ are $1$. Hence $\mathbb E\prod_{t\in S}\theta_t\le\kappa_{\rm cb}^{\,|S|}$. For (5), take $D_j$ in (1) to be the containers inside $C$ and use this estimate. ∎

Since a block of side $B$ occupies the area $B^2$, (5) gives the lower bound $\ln(1/\kappa_{\rm cb})/B^2$ on the tension (for square loops whose side is a multiple of $B$). With $L$ fixed, z reflection positivity has closed, so the gap to the area law on the slab has become only the estimate $\kappa_{\rm cb}\lt1$ Proved here (a statement with $L$ and $M$ fixed). No assumption about the distribution of the frame, such as frame decorrelation (2), is needed.

$\kappa_{\rm cb}$ is the geometric mean per block of the product of the biases when containers are placed in all blocks at once. It is not "the bias of one typical container" but "a quantity that looks at all containers at once", so the correlation of the frames is absorbed into $\kappa_{\rm cb}$. In exchange, the smallness of $\kappa_{\rm cb}$ can no longer be read as a property of one container.

(i) The slab estimate. For each face $p\in D$ of the container, the operations that flip the $z$ on the face $\{p\}\times T^2_L$ commute with one another, each flips the sector, and none changes the frame. Conditioning on their orbit gives $\mathbb E[\sigma_D\mid\text{orbit}]=\pm\prod_{p\in D}\tanh X_p$, and since $|X_p|\le\beta L^2$, $\sup_\Phi\theta_D(\Phi)\le\tanh(\beta L^2)^{|D|}$ (12-04 Proved here). From $\theta_t\le\sup\theta$,

So the area law on a slab with $L$ fixed already follows. But the tension is $-\ln\tanh(\beta L^2)\approx2e^{-2\beta L^2}$ (per unit area), which for $\beta=2.5$, $L=4$ is $2e^{-80}$. An estimate that goes through the supremum (sup) over frames gets worse superexponentially in $L$, so an estimate uniform in $L$ can only come from a mean form — a quantity like $\kappa_{\rm cb}$.

(ii) Simultaneous flips. The flow lemma (12-06) says that for any Markov kernel $K$ that preserves the conditional distribution $\mu(\cdot\mid\Phi)$ of the interior of the container, if $a(\Phi)$ is the probability that the sector changes in one step, then $\theta_D(\Phi)\le1-a(\Phi)$ Proved here (it uses only stationarity; reversibility is not needed).

Corollary 3 (simultaneous flips) Proved here With the frame held fixed, run a stationary kernel independently for one step inside each of the mutually disjoint containers $D_t$ ($t\in S$), starting from the conditional distribution. Then

Proof Fix the frame $\Phi$. The interiors of the containers are conditionally independent and the kernels also run independently, so the conditional probability that no container changes is $\prod_{t\in S}(1-a_t(\Phi))$. From the flow lemma, $\theta_t(\Phi)\le1-a_t(\Phi)$, so $\prod_t\theta_t(\Phi)\le\mathbb P(\text{none changes}\mid\Phi)$. Averaging over $\Phi$ gives (7). ∎

The right-hand side of (7) is a quantity that can be measured by Monte Carlo, so with $S=\mathcal T$, $\kappa_{\rm cb}\le\mathbb P(\text{none changes})^{1/|\mathcal T|}$ becomes a tool that estimates the upper bound on the Wilson loop in the correct direction. The natural kernel is the dynamically driven insertion (12-06, acceptance probability $\min(1,e^{-W})$).

(iii) The limits. In practice the flip probability $a$ of a one-step kernel levels off at about $\tfrac12$ (measurements with the driven-insertion kernel, 12-08 Reading). So what (7) can guarantee is of the form $\theta_t\lesssim\theta_0\sim\tfrac12$, and $\kappa_{\rm cb}$ too can only be pushed down to about $\tfrac12$. Even if this is uniform in $L$ for blocks of size $B\sim L$, the lower bound on the tension that comes out is $\ln(1/\theta_0)/L^2\to0$, which is only the naive form of Proposition 1 of 13-01 — a form that U(1) also satisfies Proved here (logic). Simultaneous flips cannot be used to test the band. To check the correct form $\theta\le e^{-cL^2}$ one needs a kernel with $a\to1$, and a one-step kernel cannot make one. Treating the measurements of simultaneous flips as "the numerical face of the band" has been withdrawn (12-08); the correct use of this tool is the lower bound on the tension of a slab with $L$ fixed — the estimation of $\kappa_{\rm cb}$.

Corollary 2 was a statement with $L$ fixed. When $L$ moves, the block size $B$ needed to show $\kappa_{\rm cb}\lt1$, and the quality of the estimate, change from scale to scale. Everything below is an estimate Reading (physics).

• The dilute regime ($\Lambda L\ll1$). For a container of side $\ell$, $\mathbb E[-\ln\theta_D]\approx\sigma_{\rm TU}\ell^2$, with $\sigma_{\rm TU}\propto\Lambda^2(\Lambda L)^{5/3}$ (13-03 §04). The influence of the frame decays over distances of about $L/\pi$. The criterion holds, but the blocks needed have $B\gtrsim\sigma_{\rm TU}^{-1/2}$, and their volume grows exponentially in $1/g^2(L)$.
• The confining regime ($\Lambda L\gg1$). In the form of the electric flux (Lemma 1 of 13-04), $\theta\approx Y(1)/Y(0)\approx c\,\sigma L^2e^{-\sigma\ell^2}$ (the factor $L^2$ is the entropy of the transverse position of the electric string). The block size needed grows only as $\sqrt{\ln L}$.
• In between ($\Lambda L\sim1$) — the band. Two uniformities are needed (§05).

As $a\to0$, both $B$ and $L$ become infinite in lattice units, so checking finitely many boxes is not enough. An inequality that connects the scales is needed, and that is the problem of the band.

Why the usual Dobrushin-type criteria cannot be used. With the frame fixed, the sectors do not interact with one another. All the dependence is in the frame field, and since a continuous field follows the boundary, the influence coefficient measured in total variation is nearly 1. Conditions of the Dobrushin, Dobrushin–Shlosman and van den Berg–Maes type are not satisfied as they stand, and we expect another input to be needed, such as contraction on good blocks (of the type of Bałaban's multiscale analysis) Reading (physics). The original papers of these criteria have not been checked in this book.

In the language of the chessboard, the band is concentrated into the following two statements of uniformity. Neither has been shown, and we set them as assumptions.

The uniform slab tension Assumption The string tension of the two-dimensional theory obtained by shrinking the transverse directions to $T^2_L$ (the slab tension, (11) of 13-02) is positive uniformly in $L$:

In the language of containers it has the form $\inf_L\mathbb E[-\ln\theta_D]/(\ell\Lambda)^2\gt0$ and corresponds to the content of the weak form (C′) of 13-02. The chessboard criterion only carries it to the area law. For U(1) the tension is $\sim e^2/L^2$ for every $L$, and it is expected to fail here only Reading (physics).

The uniform slab gap Assumption That $\kappa_{\rm cb}$ can be compared across scales: that the estimates of the correlations between biases, of the probability of bad blocks and of the fluctuations of the shells are uniform in $L$ (this corresponds to the mass gap of the slab being uniform in $L$). This is what frame decorrelation (2) of 13-02 really is when made uniform in $L$.

In this way the non-abelian content of the band is placed on the side of the uniform slab tension, and the probabilistic content of crossing scales on the side of the uniform slab gap. The logical relation between the two statements (whether one follows from the other) has not been examined.

This book has used $\kappa$ in several senses. The chessboard density of this chapter is written $\kappa_{\rm cb}$, and the subscript is never dropped. Table 2 lists the other quantities written with the same letter. The quantities are not related to one another.

The whole correspondence of symbols is in A-01.

The dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] CIT MP     container estimate |W| ≤ E∏θ                          ensemble summed over twists
[02] ASM DEC    frame decorrelation (replaced here at fixed L)
[03] PRV RP     z reflection positivity (13-04 Theorem 3)             Lean refl_pos, refl_cs (algebra)
[04] CIT CB     chessboard estimate                                   Fröhlich–Israel–Lieb–Simon
[05] PRV LOC    θ_t is a block function, reflected to θ_s             Markov property
[06] PRV KCB    Cor. 2: |W| ≤ κ_cb^n(C) ⟸ [01][03][04][05]            if L is fixed
[07] PRV SLAB   κ_cb ≤ tanh(βL²)^|D| < 1                              L fixed, tension tiny
[08] PRV FLUX   θ ≤ 1 − a                                             Lean flip_le_one_sub_abs
[09] PRV SIM    Cor. 3: E∏θ ≤ P(none changes) ⟸ [08]
[10] RDG HALF   one-step flip probability levels off near ½           measurement
[11] PRV NOTEST simultaneous flips cannot test the band ⟸ [09][10]    13-01 Proposition 1
[12] RDG ENDS   criteria and block sizes in the two end regimes       estimate
[13] RDG DOB    Dobrushin type not usable as it stands                estimate
[14] ASM UST    the uniform slab tension
[15] ASM UGAP   the uniform slab gap
[16] RDG U1     U(1) fails only in the uniform slab tension ⟸ [14][15]  estimate

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. The main conclusion of this chapter, [06], stands on paper, and its only premises are the literature rows [01] and [04] and the paper rows [03] and [05]. The assumption row [02] is not among the premises of [06] — that is the point of this chapter. [11] is a row of logic, but it has the measurement row [10] as a premise, so it is a statement within the range where the levelling off of [10] holds. The two assumptions of the band, [14] and [15], do not contain the quantity of the conclusion ($\kappa_{\rm cb}$ with $L$ fixed), and are placed in the ledger of Part 14 (14-02).

What this chapter can say

Sources and reproduction

Terms used in this chapter — frame decorrelation, the chessboard estimate, the chessboard density $\kappa_{\rm cb}$, block, simultaneous flips, the flow lemma, the uniform slab tension, the uniform slab gap, the crossover band (the band). Their meanings are in A-02 Glossary and A-06 Index of assumptions.

Next chapter: 13-06 Renormalisation comparison — a comparison that distinguishes groups — another reduction, which splits the band into β-monotonicity and the renormalisation comparison.

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