Exact winding-loop rate for the critical planar random-cluster model with $q>4$: proving equality in Duminil-Copin–Gagnebin–Harel–Manolescu–Tassion's bound (3.15) using only the random-cluster model

In their proof that the planar random-cluster and Potts models with cluster weight $q>4$ undergo a discontinuous phase transition, Duminil-Copin, Gagnebin, Harel, Manolescu and Tassion (DGHMT) related the inverse correlation length $\xi(q)^{-1}$ of the critical random-cluster model to the exponential decay rate, in the height $M$ of a large torus, of the probability that the loop representation winds vertically around the torus (the event $U(\omega)=1$). They proved by purely random-cluster arguments the lower bound (3.15) — this probability is at least $\exp(-M\xi(q)^{-1}(1+o(1)))$ — but could not prove the matching upper bound by random-cluster techniques, and instead obtained Baxter's exact formula $\xi(q)^{-1}=\lambda+2\sum_{k\ge1}\frac{(-1)^k}{k}\tanh(k\lambda)$, $\cosh\lambda=\sqrt q/2$, through a Bethe-ansatz analysis of all winding sectors of the six-vertex transfer matrix. Since the equality in (3.15) is itself a corollary of their Theorem 1.2, the open problem they pose is methodological: prove, using only probabilistic properties of the random-cluster model and not the six-vertex exact solution, that the minimal vertical-winding-loop probability has exact rate $\xi(q)^{-1}$ — equivalently, prove the reverse inequality $\limsup_{N}\limsup_{M}\frac1M\log\varphi[U(\omega)=1]\le-\xi(q)^{-1}$.

Background

The $q$-state Potts model assigns to each vertex of the square lattice $\mathbb Z^2$ a spin in $\{1,\dots,q\}$ and favors configurations in which neighboring spins agree; it is the standard lattice model of order–disorder transitions, with the Ising model as the case $q=2$. Its Fortuin–Kasteleyn (FK), or random-cluster, representation is a dependent percolation model: a configuration $\omega$ of open and closed edges of a finite graph has probability proportional to $p^{o(\omega)}(1-p)^{c(\omega)}q^{k(\omega)}$, where $o$ and $c$ count open and closed edges, $k$ counts connected components (clusters), and the cluster weight $q>0$ is a real parameter that need not be an integer. The Edwards–Sokal coupling recovers the Potts model by coloring each FK cluster with an independent uniform spin, so Potts two-point functions agree, up to $1/q$, with FK connection probabilities.

On the square lattice the random-cluster measure is self-dual, and the self-dual point $p_c=\sqrt q/(1+\sqrt q)$ is the critical point for every $q\ge1$ (Beffara–Duminil-Copin, 2012). From the exact solution of related ice-type models, Baxter predicted that the transition is continuous for $1\le q\le4$ and discontinuous (first-order: several distinct infinite-volume measures coexist at criticality) for $q>4$. The continuous half was proved by Duminil-Copin–Sidoravicius–Tassion (2017); the discontinuous half was proved by Duminil-Copin–Gagnebin–Harel–Manolescu–Tassion (2016; published 2021), called DGHMT below — the source of this problem.

Two further ingredients are needed. First, the loop representation: drawing, on the medial lattice (the graph of edge midpoints), the non-intersecting loops that separate open clusters from closed dual clusters turns a random-cluster configuration into a fully-packed loop configuration (the Temperley–Lieb representation). On a torus $\mathbb T^\diamond_{N,M}=(\mathbb Z/N\mathbb Z)\times(\mathbb Z/M\mathbb Z)$, viewed on the diagonal lattice, loops may be non-retractable, i.e. wind around the torus. DGHMT let $2U(\omega)$ denote the total number of vertical windings of all loops (necessarily an even number), so $U(\omega)=1$ is the minimal vertical-winding event, in which exactly two non-retractable loops wind once vertically around the torus. Second, the correlation length: on the rotated (diagonal) lattice $\mathbb L$, a copy of $\mathbb Z^2$ turned by $\pi/4$, it is defined by $\xi(q)^{-1}=\lim_{n\to\infty}-\frac{1}{2n}\log\varphi^0_{\mathbb L,p_c,q}[0\leftrightarrow(0,2n)]$, where $\varphi^0$ is the infinite-volume critical random-cluster measure with free boundary conditions.

DGHMT's proof for $q>4$ passes through the exact solution of the six-vertex model, an 'ice-type' model of edge orientations with two arrows entering and two leaving each vertex, related to the critical random-cluster model on the torus by the Temperley–Lieb/Baxter–Kelland–Wu correspondence. Winding sectors of the random-cluster loop configuration correspond to sectors of the six-vertex model with a fixed deficit of up arrows per row, and the sector partition functions are governed by the Perron–Frobenius eigenvalues $\Lambda_r(N)$ of diagonal blocks of the six-vertex transfer matrix, indexed by the winding number $r$; DGHMT prove, via the Bethe ansatz (the algebraic ansatz that constructs eigenvectors of exactly solvable transfer matrices), the asymptotics $\Lambda_r(N)/\Lambda_0(N)\to e^{-rR(q)}$ as $N\to\infty$ along multiples of 4, where $R(q)=\lambda+2\sum_{k\ge1}\frac{(-1)^k}{k}\tanh(k\lambda)$ and $\lambda>0$ solves $\cosh\lambda=\sqrt q/2$. Theorem 1.2 then follows by combining two purely probabilistic random-cluster lemmas with these sector asymptotics. Lemma 3.15, inequality (3.15): $\liminf_{N\to\infty}\liminf_{M\to\infty}\frac1M\log\varphi_{\mathbb T^\diamond_{N,M},p_c,q}[U(\omega)=1]\ge-\xi(q)^{-1}$ — the minimal vertical-winding event is not too rare. Lemma 3.15, inequality (3.16): $\limsup_{N\to\infty}\limsup_{M\to\infty}\frac1M\log\varphi_{\mathbb T^\diamond_{N,M},p_c,q}[U(\omega)\ge r]\le-(r-1)\xi(q)^{-1}$ for $r\ge1$. The first, combined with a partition-function sector comparison, yields $\xi(q)^{-1}\ge R(q)$; the second, with $r\to\infty$, yields $\xi(q)^{-1}\le R(q)$. As a byproduct, $\xi(q)^{-1}=R(q)$ exactly, with the correlation length diverging like $\exp(\pi^2/\sqrt{q-4})$ as $q\searrow4$ — a rigorous confirmation of Baxter's formula. The ratio $\Lambda_1/\Lambda_0$ is interpreted as twice the surface tension of the six-vertex model.

The crux left open sits in the first lemma. DGHMT's Remarks 3.16 and 3.18 state that (3.15) 'should actually be an equality' — i.e. the event $U(\omega)=1$ should have exact exponential rate $e^{-M\xi(q)^{-1}}$ in the torus height $M$ — but that 'we did not manage to derive the reverse inequality using the random-cluster model only'. Since their Theorem 1.2 pins $\xi(q)^{-1}=R(q)$, the equality is in fact known to hold; what remains open, and is explicitly posed by the authors, is a proof of it that does not pass through the exact solution: 'proving the equality in (3.15) using only the random-cluster model is an interesting question'. Such a proof would determine $\xi(q)^{-1}$ from the zero- and one-winding sectors ($\Lambda_0$ and $\Lambda_1$) alone, with no analysis of the higher sectors $\Lambda_r$, $r\ge2$. The difficulty is underscored by the state of soft methods: Ray–Spinka (2020) gave a short proof of the discontinuity itself, using the six-vertex connection but avoiding the Bethe ansatz, and noted explicitly that their argument 'does not directly yield any information' on the exponential rate of decay of correlations.

Problem Statement

Prove the equality in Duminil-Copin–Gagnebin–Harel–Manolescu–Tassion's winding-loop bound (3.15) using only the random-cluster model.

Precisely: fix a real $q>4$ and let $p_c=\sqrt q/(1+\sqrt q)$. Let $\varphi_{\mathbb T^\diamond_{N,M},p_c,q}$ be the random-cluster measure on the torus $\mathbb T^\diamond_{N,M}=(\mathbb Z/N\mathbb Z)\times(\mathbb Z/M\mathbb Z)$ viewed on the diagonal lattice, and for a configuration $\omega$ let $2U(\omega)$ be the total number of times the loops of the loop representation of $\omega$ wind vertically around the torus, so that $U(\omega)=1$ is the minimal vertical-winding event (exactly two non-retractable loops winding once vertically). Let $\xi(q)$ be the diagonal correlation length of the critical random-cluster model,

$$\xi(q)^{-1}=\lim_{n\to\infty}-\frac{1}{2n}\log\varphi^0_{\mathbb L,p_c,q}[0\leftrightarrow(0,2n)],$$

where $\mathbb L$ is the square lattice rotated by $\pi/4$ and $\varphi^0$ is the infinite-volume measure with free boundary conditions. DGHMT proved, by random-cluster arguments only, the lower bound

$$\liminf_{N\to\infty}\liminf_{M\to\infty}\frac1M\log\varphi_{\mathbb T^\diamond_{N,M},p_c,q}[U(\omega)=1]\ \ge\ -\xi(q)^{-1}.$$

The task is to prove the matching reverse inequality, by random-cluster arguments only,

$$\limsup_{N\to\infty}\limsup_{M\to\infty}\frac1M\log\varphi_{\mathbb T^\diamond_{N,M},p_c,q}[U(\omega)=1]\ \le\ -\xi(q)^{-1},$$

and hence that the limit exists and

$$\lim_{N\to\infty}\lim_{M\to\infty}\frac1M\log\varphi_{\mathbb T^\diamond_{N,M},p_c,q}[U(\omega)=1]\ =\ -\xi(q)^{-1}\ =\ -R(q),\qquad R(q)=\lambda+2\sum_{k\ge1}\frac{(-1)^k}{k}\tanh(k\lambda),\ \ \cosh\lambda=\sqrt q/2.$$

'Random-cluster arguments only' means that every input is a probabilistic property of the random-cluster measure — monotonicity and FKG (Fortuin–Kasteleyn–Ginibre) correlation inequalities, finite energy, comparison between boundary conditions, the domain Markov property, duality/self-duality, crossing-probability estimates of Russo–Seymour–Welsh (RSW) type, renormalization or subadditivity arguments — and in particular that the proof does not invoke the six-vertex model, its transfer matrix, or its exact solution by the Bethe ansatz.

Status clarification: the equality itself is not in doubt — it is a corollary of DGHMT's Theorem 1.2, which establishes $\xi(q)^{-1}=R(q)$ via the six-vertex winding-sector analysis $\Lambda_r$ with $r\to\infty$. The open content is the proof technique: a direct probabilistic derivation of the exact winding-loop rate. Per the source's Remark 3.18, such a proof would determine $\xi(q)^{-1}$ using nothing but the zero- and one-winding-sector asymptotics $\Lambda_0$ and $\Lambda_1$, requiring no control of $\Lambda_r$ for $r\ge2$.

References

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