Jiang and Lang (arXiv:2503.21147, to appear in the Annals of Probability) proved that in the Ising model on T×Z\mathbb{T}\times\mathbb{Z} and on the three-dimensional triangular lattice, where T\mathbb{T} is the two-dimensional triangular lattice, a unique infinite ++ cluster and a unique infinite −- cluster coexist at every subcritical inverse temperature β∈[0,βc)\beta\in[0,\beta_c) provided the external field is small enough, ∣h∣<hc(β)|h|<h_c(\beta). They posed as an explicitly open conjecture the same coexistence statement for the two-dimensional square matching lattice Z∗2\mathbb{Z}_*^2, the square lattice with both diagonals of every unit square added, with ferromagnetic Ising interactions on all bonds. On Z∗2\mathbb{Z}_*^2 coexistence is known only for small β\beta (an easy consequence of Higuchi's results for the model without diagonal interactions); the conjecture asks for the full range β∈[0,βc(Z∗2))\beta\in[0,\beta_c(\mathbb{Z}_*^2)) with a nontrivial field window. As of August 2026, no proof or refutation of the conjecture appears in the literature: it remains open.

Background

The Ising model is the classical lattice model of ferromagnetism. To each vertex xx of a graph one attaches a spin σx∈{−1,+1}\sigma_x\in\{-1,+1\}; at inverse temperature β≥0\beta\ge 0 and external field hh, the Gibbs measure of a finite subgraph weights a configuration σ\sigma proportionally to exp⁡(β∑xyσxσy+h∑xσx)\exp(\beta\sum_{xy}\sigma_x\sigma_y+h\sum_x\sigma_x), the first sum running over the edges of the graph; note that the external field is not rescaled by β\beta (this is Equation (1) of the source). On the square lattice Z2\mathbb{Z}^2 (nearest-neighbor edges only) the infinite-volume model has a critical inverse temperature βc(Z2)∈(0,∞)\beta_c(\mathbb{Z}^2)\in(0,\infty): for β>βc(Z2)\beta>\beta_c(\mathbb{Z}^2) and h=0h=0 the system is magnetized, while for β<βc(Z2)\beta<\beta_c(\mathbb{Z}^2) it is disordered.

A geometric question, independent of magnetization, is percolation of spin clusters: in an infinite-volume spin configuration, the ++ clusters and −- clusters are the maximal connected sets of equal sign (connectivity following the graph's edges), and one asks whether an infinite cluster of a given sign exists. For translation-invariant measures with finite energy — every spin takes each value with probability bounded away from 00 on every finite set, which the Ising model satisfies — an infinite cluster, when it exists, is unique (Burton and Keane, 1989).

In two dimensions at zero field the percolation picture is sharp, and it involves a second graph. The matching lattice (or star lattice) Z∗2\mathbb{Z}_*^2 of Z2\mathbb{Z}^2 is obtained by adding, to the nearest-neighbor edges, both diagonals of every unit square, so that same-sign spins two diagonal steps apart are declared adjacent; clusters read with this connectivity are called star clusters. Coniglio, Nappi, Peruggi and Russo (1976) predicted, and Higuchi (1982, 1993) proved, that for the Ising model on Z2\mathbb{Z}^2 at h=0h=0 and every β∈[0,βc(Z2))\beta\in[0,\beta_c(\mathbb{Z}^2)) no infinite ++ or −- cluster exists with nearest-neighbor connectivity, yet an infinite ++ star cluster and an infinite −- star cluster coexist. (The 1982 paper establishes the Bernoulli-site-percolation case: at occupation density 1/21/2 on Z2\mathbb{Z}^2, the star clusters of both signs are infinite; in particular the site-percolation threshold of Z∗2\mathbb{Z}_*^2, i.e. the occupation density above which an infinite occupied cluster first appears, lies strictly below 1/21/2.) Crucially, in these results the diagonal bonds of Z∗2\mathbb{Z}_*^2 serve for connectivity only; they carry no interaction. On the triangular lattice T\mathbb{T} — which, unlike Z2\mathbb{Z}^2, is its own matching graph and has site-percolation threshold exactly 1/21/2 — the behavior is the opposite: Bálint, Camia and Meester (2010) proved that for β<βc(T)\beta<\beta_c(\mathbb{T}) the model sits exactly at the percolation threshold of its natural two-parameter generalization (a percolation transition at spin density 1/21/2), and consequently at h=0h=0 there is no infinite cluster of either sign.

In three dimensions both signs can percolate with genuine interactions on every bond. Campanino and Russo (1985) proved that on the cubic lattice Z3\mathbb{Z}^3 an infinite ++ cluster and an infinite −- cluster coexist when β\beta is small and ∣h∣|h| is small; Aizenman, Bricmont and Lebowitz (1987) showed that in sufficiently high dimension the minority spins percolate even inside the opposite phase at temperatures slightly below critical. Jiang and Lang (2025) pushed the three-dimensional coexistence all the way to criticality for two lattices: for the Ising model on T×Z\mathbb{T}\times\mathbb{Z} (the Cartesian product of the triangular lattice with the line) and on the three-dimensional triangular lattice, for every β∈[0,βc)\beta\in[0,\beta_c) there exists hc(β)>0h_c(\beta)>0 such that a unique infinite ++ cluster and a unique infinite −- cluster coexist whenever ∣h∣<hc(β)|h|<h_c(\beta). Their proof introduces an auxiliary percolation parameter pp (in the T×Z\mathbb{T}\times\mathbb{Z} argument, the probability that a given vertex can be used for an increasing event, realized through an independent Bernoulli site percolation; in the Z∗2\mathbb{Z}_*^2 sketch of Remark 4, the probability with which each added diagonal edge is selected) and establishes a differential inequality — a uniform-in-volume bound on the ratio of the hh- and pp-partial derivatives of the probability of increasing events (events closed under flipping −- spins to ++) — from which coexistence follows.

Jiang and Lang call Higuchi's two-dimensional coexistence result "somewhat artificial", since its diagonal bonds carry no interaction, and raise the natural question for the fully interacting model: the Ising model on Z∗2\mathbb{Z}_*^2 with ferromagnetic interactions on all nearest-neighbor and diagonal bonds. They record that coexistence is easy to derive from Higuchi's results when β\beta is small, conjecture that it persists for all β∈[0,βc(Z∗2))\beta\in[0,\beta_c(\mathbb{Z}_*^2)) with the same small-field window, and note that their method "should be applicable" because Z∗2\mathbb{Z}_*^2 is the triangular lattice together with one additional family of diagonal edges. No proof is given, and none has appeared since.

Problem Statement

The problem is the conjecture raised in Remark 4 of Jiang and Lang (arXiv:2503.21147). Consider the ferromagnetic Ising model on the square matching lattice Z∗2\mathbb{Z}_*^2 — the graph with vertex set Z2\mathbb{Z}^2 whose edges join every nearest-neighbor pair as well as both diagonal pairs of each unit square — with a uniform ferromagnetic coupling on every edge, at inverse temperature β≥0\beta\ge 0 and external field hh, with Gibbs weights exp⁡(β∑xyσxσy+h∑xσx)\exp(\beta\sum_{xy}\sigma_x\sigma_y+h\sum_x\sigma_x) in finite volume (the field term is not multiplied by β\beta, following Equation (1) of the source) and the corresponding infinite-volume Gibbs state; let βc(Z∗2)\beta_c(\mathbb{Z}_*^2) denote the critical inverse temperature of this model at h=0h=0. Prove, or refute, the following statement: for every β∈[0,βc(Z∗2))\beta\in[0,\beta_c(\mathbb{Z}_*^2)) there exists hc(β)>0h_c(\beta)>0 such that whenever ∣h∣<hc(β)|h|<h_c(\beta), the infinite-volume state contains almost surely a unique infinite cluster of ++ spins and a unique infinite cluster of −- spins. The already-known small-β\beta regime, which follows from Higuchi's results, is part of the claim; the open content is the full range of subcritical inverse temperatures up to βc(Z∗2)\beta_c(\mathbb{Z}_*^2), including the zero-field slice h=0h=0 for intermediate β\beta.

References

  1. Jianping Jiang and Sike Lang, 'Percolation of both signs in a triangular-type 3D Ising model above TcT_c', arXiv:2503.21147 (v1 March 2025, v2 December 2025; to appear in the Annals of Probability), DOI: 10.48550/arXiv.2503.21147, https://arxiv.org/abs/2503.21147
  2. Yasunari Higuchi, 'Coexistence of the infinite (*) clusters: a remark on the square lattice site percolation', Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 61 (1982), 75–81, DOI: 10.1007/BF00537226, https://doi.org/10.1007/BF00537226
  3. Yasunari Higuchi, 'Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions', Probability Theory and Related Fields 97 (1993), 1–33, DOI: 10.1007/BF01199310, https://doi.org/10.1007/BF01199310
  4. Antonio Coniglio, Chiara Rosanna Nappi, Fulvio Peruggi and Lucio Russo, 'Percolation and phase transitions in the Ising model', Communications in Mathematical Physics 51 (1976), 315–323, DOI: 10.1007/BF01617925, https://doi.org/10.1007/BF01617925
  5. András Bálint, Federico Camia and Ronald Meester, 'The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model', Journal of Statistical Physics 139 (2010), 122–138, DOI: 10.1007/s10955-010-9930-y, https://doi.org/10.1007/s10955-010-9930-y (preprint arXiv:0806.3020, https://arxiv.org/abs/0806.3020)
  6. M. Campanino and L. Russo, 'An Upper Bound on the Critical Percolation Probability for the Three-Dimensional Cubic Lattice', The Annals of Probability 13 (1985), 478–491, DOI: 10.1214/aop/1176993004, https://doi.org/10.1214/aop/1176993004
  7. M. Aizenman, J. Bricmont and J. L. Lebowitz, 'Percolation of the minority spins in high-dimensional Ising models', Journal of Statistical Physics 49 (1987), 859–865, DOI: 10.1007/BF01009363, https://doi.org/10.1007/BF01009363
  8. R. M. Burton and M. Keane, 'Density and uniqueness in percolation', Communications in Mathematical Physics 121 (1989), 501–505, DOI: 10.1007/BF01217735, https://doi.org/10.1007/BF01217735