The high temperature Ising model on the triangular lattice is a critical percolation model

TL;DR

This paper demonstrates that the high temperature Ising model on the triangular lattice exhibits a percolation phase transition at a critical point, supporting the conjecture that it belongs to the percolation universality class and relates to SLE$_6$ scaling limits.

Contribution

It generalizes the FK random-cluster representation by introducing a dependent percolation model with a tunable parameter, revealing critical behavior and phase transition properties.

Findings

Percolation phase transition at r=1/2 for β<β_c

Uniqueness of the infinite +1 cluster for r>1/2

Exponential decay of cluster sizes for r<1/2

Abstract

The Ising model at inverse temperature $\beta$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-\beta}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $\beta$, this generates a dependent (spin) percolation model with parameter $r$. We show that, on the triangular lattice and for $\beta<\beta_c$, this model has a percolation phase transition at $r=1/2$, corresponding to the Ising model. This sheds some light on the conjecture that the high temperature Ising model on the triangular lattice is in the percolation universality class and that its scaling limit can be described in terms of SLE$_6$. We also prove uniqueness of the infinite +1 cluster for $r>1/2$, sharpness of the percolation phase transition (by showing exponential decay of the cluster size distribution for $r<1/2$), and continuity of the percolation function for all $r \in [0,1]$.