Percolation of both signs in a triangular-type 3D Ising model above $T_c$

Abstract

Let $\mathbb{T}$ be the two-dimensional triangular lattice, and $\mathbb{Z}$ the one-dimensional integer lattice. Let $\mathbb{T}\times \mathbb{Z}$ denote the Cartesian product graph. Consider the Ising model defined on this graph with inverse temperature $\beta$ and external field $h$, and let $\beta_c$ be the critical inverse temperature when $h=0$. We prove that for each $\beta\in[0,\beta_c)$, there exists $h_c(\beta)>0$ such that both a unique infinite $+$cluster and a unique infinite $-$cluster coexist whenever $|h|<h_c(\beta)$. The same coexistence result also holds for the three-dimensional triangular lattice.