Ising percolation in the hyperbolic plane

Abstract

We study infinite “+” or “−” clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph G with finite vertex degree. If the critical percolation probability pcsite for the independent identically distributed (IID). Bernoulli site percolation on G is less than 12, we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If pcsite>12, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of pcsite. We also obtain an explicit region for the coupling constant when the XOR Ising model (the product of two IID Ising models) does not have a unique infinite contour a.s. and an explicit region for the coupling constant when the XOR Ising model has infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters.