Ising Percolation on Nonamenable Planar Graphs

TL;DR

This paper investigates the percolation properties of the Ising model on non-amenable planar graphs, identifying conditions for the coexistence of infinite clusters of opposite spins and providing bounds related to percolation thresholds.

Contribution

It provides explicit conditions for the existence of multiple infinite clusters in the Ising model on non-amenable graphs, linking percolation thresholds to cluster behavior.

Findings

Existence of regions with infinitely many infinite + and - clusters under certain conditions.

No infinite 1-clusters in the random cluster representation within these regions.

Lower bounds for critical probabilities in the random cluster model based on site percolation thresholds.

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 $p_c^{site}$ for the i.i.d.~Bernoulli site percolation on $G$ is less than $\frac{1}{2}$, 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 $p_c^{site}>\frac{1}{2}$, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of $p_c^{site}$.