Sharpness of the Phase Transition for the Corrupted Compass Model on Transitive Graphs

TL;DR

This paper proves that the phase transition in the corrupted compass model on transitive graphs is sharp, with finite clusters exponentially small below the critical point and a mean-field lower bound above it, using the OSSS inequality.

Contribution

It establishes the sharpness of the phase transition in the corrupted compass model and demonstrates the application of the OSSS inequality in this context.

Findings

Subcritical clusters are exponentially small.

Existence of a mean-field lower bound in the supercritical regime.

Sharp phase transition confirmed for the model.

Abstract

In the corrupted compass model on a vertex-transitive graph, a neighbouring edge of every vertex is chosen uniformly at random and opened. Additionally, with probability $p$, independently for every vertex, every neighbouring edge is opened. We study the size of open clusters in this model. Hirsch et al. have shown that for small $p$ all open clusters are finite almost surely, while for large $p$, depending on the underlying graph, there exists an infinite open cluster almost surely. We show that the corresponding phase transition is sharp, i.e., in the subcritical regime, all open clusters are exponentially small. Furthermore we prove a mean-field lower bound in the supercritical regime. The proof uses the by now well established method using the OSSS inequality. A second goal of this note is to showcase this method in an uncomplicated setting.