Sharp thresholds for the random-cluster and Ising models

TL;DR

This paper proves a sharp-threshold theorem for crossing probabilities in the square lattice for models like the random-cluster and Ising models, extending influence theorems without symmetry assumptions.

Contribution

It introduces a new sharp-threshold result for these models, applicable near critical points and with no symmetry constraints.

Findings

Established sharp thresholds for crossing probabilities

Extended influence theorem to non-symmetric measures

Applied results to models near criticality

Abstract

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point $p_{\mathrm {sd}}(q)=\sqrt{q}/(1+\sqrt{q})$, the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.