Separated-occurrence inequalities for dependent percolation and Ising models

TL;DR

This paper establishes separated-occurrence inequalities for dependent lattice models like FK and Ising models, quantifying how the probability of events occurring at large separation decays exponentially.

Contribution

It extends separated-occurrence inequalities to certain dependent models, including subcritical FK and specific supercritical Ising models with external fields.

Findings

Inequalities hold for subcritical FK models.

Inequalities hold for certain supercritical Ising models.

Exponential decay of joint event probabilities at large separation.

Abstract

Separated-occurrence inequalities are variants for dependent lattice models of the van den Berg-Kesten inequality for independent models. They take the form $P(A \circ_r B) \leq (1 + ce^{-\epsilon r})P(A)P(B)$, where $A \circ_r B$ is the event that $A$ and $B$ occur at separation $r$ in a configuration $\omega$, that is, there exist two random sets of bonds or sites separated by at least distance $r$, one set responsible for the occurrence of the event $A$ in $\omega$, the other for the occurrence of $B$. We establish such inequalities for certain subcritical FK models, and for certain Ising models which are at supercritical temperature or have an external field, with $A$ and $B$ increasing or decreasing events.