Random-cluster correlation inequalities for Gibbs fields

TL;DR

This paper establishes a correlation inequality for Gibbs fields using a random cluster representation, linking percolation properties to the uniqueness of Gibbs measures and potentially addressing the critical temperature in spin glasses.

Contribution

It introduces a novel correlation inequality for Gibbs fields based on hyperbond connectivity, connecting percolation to Gibbs measure uniqueness.

Findings

Absence of blue bond percolation implies Gibbs measure uniqueness.

In 2D, this approach may help prove the critical temperature is zero.

Provides a new tool for analyzing phase transitions in spin glasses.

Abstract

In this note we prove a correlation inequality for local variables of a Gibbs field based on the connectivity by active hyperbonds in a random cluster representation of the non overlap configuration distribution of two independent copies of the field. As a consequence, we show that absence of Machta-Newman-Stein blue bonds percolation implies uniqueness of Gibbs distribution in EA Spin Glasses. In dimension two this could constitute a step towards a proof that the critical temperature is zero.