No phase transition for Gaussian fields with bounded spins

TL;DR

This paper proves the absence of phase transitions in Gaussian fields with bounded spins by demonstrating the uniqueness of the Gibbs measure, leveraging the attractiveness and uniqueness of the Gibbs sampler's invariant measure.

Contribution

It establishes the uniqueness of the Gibbs measure for a broad class of Gaussian fields with bounded spins, extending understanding of phase transition phenomena.

Findings

Unique Gibbs measure for the specified Gaussian field.

Gibbs sampler is attractive and has a unique invariant measure.

No phase transition occurs in this setting.

Abstract

Let a<b, \Omega=[a,b]^{\Z^d} and H be the (formal) Hamiltonian defined on \Omega by H(\eta) = \frac12 \sum_{x,y\in\Z^d} J(x-y) (\eta(x)-\eta(y))^2 where J:\Z^d\to\R is any summable non-negative symmetric function (J(x)\ge 0 for all x\in\Z^d, \sum_x J(x)<\infty and J(x)=J(-x)). We prove that there is a unique Gibbs measure on \Omega associated to H. The result is a consequence of the fact that the corresponding Gibbs sampler is attractive and has a unique invariant measure.