Regularity of Gibbs measures for unbounded spin systems on general graphs

TL;DR

This paper studies Gibbs measures for unbounded spin systems on general graphs, establishing regularity and construction methods for the plus measure, with broad applicability beyond lattice models.

Contribution

It introduces new regularity estimates for Gibbs measures with unbounded spins on general graphs, extending previous results to more general boundary conditions and graph structures.

Findings

Constructed the plus measure as a limit of finite-volume Gibbs measures.

Proved regularity and tightness of finite-volume Gibbs measures under various boundary conditions.

Developed a measure change control function analogous to Cameron-Martin theorem for non-Gaussian fields.

Abstract

We consider a general class of spin systems with potentially unbounded real-valued spins, defined via a single-site potential with super-Gaussian tails on general graphs, allowing for both short- and long-range interactions. This class includes all $P(\varphi)$ models, in particular the well-studied $\varphi^4$ model. We construct an infinite-volume extremal measure called the plus measure as the limit of finite-volume Gibbs measures with weakly growing boundary conditions and show that it is regular, in the sense that it admits a bounded Radon-Nikodym derivative with respect to a product measure of single-site distributions with super-Gaussian tails. Moreover, we provide an alternative construction of the plus measure as the limit of finite-volume Gibbs measures that are regular up to the boundary. As a key intermediate step, we establish regularity and tightness of finite-volume Gibbs measures for a large class of growing boundary conditions $\xi$. Our regularity estimates are encoded in terms of a function $A(\xi)$, which provides precise control on the change of measure induced by boundary perturbations, and can thus be viewed as an analogue of the Cameron-Martin theorem for non-Gaussian fields. In the nearest-neighbour case, this class includes boundary conditions that grow at most double-exponentially in the distance to the boundary when the single-site measure has tails of the form $e^{-a|u|^n}$ for some $n>2$.Our results apply to arbitrary graphs and improve upon earlier results of Lebowitz and Presutti, and Ruelle, which apply in the context of $\mathbb{Z}^d$ and allow only logarithmically growing boundary conditions, as well as subsequent extensions to vertex-transitive graphs of polynomial growth.