Uniqueness of Gibbs fields with unbounded random interactions on unbounded degree graphs

TL;DR

This paper proves the uniqueness of Gibbs fields with unbounded, random interactions on graphs with unbounded degrees, under high-temperature conditions and specific growth and distribution constraints.

Contribution

It establishes high-temperature uniqueness for Gibbs fields with unbounded, random interactions on graphs with unbounded degrees, extending previous results to more general settings.

Findings

High-temperature uniqueness is proved under tempered degree growth.

Interaction potentials are independent, identically distributed, and exponentially integrable.

Results apply to graphs with unbounded vertex degree and continuous spins.

Abstract

Gibbs fields with continuous spins are studied, the underlying graphs of which can be of unbounded vertex degree and the spin-spin pair interaction potentials are random and unbounded. A high-temperature uniqueness of such fields is proved to hold under the following conditions: (a) the vertex degree is of tempered growth, i.e., controlled in a certain way; (b) the interaction potentials $W_{xy}$ are such that $\|W_{xy}\|=\sup_{\sigma,\sigma'} |W_{xy}(\sigma, \sigma')|$ are independent (for different edges $\langle x, y \rangle$), identically distributed and exponentially integrable random variables.