Magnetization bound for classical spin models on graphs

TL;DR

This paper establishes the existence of phase transitions at finite temperature for classical ferromagnetic spin models on a class of infinite graphs called infrared finite graphs, extending known results beyond regular lattices.

Contribution

It proves phase transitions for O(n) models on infrared finite graphs, generalizing classical lattice results to a broader class of graph structures.

Findings

Phase transitions exist at finite temperature for models on infrared finite graphs.

The proof is based on a rigorous bound on the average magnetization.

Includes the Ising model as a special case.

Abstract

In this paper we prove the existence of phase transitions at finite temperature for O(n) classical ferromagnetic spin models on infrared finite graphs. Infrared finite graphs are infinite graphs with $\lim {m\to 0^+} {\bar Tr (L+m)^{-1} < \infty$, where $L$ is the Laplacian operator of the graph. The ferromagnetic couplings are only requested to be bounded by two positive constants. The proof, inspired by the classical result of Fr\"ohlich, Simon and Spencer on lattices, is given through a rigorous bound on the average magnetization. The result holds for $n\ge 1$ and it includes as a particular case the Ising model.