A few observations around Gaussian domination and continuous symmetry breaking for spin O(N) model

TL;DR

This paper explores Gaussian domination in the spin O(N) model on finite graphs, providing new insights into correlation inequalities, long-range order, and the effects of graph structure without relying solely on reflection positivity.

Contribution

It offers new perspectives on Gaussian domination at various temperatures and examines how local graph modifications influence correlation properties, expanding understanding beyond traditional methods.

Findings

Established a general inequality for spin correlations under Gaussian domination.

Demonstrated implications for long-range order at low temperatures.

Provided counterexamples showing the impact of graph structure on Gaussian domination.

Abstract

We investigate the notion of Gaussian domination for the spin $O(N)$ model on general finite graphs. We begin by proving a general inequality for spin correlations under the assumption of Gaussian domination, which directly implies long-range order at low temperatures for graphs with bounded Green's function. Usually, Gaussian domination is proved via reflection positivity, but this requires strict symmetries and is very rigid. In this article we also probe the boundaries of elementary methods for proving Gaussian domination. Although we did not find a way to get uniform bounds, we do offer new views for Gaussian domination at low and high temperatures for finite graphs, and a few counterexamples illustrating the interplay between correlation estimates and Gaussian domination and how local changes in the graph structure can affect the presence of Gaussian domination.