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

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.