The supercritical phase of the $\varphi^4$ model is well behaved

TL;DR

This paper investigates the supercritical phase of the $^4$ model on $Z^d$, demonstrating well-behaved cluster properties that enable analysis of large deviations and spectral gaps in the supercritical regime.

Contribution

It establishes local uniqueness of macroscopic clusters in the supercritical phase using a random cluster representation, facilitating further analysis of the model's properties.

Findings

Local uniqueness of macroscopic clusters in supercritical phase

Exponential bounds for large deviations of empirical magnetisation

Bounds on spectral gaps of dynamical $^4$ models

Abstract

In this article, we analyse the $\varphi^4$ model on $\mathbb Z^d$ in the supercritical regime $\beta > \beta_c$. We consider a random cluster representation of the $\varphi^4$ model, which corresponds to an Ising random cluster model on a random environment. We prove that the supercritical phase of this percolation model on $\mathbb Z^d$ ($d\geq 2$) is well behaved in the sense that, for every $\beta>\beta_c$, local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This result provides the basis for renormalisation techniques used to study several fine properties of the supercritical phase. As applications, we prove surface order exponential bounds for the (lower) large deviations of the empirical magnetisation as well as for the spectral gaps of dynamical $\varphi^4$ models in the entire supercritical regime.