Small deviations of Gaussian multiplicative chaos and the free energy of the two-dimensional massless Sinh--Gordon model

TL;DR

This paper establishes small deviation bounds for Gaussian multiplicative chaos measures on tori and analyzes the free energy of the massless Sinh--Gordon model, revealing its infinite volume limit.

Contribution

It provides a new decomposition for log-correlated Gaussian fields and applies it to study the free energy of the 2D massless Sinh--Gordon model.

Findings

Derived new small deviation bounds for Gaussian multiplicative chaos.

Proved the existence of a non-zero finite subsequential limit of the free energy.

Connected Gaussian chaos bounds to the free energy of the Sinh--Gordon model.

Abstract

We prove a global decomposition result for $\log$-correlated Gaussian fields on the $d$-dimensional torus and use this to derive new small deviations bounds for a class of Gaussian multiplicative chaos measures obtained from Gaussian fields with zero spatial mean on the $d$-dimensional torus. The upper bound is obtained by a modification of the method that was used in \cite{LRV}, and the lower bound is obtained by applying the Donsker--Varadhan variational formula. We also give the probabilistic path integral formulation of the massless Sinh--Gordon model on a torus of side length $R$, and study its partition function as $R$ tends to infinity. We apply the small deviation bounds for Gaussian multiplicative chaos to obtain lower and upper bounds for the logarithm of the partition function, leading to the existence of a non-zero and finite subsequential infinite volume limit for the free energy.