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

Abstract: We derive new small deviations bounds for a class of Gaussian multiplicative chaos measures obtained from Gaussian fields with zero spatial average. The upper bound holds for a class of fields closely related to the $\star$-scale invariant Gaussian fields in arbitrary number of dimensions, and the lower bound holds for the Gaussian free field with zero spatial average on the 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 $R$ 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.