On small deviations of Gaussian multiplicative chaos with a strictly logarithmic covariance on Euclidean ball

Abstract

Recognizing the regime of positive definiteness for a strictly logarithmic covariance kernel, we prove that the small deviations of a related Gaussian multiplicative chaos (GMC) $M_\gamma$ are for each natural dimension $d$ always of lognormal type, i.e. the upper and lower limits as $t\to \infty$ of $$ -\ln\Big(\mathbb{P}(M_\gamma(B(0,r))\le \delta \Big)/(\ln \delta)^2 $$ are finite and bounded away from zero. We then place the small deviations in the context of Laplace transforms of $M_\gamma$ and discuss the explicit bounds on the associated constants. We also provide some new representations of the Laplace transform of GMC related to a strictly logarithmic covariance kernel.