Gaussian multiplicative chaos on a ball

Anna Talarczyk-Noble, Maciej Wiśniewolski

Published 2024-01-15Version 1

We provide representations of the Laplace transforms of GMC on Euclidean balls $B(0,r)$, $r\in (0,1]$. Up to non-vanishing spatial mean, the small deviations of Gaussian multiplicative chaos $M$ associated to the Lebesgue measure on ball are of log-normal type $$ \ln\Big(\mathbb{E}e^{-e^tM}\Big) = -c t^2 + o(t^2), \quad t\rightarrow \infty. $$ We find the bounds on optimal constants $c = c(r)$ associated to small deviations of GMC on balls.