On Large Values of Non-Gaussian Logarithmically Correlated Fields

Abstract

Random field models with logarithmic correlation structure appear widely across probability theory and there has been great progress in understanding the large value statistics of many Gaussian (or nearly Gaussian) log-correlated fields in the past decades. Studying the universality and non-universality for non-Gaussian log-correlated field models is a natural follow-up question, and recent studies on the characteristic polynomials of the sparse random matrices raised people's attention to Poissonian log-correlated field models. In this dissertation, we establish results for two classes of non-Gaussian log-correlated field models.(i) Poissonian logarithmically correlated fields arise naturally in the study of the characteristic polynomials of the sparse random matrices, including random permutation matrices, their finite independent sums, and the sparse IID Bernoulli random matrices. We introduce a prototypical random function series model with log-correlation and Poissonian tail, make a connection with a branching random walk model with random time-varying branching law, and establish an asymptotic upper bound to the third order on the maximum of this random function series on the unit circle.(ii) Gaussian holomorphic multiplicative chaos (HMC) is a random distribution arises from the study of the characteristic polynomials of circular ensembles and the partial sums of the random multiplicative functions. In particular, the fractional moments of the Fourier coefficients of Gaussian HMC and the aforementioned related models are shown to have a ``better-than-square-root" cancellation. We study the fractional moments of the Fourier coefficients of the non-Gaussian version of HMC and obtain the full universality result for the ``better-than-square-root" cancellation phenomenon in our setting, along with a double-layer phase transition.