Maximum of the characteristic polynomial of the Ginibre ensemble

Abstract

We compute the leading asymptotics of the maximum of the (centered) logarithm of the absolute value of the characteristic polynomial, denoted $\Psi_N$, of the Ginibre ensemble as the dimension $N$ of the random matrix tends to infinity. The method relies on the log-correlated structure of the field $\Psi_N$ and we obtain the lower-bound for the maximum by constructing a family of Gaussian multiplicative chaos measures associated with certain regularization of $\Psi_N$ at small mesoscopic scales. We also obtain the leading asymptotics for the dimensions of the sets of thick points and verify that they are consistent with the predictions coming from the Gaussian Free Field. A key technical input is the approach from Ameur-Hedenmalm-Makarov to derive the necessary asymptotics, as well as the results from Webb-Wong.