Maximum of the Characteristic Polynomial of the Ginibre Ensemble

Abstract

We compute the leading asymptotics for the maximum of the (centered) logarithm of the absolute value of the characteristic polynomial, denoted $$\Psi _N$$ , of the Ginibre ensemble as the dimension of the random matrix N tends to $$\infty $$ . 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 et al. (Ann Probab 43(3):1157–1201, 2015) to derive the necessary asymptotics, as well as the results from Webb and Wong (Proc Lond Math Soc (3) 118(5):1017–1056, 2019).