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.

arXiv commentTypos corrected and some proofs have been clarified thanks to the referee's comments. 3 figures and references added. The appendix on the second moment of the characteristic polynomial have been removed since this result is already in the work of Akemann-Vernizzi [1]. Version accepted for publication in Comm. Math. Phys

References in corpus (14)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Fluctuations of eigenvalues of random normal matrices
• Characteristic Polynomials of Complex Random Matrix Models
• Maximum of the characteristic polynomial of random unitary matrices
• Random normal matrices and Ward identities
• On the maximum of the C$\beta$E field
• The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem
• Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities
• On the distribution of maximum value of the characteristic polynomial of GUE random matrices
• On absolute moments of characteristic polynomials of a certain class of complex random matrices
• On the moments of the characteristic polynomial of a Ginibre random matrix
• Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
• Local density for two-dimensional one-component plasma
• Matrix averages relating to the Ginibre ensembles

Cited by in corpus (4)

• The random normal matrix model: insertion of a point charge
• Quantum statistics transmutation via magnetic flux attachment
• Maxima of log-correlated fields: some recent developments
• Global rigidity and exponential moments for soft and hard edge point processes