Stanford University

Tue 29 Jan 2019, 3:30pm
Probability Seminar
MATH 126

Maximum of the characteristic polynomial for a random permutation matrix

MATH 126
Tue 29 Jan 2019, 3:30pm4:30pm
Abstract
Statistics of the characteristic polynomial for large Haar unitary matrices U at points on the unit circle have received considerable attention due to similarities with the statistics of the Riemann zeta function far up the critical axis. While the bestknown instances of this universality phenomenon concern statistics of \emph{zeros} for these functions (eigenvalues of U), there is strong evidence that the analogy also applies to extreme values.
Towards the more modest goal of understanding this universality phenomenon within the class of distributions on the unitary group, in this talk we consider the characteristic polynomial \chi_N(z) for an N\times N Haar permutation matrix. Our main result is a law of large numbers for (the logarithm of) the maximum modulus of \chi_N(z) over the unit circle. The main idea is to uncover a multiscale structure in the distribution of the random field \chi_N(z), and to adapt a wellknown second moment argument for the maximum of a branching random walk. Unlike the analogous problem for the Haar unitary, the distribution of \chi_N(z) is sensitive to Diophantine properties of the argument of z. To deal with this we borrow tools from the HardyLittlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.
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