EVENT DETAILS AND ABSTRACT

Title: The maximum of the characteristic polynomial for a random permutation matrix
Speaker: Nick Cook
Speaker Info: UCLA
Brief Description:
Special Note:
Abstract:

Let $\chi_N(z)= \det(zI - P)$ be the characteristic polynomial of a uniform random permutation matrix $P$ of dimension $N$. We prove a law of large numbers for the maximum over the unit circle of the modulus of $\chi_N$, showing it is $N^{c+o(1)}$ with high probability for a certain constant $c$. The main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_N$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. Unlike the well-studied CUE field in which $P_N$ is replaced with a 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 Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.