##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability & Statistics

## Nick Cook

#### UCLA

## The maximum of the characteristic polynomial for a random permutation matrix

##### Abstract:

Let $P$ be a uniform random permutation matrix of size $N$ and let $\chi_N(z)= \det(zI - P)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically, \[ \sup_{|z|=1}|\chi_N(z)|= N^{x_c + o(1)} \] with probability tending to one as $N\to \infty$, for a numerical constant $x_c\approx 0.677$. 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 \emph{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.

Host: Tianyi Zheng

### April 19, 2018

### 10:00 AM

### AP&M 7218

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