Maximum of the characteristic polynomial for a random permutation matrix

Abstract

Let $P_N$ be a uniform random $N\times N$ permutation matrix and let $\chi_N(z)=\det(zI_N- P_N)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically, \[ \sup_{|z|=1}|\chi_N(z)|= N^{x_0 + o(1)} \] with probability tending to one as $N\to \infty$, for a numerical constant $x_0\approx 0.652$. The main idea of the proof is to uncover a logarithmic correlation structure for 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 $\chi_N(e^{2\pi it})$ is sensitive to Diophantine properties of the point $t$. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory.