Maximum of the characteristic polynomial for a random permutation matrix

Let $P_N$ be a uniform random $N\times N$ permutation matrix and let ${\chi }_N(z)=\det (z{I}_N-P_N)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of ${\chi }_N$ on the unit circle, specifically, $$\underset{\mathrm{\mid }z\mathrm{\mid }=1}{\sup }\mathrm{\mid }{\chi }_N(z)\mathrm{\mid }=N^{x_0+o(1)}$$ with probability tending to one as $N\to \mathrm{\infty }$, 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 $\chi_N(e^{2\pi it})$issensitivetoDiophantinepropertiesofthepoint$t$. To deal with this we borrow tools from the Hardy–Littlewood circle method in analytic number theory.