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.

中文翻译：

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随机置换矩阵的特征多项式的最大值

令$P_N$ 是一个均匀随机$N\times N$ 置换矩阵，并令$\chi_N(z)=\det(zI_N-P_N)$ 表示其特征多项式。我们证明了单位圆上$\chi_N$的最大模数的大数定律，具体来说，\[ \sup_{|z|=1}|\chi_N(z)|= N^{x_0 + o(1 )} \] 的概率趋于 1 为 $N\to \infty$，对于数值常数 $x_0\approx 0.652$。证明的主要思想是揭示 $\chi_N$（的对数）分布的对数相关结构，将其视为圆上的随机场，并适应一个众所周知的二阶矩参数以获得最大值分支随机游走。与经过充分研究的 \emph{CUE field} 不同，其中 $P_N$ 被 Haar 酉代替，$\chi_N(e^{2\pi it})$ 的分布对点 $t 的丢番图性质敏感$.