{"paper":{"title":"Maximum of the characteristic polynomial for a random permutation matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nicholas Cook, Ofer Zeitouni","submitted_at":"2018-06-20T04:46:23Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07549","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}