The Characteristic Polynomial of Sums of Random Permutations and Regular Digraphs

被引：1

作者：

Coste, Simon ^{[1
]}

Lambert, Gaultier ^{[2
]}

Zhu, Yizhe ^{[3
]}

机构：

[1] ENS PSL Univ, Dept Informat, 45 Rue Ulm, F-75230 Paris 05, France

[2] Univ Zurich, Inst Math, 190 Winterthurerstr, CH-8057 Zurich, Switzerland

[3] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA

来源：

INTERNATIONAL MATHEMATICS RESEARCH NOTICES | 2024年 / 2024卷 / 03期

基金：

美国国家科学基金会; 瑞士国家科学基金会;

关键词：

CIRCULAR LAW; SPECTRAL GAP; MATRICES; EIGENVALUES;

D O I：

10.1093/imrn/rnad182

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Let A(n) be the sum of d permutation matrices of size n x n, each drawn uniformly at random and independently. We prove that the normalized characteristic polynomial 1/root d det(I-n - zA(n)/root d) converges when n -> infinity towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs. Our results are valid both in the regime where d is fixed and for d slowly growing with n.

引用

页码：2461 / 2510

页数：50

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