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November 2019 The Circular Law for random regular digraphs
Nicholas Cook
Ann. Inst. H. Poincaré Probab. Statist. 55(4): 2111-2167 (November 2019). DOI: 10.1214/18-AIHP943

Abstract

Let $\log^{C}n\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_{n}$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_{n}$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_{n}$.

Soit $\log^{C}n\le d\le n/2$ pour une constante suffisamment grande $C>0$. Notons $A_{n}$ la matrice d’adjacence d’un graphe dirigé aléatoire $d$-régulier sur $n$ sommets. Nous montrons que lorsque $n$ tend vers l’infini, la distribution empirique des valeurs propres de $A_{n}$, convenablement normalisée, suit la loi du cercle. Une étape cruciale consiste à obtenir une borne inférieure quantitative asymptotique pour la plus petite valeur singulière de perturbations additives de $A_{n}$.

Citation

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Nicholas Cook. "The Circular Law for random regular digraphs." Ann. Inst. H. Poincaré Probab. Statist. 55 (4) 2111 - 2167, November 2019. https://doi.org/10.1214/18-AIHP943

Information

Received: 7 August 2017; Revised: 14 September 2018; Accepted: 4 October 2018; Published: November 2019
First available in Project Euclid: 8 November 2019

zbMATH: 07161500
MathSciNet: MR4029149
Digital Object Identifier: 10.1214/18-AIHP943

Subjects:
Primary: 15B52
Secondary: 05C80, 60B20

Rights: Copyright © 2019 Institut Henri Poincaré

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Vol.55 • No. 4 • November 2019
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