Open Access
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 logCndn/2 for a sufficiently large constant C>0 and let An 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 An, 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 An.

Soit logCndn/2 pour une constante suffisamment grande C>0. Notons An 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 An, 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 An.

Citation

Download Citation

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

Keywords: Directed graph , Logarithmic potential , Non-normal matrix , Random matrix , singular values , Universality

Rights: Copyright © 2019 Institut Henri Poincaré

Vol.55 • No. 4 • November 2019
Back to Top