# Spectral radii of sparse random matrices

Abstract : We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erd\H{o}s-R\'enyi graphs. For the Erd\H{o}s-R\'enyi graph $G(n,d/n)$, our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that $d \gg \log n$. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime $d \ll \log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d \sim \log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs, as well as so-called structured random matrices. The proof combines (i) a new inequality between the spectral radius of a matrix and the spectral radius of its nonbacktracking version together with (ii) a new application of the method of moments for nonbacktracking matrices.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-01528786
Contributor : Florent Benaych-Georges <>
Submitted on : Monday, May 29, 2017 - 4:55:17 PM
Last modification on : Friday, January 10, 2020 - 9:09:00 PM

### Identifiers

• HAL Id : hal-01528786, version 1
• ARXIV : 1704.02945

### Citation

Florent Benaych-Georges, Charles Bordenave, Antti Knowles. Spectral radii of sparse random matrices. 2016. ⟨hal-01528786⟩

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