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arXiv · 1704.02945

Spectral radii of sparse random matrices

Florent Benaych-Georges·Charles Bordenave·Antti Knowles
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Abstract

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. Our error bounds are sharp for a large class of sparse random matrices. In particular, for the Erdős-Rényi 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. 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.

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Florent Benaych-Georges, Charles Bordenave, Antti Knowles. 2021-01-22. Spectral radii of sparse random matrices. https://arxiv.org/abs/1704.02945

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