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Spectral radii of sparse random matrices

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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 $d \gg \log n$. This establishes a crossover in the behaviour of the extremal eigenvalues around $d \sim \log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs. Joint work with Florent Benaych-Georges and Charles Bordenave.

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