2017/04/10 by Florent Benaych-Georges, Benaych-Georges, Florent, Charles Bordenave +3 · 8 citations
Mathematics · #Graph theory and applications #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.PR #msc:05C80 #msc:15B52 #msc:60B20
paper · pdf · doi:10.48550/arxiv.1704.02945
arxiv created 2021/01/22 · arxiv updated 2021/01/25
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 ≫ log n. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime d ≪ log n, this establishes a crossover in the behaviour of the extreme eigenvalues around d ∼ 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.