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Spectral properties of non-Hermitian random matrices

Abstract

This thesis presents new results concerning the spectral properties of certain families of large random matrices. The overarching goal is to extend some well-known results for matrices with independent and identically distributed (iid) entries to random matrices whose entries are either dependent or non-identically distributed. Particular attention is given to the adjacency matrix of a random regular digraph.

Making use of the method of exchangeable pairs, we establish concentration bounds for codegrees and edge densities of induced subgraphs of a random regular digraph. We apply these bounds along with other coupling techniques to prove that the associated adjacency matrix is invertible with high probability, assuming a mild growth condition on the degree of the graph.

We next prove lower tail estimates for the smallest singular value of matrices with independent but non-identically distributed entries, including matrices with most entries set to zero deterministically. In particular, we show that for small diagonal perturbations of centered random matrices with independent entries of bounded variance, the smallest singular value is of inverse-polynomial order with high probability.

Finally, we prove that the circular law holds for a random matrix obtained from the adjacency matrix of a dense random regular digraph by multiplying each entry by a random sign.

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