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Spectral Statistics of Sparse Random Graphs with a General Degree Distribution

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arxiv 1509.03368 v1 pith:SUMOXHAK submitted 2015-09-11 math.PR

classification math.PR
keywords provecitedistributiongraphsrandomsparseaddedadjacency
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abstract

We consider the adjacency matrices of sparse random graphs from the Chung-Lu model, where edges are added independently between the $N$ vertices with varying probabilities $p_{ij}$. The rank of the matrix $(p_{ij})$ is some fixed positive integer. We prove that the distribution of eigenvalues is given by the solution of a functional self-consistent equation. We prove a local law down to the optimal scale and prove bulk universality. The results are parallel to \cite{Erdos2013b} and \cite{Landon2015}.

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  1. Universality of the least singular value for the sum of random matrices

    math.PR 2019-08 conditional novelty 7.0 of 10

    The least singular value of R*XT + U*YV, after scaling by N, converges in distribution to 1 - e^{-r^2}, the same limit as for a complex Gaussian matrix.

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