Random regular digraphs: singularity and spectrum

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Nicholas Cook , UCLA

Please note special day (Tuesday).听 听听We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights.听Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices.听Towards establishing the same result for the unweighted adjacency matrix, we prove that it is invertible with high probability, even for sparse digraphs with degree growing only poly-logarithmically.