[Combprob] Thursday: Nick Cook (UCLA): "Random regular digraphs: singularity and spectrum"

[Daniel Spielman]

Thursday, Oct 29 at 4:00, in LOM 215.

Nick Cook (UCLA): "Random regular digraphs: singularity and spectrum"

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. 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. 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.
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