for iid Poisson matrices this is covered by the circular law (for the bulk distribution), see https://arxiv.org/abs/1109.3343 . Local spacing universality is still open for this (the best result so far, due to Van and myself, requires four matching moments with a Gaussian). For symmetric Poisson matrices the semicircular law is classical (due to Pastur?) and the universality of spacing is also known (see e.g. https://arxiv.org/abs/1407.5606 ).

]]>There is now a quite precise understanding of the distribution of the extreme eigenvalues of an Erdos-Renyi adjacency matrix, see e.g., https://arxiv.org/abs/1103.3869

]]>You wrote “almost surely have simple spectrum”, did you mean “with high probability”?

*[Clarified to “asymptotically almost surely” – T.]*

Is it possible to (effectively) compute an upper bound for the ratio (for all sufficiently large n)?

*[Yes; see the remark after Theorem 2.6 of the paper. -T.]*

The techniques in this paper break down (among other things, they rely on the Cauchy interlacing law, which breaks down in non-Hermitian settings), but I have a student looking into this problem who is trying some other methods.

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