Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erdős-Rényi Random Graphs

Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs

“…≥ λ N (A N ) be the eigenvalues of A N . It was shown in [8] (see also [22] for a graphon approach) that the empirical distribution of the centered adjacency matrix converges, after scaling with √ N ε N , to a compactly supported measure µ f . When f ≡ 1, the limiting law µ f turns out to be the semicircle law.…”

Regular variation and free regular infinitely divisible laws

“…It is a pertinent question that whether the conclusion involving Marchenko-Pastur law can be replaced by the standard Wigner's semicircle law, w. The measures of the form w ⊠ ρ for some ρ ∈ M + has appeared as the limiting spectral distributions of random matrices (see Anderson and Zeitouni [2008], Chakrabarty et al [2016Chakrabarty et al [ , 2018b), free type W distributions (see Pérez-Abreu and Sakuma [2012]) and in several other places.…”