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In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996). Our main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs. We then relate the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hypergraphs. We also prove the spectral gap for the non-backtracking operator of a random regular hypergraph introduced in Angelini et al. (2015). Finally, we obtain the convergence of the empirical spectral distribution (ESD) for random regular hypergraphs in different regimes. Under certain conditions, we can show a local law for the ESD.more » « less
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null (Ed.)We describe the non-backtracking spectrum of a stochastic block model with connection probabilities [Formula: see text]. In this regime we answer a question posed in [L. Dall’Amico, R. Couillet and N. Tremblay, Revisiting the Bethe–Hessian: Improved community detection in sparse heterogeneous graphs, in Advances in Neural Information Processing Systems (2019), pp. 4039–4049] regarding the existence of a real eigenvalue “inside” the bulk, close to the location [Formula: see text]. We also introduce a variant of the Bauer–Fike theorem well suited for perturbations of quadratic eigenvalue problems, which could be of independent interest.more » « less
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We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsityω(1/n): inhomogeneous random graphs with roughly equal expected degrees,W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.more » « less
