Spectra of random regular hypergraphs (Q2048572)

Summary: In this paper, we study the spectra of regular hypergraphs following the definitions from \textit{K. Feng} and \textit{W.-C. W. Li} [J. Number Theory 60, No. 1, 1--22 (1996; Zbl 0874.05041)]. 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 [\textit{M. C. Angelini} et al., ``Spectral detection on sparse hypergraphs'', in: Proceedings of the 53rd Annual Allerton conference on communication, control, and computing, Allerton 2015. Los Alamitos: IEEE Computer Society. 66--73 (2015; \url{doi:10.1109/ALLERTON.2015.7446987})]. 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.