  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.

Combinatorics;Probability