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We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a bipartite block structure. We obtain probabilistic upper (respectively, lower) bounds for the largest (respectively, smallest) singular values of a large rectangular random matrix X. These bounds are given in terms of the maximal and minimal 2-norms of the rows and columns of the variance profile of X. The proofs involve finding probabilistic upper bounds on the spectral radius of an associated non-backtracking matrix B. The two-sided bounds can be applied to the centered adjacency matrix of sparse inhomogeneous Erd˝os-Rényi bipartite graphs for a wide range of sparsity, down to criticality. In particular, for Erd˝os-Rényi bipartite graphs G(n,m, p) with p = ω(log n)/n, and m/n→ y ∈ (0,1), our sharp bounds imply that there are no outliers outside the support of the Marˇcenko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices. 

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent interest. We also prove a semicircle law for random [Formula: see text]-biregular bipartite graphs when [Formula: see text]. As an application, we translate the results to adjacency matrices of uniformly distributed random regular hypergraphs. 

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.