Eigenvalues and spectral gap in sparse random simplicial complexes

We consider the adjacency operator A of the Linial–Meshulam model X(d, n, p) for random d-dimensional simplicial complexes on n vertices, where each d-cell is added independently with probability p ∈ [0, 1] to the complete (d − 1)-skeleton. We consider sparse random matrices H, which are generalizations of the centered and normalized adjacency matrix A := (np(1−p))^−1/2 · (A − E[A]), obtained by replacing the Bernoulli(p) random variables used to construct A with arbitrary bounded distribution Z. We obtain bounds on the expected Schatten norm of H, which allow us to prove results on eigenvalue confinement and in particular that H₂ converges to 2√d both in expectation and P-almost surely as n → ∞, provided that Var(Z)>>(Formula presented) . The main ingredient in the proof is a generalization of (Invent. Math. 214 (2018) 1031–1080, Theorem 4.8) to the context of high-dimensional simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.