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 $p\in[0,1]$ to the complete $(d-1)$-skeleton. We consider sparse random matrices $H$, which are generalizations of the centered and normalized adjacency matrix $\mathcal{A}:=(np(1-p))^{-1/2}\cdot(A-\mathbb{E}\left[A\right])$, 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 $\left\Vert H\right\Vert _{2}$ converges to $2\sqrt{d}$ both in expectation and $\mathbb{P}-$almost surely as $n\to\infty$, provided that $\mathrm{Var}(Z)\gg\frac{\log n}{n}$. The main ingredient in the proof is a generalization of [LVHY18,Theorem 4.8] to the context of high-dimensional simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.

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