Eigenvalues and spectral gap in sparse random simplicial complexes

TL;DR

This paper studies the spectral properties of adjacency operators in sparse random simplicial complexes, extending random matrix theory to high-dimensional structures with dependent entries and establishing eigenvalue bounds.

Contribution

It introduces bounds on the Schatten norm of generalized adjacency matrices in high-dimensional complexes, extending spectral analysis techniques to dependent, sparse random matrices.

Findings

Eigenvalue norm converges to 2√d as n→∞

Provides bounds on the expected Schatten norm of the adjacency operator

Extends spectral confinement results to high-dimensional simplicial complexes

Abstract

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.