Eigenvalue Distribution of the Laplacian on Random Complexes

Abstract

We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model $Y_{q+1}(n, p)$. We first give a definition of a Gaussian--semicircle law: $\mathcal{N}(0, \sigma^2)\boxplus \operatorname{SC}(s\sigma^2)$, and give a combinatorial formula of its moments in terms of pairing partitions. In addition to that, we also prove that the limiting empirical eigenvalue distribution of this standardized up Laplacian follows a Gaussian--semicircle law, in the sense of almost surely weak convergence.