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: $\mathcal{N}(0,{\sigma }^2)⊞\mathrm{SC}(s{\sigma }^2)$, 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.

24 pages, 7 figures