Semicircle law for multi-parameter random simplicial complexes

Abstract

In this paper, we consider the multi-parameter random simplicial complex model, which generalizes the Linial-Meshulam model and random clique complexes by allowing simplices of different dimensions to be included with distinct probabilities. For $n,d\in \mathbb{N}$ and $\mathbf{p}=(p_1,p_2,\dots ,p_d)$ such that $p_i\in (0,1]$ for all $1\le i\le d$, the multi-parameter random simplicial complex $Y_d(n,\mathbf{p})$ is constructed inductively. Starting with $n$ vertices, edges (1-cells) are included independently with probability $p_1$, yielding the Erdős-Rényi graph $\mathcal{G}(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2\le k\le d$. We study the signed and unsigned adjacency matrices of $d$-dimensional multi-parameter random simplicial complexes $Y_d(n,\mathbf{p}),$ under the assumptions ${\min }_{i=1,\dots d-1}\mathrm{liminf}{p}_i>0$ and $n{p}_d\to \infty$ with $p_d=o(1)$. In general, these matrices have random dimensions and exhibit dependency among its entries. We prove that the empirical spectral distributions of both matrices converge weakly to the semicircle law in probability.

22 pages, 2 Figures