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 $\mathbf{p}=(p_1,p_2,\ldots, p_d)\in (0,1]^d$, 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 Erd\H{o}s-R\'enyi graph $G(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2 \leq k \leq d$. We study the signed and unsigned adjacency matrices of $d$-dimensional multi-parameter random simplicial complexes $Y_d(n,\mathbf{p})$, under the assumptions $\min_{i=1,\ldots d-1}\liminf p_i >0$ and $np_d \rightarrow \infty$ with $p_d=o(1)$. In general, these matrices have random dimensions and exhibit dependency among their entries. We prove that the empirical spectral measures of both matrices converge weakly to the semicircle law in probability. We also prove that the limiting spectral measure of the signed and unsigned adjacency matrices of another associated model, the multi-parameter upper model, is the semicircle law, under the assumption $np_d(1-p_d) \rightarrow \infty$. Further, we establish convergence results for the number of $(d-1)$-cells and the number of maximal $(d-1)$-cells of multi-parameter simplicial complexes.