Semicircle law for multi-parameter random simplicial complexes

TL;DR

This paper proves that the spectral distributions of adjacency matrices in multi-parameter random simplicial complexes converge to the semicircle law, extending random matrix theory to complex high-dimensional structures.

Contribution

It establishes the semicircle law for the spectra of adjacency matrices in multi-parameter random simplicial complexes, a significant generalization of previous models.

Findings

Spectral distributions converge to the semicircle law.

Results hold under specific probability conditions.

Extends random matrix theory to high-dimensional 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 $\mathbf{p}=(p_1,p_2,\ldots, p_d)$ such that $p_i \in (0,1]$ for all $1 \leq i \leq d$, 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 $\mathcal{G}(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2 \leq k \leq d$. 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 its entries. We prove that the empirical spectral distributions of both matrices converge weakly to the semicircle law in probability.