Semicircle law for multi-parameter random simplicial complexes

TL;DR

This work analyzes the spectra of unsigned and signed adjacency matrices of the d-dimensional multi-parameter random simplicial complex $Y_d(n,\mathbf{p})$ in a dense regime with $\min_{i\le d-1}\liminf p_i>0$ and $p_d=o(1)$ while $np_d\to\infty$. The authors introduce an extended adjacency matrix $\widehat{A}_n$ and a centered Hadamard product $B_n$, proving that the normalized centered matrices $H_n$ and $H_n^+$ have limiting spectral distribution $\mathrm{Ber}(c) \otimes \mu_{sc}$, where $c$ is the limiting fraction of $(d-1)$-cells. They then bootstrap from the centered case to establish that the (unsigned and signed) adjacency matrices, after the same normalization, have the semicircle law as their limiting spectral distribution in probability, and that the number of $(d-1)$-cells concentrates to $c$. The paper also extends these results to the multi-parameter upper model and discusses the connection to random matrix theory and higher-dimensional topology, providing a robust null-model framework for topological data analysis. Overall, it broadens semicircular universality to random simplicial complexes with random dimensions and dependencies, linking combinatorial topology with spectral randomness.

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 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 $2 \leq k \leq d$. 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.

Figures (2)

• Figure 1: Figure (a) shows the orientation induced by the positive orientation of $[0,1,2]$ on its boundary. Figure (b) shows the 1-cell $[0,1]\stackrel{X}{\sim} [0,2]$ and $[0,1]\stackrel{X}{\sim} [2,1]$.
• Figure 2: The empirical spectral distributions of: (top-left) extended adjacency matrix $\widehat{A}_n$ normalized by $\frac{1}{\sqrt{n\prod_{i=1}^dp_i^{\beta_i}(1-p_d)}}$, (top-right) its centered version $H_n$, (bottom-left) the normalized unsigned adjacency matrix $\widetilde{A}_n$ and (bottom-right) the normalized signed adjacency matrix $\widetilde{A}_n^+$. The histograms are for $n=40,d=2$ and $p=[0.8,0.7]$, averaged over 10 realizations.

Theorems & Definitions (34)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Theorem 6
• Remark 7
• Corollary 8
• Theorem 9
• Definition 10
• Definition 11