On the spectrum of Random Simplicial Complexes in Thermodynamic Regime

TL;DR

This paper investigates the spectral properties of the Linial-Meshulam complex, a higher-dimensional generalization of Erdős-Rényi graphs, revealing a limiting spectral distribution and its behavior under various regimes.

Contribution

It establishes the existence of a non-random limiting spectral distribution for the adjacency matrices of the complex and analyzes its properties and asymptotic behavior.

Findings

LSD exists and is non-symmetric around zero

Under normalization, LSD converges to the semicircle law as λ increases

The local weak limit of the line graph influences the LSD's continuous part

Abstract

Linial-Meshulam complex is a random simplicial complex on $n$ vertices with a complete $(d-1)$-dimensional skeleton and $d$-simplices occurring independently with probability p. Linial-Meshulam complex is one of the most studied generalizations of the Erdos-Renyi random graph in higher dimensions. In this paper, we discuss the spectrum of adjacency matrices of the Linial-Meshulam complex when $np \rightarrow \lambda$. We prove the existence of a non-random limiting spectral distribution(LSD) and show that the LSD of signed and unsigned adjacency matrices of Linial-Meshulam complex are reflections of each other. We also show that the LSD is unsymmetric around zero, unbounded and under the normalization $1/\sqrt{\lambda d}$, converges to standard semicircle law as $\lambda \rightarrow \infty$. In the later part of the paper, we derive the local weak limit of the line graph of the Linial-Meshulam complex and study its consequence on the continuous part of the LSD.