Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes

TL;DR

This paper establishes limit theorems for higher-dimensional Betti numbers in multiparameter random simplicial complexes, revealing phase transitions and large deviation behaviors in their topology.

Contribution

It provides the first rigorous probabilistic analysis of Betti numbers in multiparameter complexes beyond the critical dimension, including laws of large numbers and central limit theorems.

Findings

Betti numbers follow strong laws of large numbers

Central limit theorems characterize fluctuations of Betti numbers

Phase transitions occur depending on scaling constants

Abstract

We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the critical dimension. We show that the higher-dimensional Betti numbers satisfy strong laws of large numbers and central limit theorems. Moreover, lower tail large deviations for these Betti numbers are also discussed. Some of our results indicate an occurrence of phase transitions in terms of the scaling constants of the central limit theorem, and the exponentially decaying rate of convergence of lower tail large deviation probabilities.