Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

TL;DR

This paper studies the probabilities of rare topological structures in multi-parameter random simplicial complexes, extending classical graph theory to higher dimensions and analyzing large deviations of subcomplex counts and Betti numbers.

Contribution

It derives the order of magnitude for large deviation probabilities of subcomplex counts and Betti numbers in multi-parameter simplicial complexes, advancing understanding of their topological fluctuations.

Findings

Established large deviation estimates for subcomplex counts.

Derived probability bounds for Betti numbers in critical dimensions.

Analyzed the occurrence of unusual topological features in high-dimensional complexes.

Abstract

We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.