On the Dimension of Random Simplicial Complexes

TL;DR

This paper investigates the maximal dimension of random simplicial complexes, extending theoretical results across different probabilistic regimes and providing new insights into their distribution and large deviations.

Contribution

It extends existing results to the Poisson setting, introduces new expectation and large deviation results, and provides a precise distribution in the dense regime.

Findings

Extended results to Poisson processes in sparse graphs

Derived new expectation and large deviation principles

Provided a precise distribution in the dense case

Abstract

The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique number of geometric graphs or scan statistics. We extend existing results in the binomial point process case to the Poisson setting in sparse graphs, give new ones about expectations and large deviation principles in all regimes, as well as give a first precise distribution result in the dense case.