Bounds on the mod 2 homology of random 2-dimensional determinantal hypertrees

TL;DR

This paper proves that the normalized first homology dimension of random 2-dimensional determinantal hypertrees converges to zero, confirming conjectures and employing large deviation principles from Erdős-Rényi graph theory.

Contribution

It establishes the asymptotic behavior of homology in random 2D hypertrees, confirming conjectures by Kahle, Newman, Linial, and Peled.

Findings

Normalized homology dimension converges to zero in probability.

Results confirm conjectures on random 2D complexes.

Uses large deviation principles from Erdős-Rényi graph theory.

Abstract

As a first step towards a conjecture of Kahle and Newman, we prove that if $T_n$ is a random $2$-dimensional determinantal hypertree on $n$ vertices, then \[\frac{\dim H_1(T_n,\mathbb{F}_2)}{n^2}\] converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the $1$-out $2$-complex. Our proof relies on the large deviation principle for the Erd\H{o}s-R\'enyi random graph by Chatterjee and Varadhan.