Bounds on the Mod 2 Homology of Random 2-Dimensional Determinantal Hypertrees

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

$$\begin{aligned} \frac{\dim H_1(T_n,\mathbb {F}_2)}{n^2} \end{aligned}$$

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 Erdős–Rényi random graph by Chatterjee and Varadhan.