Bounds on the mod 2 homology of random 2-dimensional determinantal hypertrees
András Mészáros
TL;DR
This work proves that for a random $2$-dimensional determinantal hypertree $T_n$ on $n$ vertices, the mod $2$ first homology satisfies $\dim H_1(T_n, \mathbb{F}_2)/n^2 \to 0$ in probability, and it establishes the same scaling for the $1$-out $2$-complex $S_2(n,1)$. The authors recast cochains via the complete $1$-skeleton as graphs and bound the expected number of $1$-cochains in the coboundary by leveraging the dense graph limit framework and the Chatterjee–Varadhan large deviation principle in graphon space, introducing the auxiliary map $Z_W=2W\circ(\mathbbm{1}-W)$ and an upper semicontinuous functional $f$ to control coboundary events. By deriving a tight exponential bound on the count of cocycles and applying Markov's inequality, they show the $n^2$-scale vanishing of $H_1$-dimension in probability, and they extend the result to the $S_2(n,1)$ model, confirming a Linial–Peled type phenomenon in the $\,\mathbb{F}_2$ setting. The methods connect high-dimensional random simplicial topology with dense graph limit theory and LDP techniques, highlighting a powerful approach to understanding the asymptotic homology of random complexes in the dense regime.
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 Erdős-Rényi random graph by Chatterjee and Varadhan.