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

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.