  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.
