For every prime p, the dimension of H_1(T_n, F_p) of a random 2-dimensional determinantal hypertree and of a random 1-out 2-complex is o(n^2) in probability.
The continuum random tree III.The Annals of Probability, pages 248–289, 1993
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Using dense graph limit theory to count cocycles of random simplicial complexes
For every prime p, the dimension of H_1(T_n, F_p) of a random 2-dimensional determinantal hypertree and of a random 1-out 2-complex is o(n^2) in probability.