In the medial regime, upper random simplicial complexes have nonzero Betti numbers only in dimensions k+c < n-j < k+log2 k +c' (k=log2 ln n) while lower ones are (k+a)-connected with dimension ~k+log2 k.
Kahle, Topology of random simplicial complexes: a survey
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Random Simplicial Complexes in the Medial Regime
In the medial regime, upper random simplicial complexes have nonzero Betti numbers only in dimensions k+c < n-j < k+log2 k +c' (k=log2 ln n) while lower ones are (k+a)-connected with dimension ~k+log2 k.