Determinantal Hypertrees

Abstract

We deduce a structurally inductive description of Kalai's famous tree generalization, the Q-acyclic complex, which extends to an analogous description of the determinantal probability measure associated with Kalai's celebrated enumeration result. Along the way, we generalize the homological notion of a simplicial complex being Q-acyclic to a relative-homological notion of a pair of simplicial complexes being Q-acyclic relative to each other. We also apply these new results to random topology and the spectral analysis of random graphs.