Homology of Multi-Parameter Random Simplicial Complexes

摘要

We consider a multi-parameter model for randomly constructing simplicial complexes that interpolates between random clique complexes and Linial-Meshulam random k-dimensional complexes. Unlike these models, multi-parameter complexes exhibit nontrivial homology in numerous dimensions simultaneously. We establish upper and lower thresholds for the appearance of nontrivial cohomology in each dimension and characterize the behavior at criticality.