Random Permutations and Simplicial Complexes

Abstract

We study the asymptotic behavior of distributions on two different combinatorial objects, permutations and simplicial complexes. First we study strong α-logarithmic measures on the symmetric group, including the well- studied Ewens sampling formula (see [21] and [20] for reference). We show that, for almost every α, precisely ⌈(1 − α log 2)−1⌉ are need to invariably generate Sn asymptotically with positive probability. The corollary is that any fewer permutations will with high probability not invariably generate Sn. In particular, for several measures on Sn no finite number is sufficient. Then we direct our attention to a multi-parameter measure on simplicial complexes. This measure is an interpolation between random clique complexes and Linial-Meshulam random k-dimensional complexes, both subjects of considerable attention over the last two decades. We extend results for each into this new model, establishing upper and lower thresholds for the appearance of nontrivial cohomology in each dimension and characterize the behavior at one window of criticality. We also manage to establish the size of homology within these regimes. Notably, unlike these other distributions, multi-parameter complexes can exhibit nontrivial homology in numerous dimensions simultaneously.