Algebraic and combinatorial expansion in random simplicial complexes

Abstract

In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a d-dimensional Linial-Meshulam random simplicial complex, above the cohomological connectivity threshold. We consider the spectral gap of the Laplace operator and the Cheeger constant as this was introduced by Parzanchevski, Rosenthal, and Tessler. We show that with high probability the spectral gap of the random simplicial complex as well as the Cheeger constant are both concentrated around the minimum co-degree of among all (𝑑 − 1)-faces. Furthermore, we consider a random walk on such a complex, which generalizes the standard random walk on a graph. We show that the associated conductance is with high probability bounded away from 0, resulting in a bound on the mixing time that is logarithmic in the number of vertices of the complex.

Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes

“…[17,22,[26][27][28][29]33]), and several notions of connectedness have been analysed (see e.g. [3,4,25,34,38]), as well as related concepts such as expansion [24] and bootstrap percolation [23]. In this paper, we consider a model of random simplicial complexes generated from non-uniform random hypergraphs, in which edges may have different sizes, and study cohomology groups over an arbitrary (not necessarily finite) abelian group R.…”

On the spectrum of Random Simplicial Complexes in Thermodynamic Regime

“…Linial-Meshulam complex introduced in [31] is the earliest random simplicial complex studied and is a generalization of Erdős-Rényi random graphs to higher dimensions d ≥ 2. A Linial-Meshulam complex is a random d-dimensional complex on n vertices with a complete (d − 1)-dimensional skeleton and d-cells occurring independently with probability p. Linial-Meshulam complex has attracted considerable attention from the mathematical community and is one of the most heavily studied generalizations of Erdős-Rényi random graph [19], [29], [32].…”