Inside the critical window for cohomology of random k-complexes

2013/01/07 by Matthew Kahle, Kahle, Matthew, Boris Pittel +1 · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1301.1324

openalex publication_date 2013/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove sharper versions of theorems of Linial-Meshulam and Meshulam-Wallach which describe the behavior for (Z/2)-cohomology of a random k-dimensional simplicial complex within a narrow transition window. In particular, we show that within this window the (k-1)st Betti number is in the limit Poisson distributed. For k=2 we also prove that in an accompanying growth process, with high probability, first cohomology vanishes exactly at the moment when the last isolated (k-1)-simplex gets covered by a k-simplex.

Cited by

• Using dense graph limit theory to count cocycles of random simplicial complexes

Related

• Vanishing of cohomology groups of random simplicial complexes
• Generalized Random Simplicial Complexes
• Phase transition in cohomology groups of non-uniform random simplicial\n complexes
• Random clique complex process inside the critical window
• Sharp vanishing thresholds for cohomology of random flag complexes