Inside the critical window for cohomology of random k‐complexes1. Issue 1 (29th January 2015)

• Record Type: Journal Article
• Authors: Kahle, Matthew; Pittel, Boris
• Is Part Of: Random structures & algorithms. Volume 48:Issue 1(2016)
• Page Start: 102
• Page End: 124
• Publication Date: 2015-01-29
• Subjects: random simplicial complexes -- phase transitions; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20577
• Languages: English
• ISSNs: 1042-9832

Abstract

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 if Y is a random k ‐dimensional simplicial complex with each k ‐simplex appearing i.i.d. with probability p = k log n + c n, with k ≥ 1 and c ∈ ℝ fixed, then the dimension of cohomology β k − 1 ( Y ) is asymptotically Poisson distributed with mean e − c / k ! . In the k = 2 case we also prove that in an accompanying growth process, with high probability, H k − 1 ( Y, ℤ / 2 ) vanishes exactly at the moment when the last ( k − 1 ) ‐simplex gets covered by a k ‐simplex, a higher‐dimensional analogue of a "stopping time" theorem about connectivity of random graphs due to Bollobás and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016