Inside the critical window for cohomology of random k-complexes

• Published in: Random structures & algorithms Vol. 48; no. 1; pp. 102 - 124
• Main Authors: Kahle, Matthew, Pittel, Boris
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01.01.2016; Wiley Subscription Services, Inc
• Subjects: phase transitions; random simplicial complexes
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20577

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=klogn+cn, with k≥1 and c∈ℝ fixed, then the dimension of cohomology βk−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, Hk−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