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Inside the critical window for cohomology of random k ‐complexes
Author(s) -
Kahle Matthew,
Pittel Boris
Publication year - 2016
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20577
Subject(s) - mathematics , simplex , struct , cohomology , dimension (graph theory) , combinatorics , random graph , poisson distribution , discrete mathematics , pure mathematics , computer science , statistics , graph , programming language
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 cohomologyβ k − 1 ( Y ) is asymptotically Poisson distributed with meane − 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