Inside the critical window for cohomology of random k -complexes
Matthew Kahle, Boris G. Pittel · Random Structures and Algorithms · 2015
We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for -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 with and fixed, then the dimension of cohomology is asymptotically Poisson distributed with mean . In the k = 2 case we also prove that in an accompanying growth process, with high probability, vanishes exactly at the moment when the last -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