Inside the critical window for cohomology of random k-complexes
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‐sim...
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Published in | Random structures & algorithms Vol. 48; no. 1; pp. 102 - 124 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Hoboken
Blackwell Publishing Ltd
01.01.2016
Wiley Subscription Services, Inc |
Subjects | |
Online Access | Get full text |
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Summary: | 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 |
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Bibliography: | DARPA - No. N66001-12-1-4226 ArticleID:RSA20577 istex:F802066F312AC2D20722F9B3C138B4AEDBBC7070 ark:/67375/WNG-KCR94CR2-R NSF (DMS-1101237) Supported by DARPA (N66001-12-1-4226) (to M.K.); NSF (DMS-1101237) (to B.P.). Supported by DARPA (N66001‐12‐1‐4226) (to M.K.); NSF (DMS‐1101237) (to B.P.). |
ISSN: | 1042-9832 1098-2418 |
DOI: | 10.1002/rsa.20577 |