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On the vanishing of homology in random Čech complexes
Author(s) -
Bobrowski Omer,
Weinberger Shmuel
Publication year - 2017
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.20697
Subject(s) - torus , homology (biology) , struct , phase transition , mathematics , homogeneous , singular homology , persistent homology , poisson distribution , combinatorics , physics , chemistry , topology (electrical circuits) , condensed matter physics , geometry , algorithm , computer science , amino acid , statistics , biochemistry , programming language
We compute the homology of random Čech complexes over a homogeneous Poisson process on the d ‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős ‐Rényi phase transition, where the Čech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017