Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time

  • For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335–358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551–582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267–1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of criticalFor critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335–358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551–582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267–1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdős-Rényi random graphs.show moreshow less

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Metadaten
Author:Markus HeydenreichORCiDGND, Remco van der Hofstad
URN:urn:nbn:de:bvb:384-opus4-1037535
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/103753
ISSN:0178-8051OPAC
ISSN:1432-2064OPAC
Parent Title (English):Probability Theory and Related Fields
Publisher:Springer Science and Business Media LLC
Place of publication:Berlin
Type:Article
Language:English
Year of first Publication:2011
Publishing Institution:Universität Augsburg
Release Date:2023/04/21
Tag:Statistics, Probability and Uncertainty; Statistics and Probability; Analysis
Volume:149
Issue:3-4
First Page:397
Last Page:415
Note:
Correction published at: https://doi.org/10.1007/s00440-019-00929-x
DOI:https://doi.org/10.1007/s00440-009-0258-y
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Sonstige Open-Access-Lizenz