Hypercube percolation

R.W. van der Hofstad; A. Nachmias

doi:10.4171/JEMS/679

Hypercube percolation

R.W. van der Hofstad, A. Nachmias

Research output: Contribution to journal › Article › Academic › peer-review

17 Citations (Scopus)

Abstract

We study bond percolation on the Hamming hypercube f0; 1gm around the critical probability pc. It is known that if p D pc.1 C O.2-m=3//, then with high probability the largest connected component Here we show that for any sequence ".m/such that ".m/D o.1/but ϵ.m/percolation on the hypercube at pc.1 C ϵm//has with high probability, where C2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [18].

Original language	English
Pages (from-to)	725-814
Number of pages	90
Journal	Journal of the European Mathematical Society
Volume	19
Issue number	3
DOIs	https://doi.org/10.4171/JEMS/679
Publication status	Published - 2017

Keywords

Birth of the giant component
Critical behavior
Hypercube
Mean-field results
Mixing time
Non-backtracking random walk
Percolation
Scaling window

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10.4171/JEMS/679

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AB - We study bond percolation on the Hamming hypercube f0; 1gm around the critical probability pc. It is known that if p D pc.1 C O.2-m=3//, then with high probability the largest connected component Here we show that for any sequence ".m/such that ".m/D o.1/but ϵ.m/percolation on the hypercube at pc.1 C ϵm//has with high probability, where C2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [18].

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KW - Critical behavior

KW - Hypercube

KW - Mean-field results

KW - Mixing time

KW - Non-backtracking random walk

KW - Percolation

KW - Scaling window

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Hypercube percolation

Abstract

Keywords

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