Hypercube percolation

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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 languageEnglish
Pages (from-to)725-814
Number of pages90
JournalJournal of the European Mathematical Society
Volume19
Issue number3
DOIs
Publication statusPublished - 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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