Hypercube percolation

  • Remco van der Hofstad

    TU Eindhoven, The Netherlands
  • Asaf Nachmias

    University of British Columbia, Vancouver, Canada

Abstract

We study bond percolation on the Hamming hypercube {0,1} around the critical probability . It is known that if , then with high probability the largest connected component is of size . Here we show that for any sequence such that but percolation on the hypercube at has

with high probability, where is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [18].

Cite this article

Remco van der Hofstad, Asaf Nachmias, Hypercube percolation. J. Eur. Math. Soc. 19 (2017), no. 3, pp. 725–814

DOI 10.4171/JEMS/679