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Unlacing hypercube percolation: a survey

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  • Remco Hofstad
  • Asaf Nachmias

Abstract

The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube $$\{0,1\}^{m}$$ { 0 , 1 } m obtained in the series of papers (Borgs et al. in Random Struct Algorithms 27:137–184, 2005 ; Borgs et al. in Ann Probab 33:1886–1944, 2005 ; Borgs et al. in Combinatorica 26:395–410, 2006 ; van der Hofstad and Nachmias in Hypercube percolation, Preprint 2012 ). Secondly, we explain how this study can be performed without the use of the so-called “lace expansion” technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Remco Hofstad & Asaf Nachmias, 2014. "Unlacing hypercube percolation: a survey," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 23-50, January.
    • Handle: RePEc:spr:metrik:v:77:y:2014:i:1:p:23-50
    DOI: 10.1007/s00184-013-0473-5
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