TY - JOUR
T1 - Unlacing hypercube percolation
T2 - A survey
AU - van der Hofstad, Remco
AU - Nachmias, Asaf
N1 - Funding Information:
The work of RvdH was supported in part by the Netherlands Organization for Scientific Research (NWO). The work of AN was partially supported by NSF Grant No. 6923910 and NSERC Grants.
PY - 2014/1
Y1 - 2014/1
N2 - The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {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.
AB - The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {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.
KW - Hypercube
KW - Percolation
KW - Phase transition
UR - http://www.scopus.com/inward/record.url?scp=84892679735&partnerID=8YFLogxK
U2 - 10.1007/s00184-013-0473-5
DO - 10.1007/s00184-013-0473-5
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AN - SCOPUS:84892679735
SN - 0026-1335
VL - 77
SP - 23
EP - 50
JO - Metrika
JF - Metrika
IS - 1
ER -