Unlacing hypercube percolation: a survey

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.

[1]  G. Kozma,et al.  Arm exponents in high dimensional percolation , 2009, 0911.0871.

[2]  Nicholas C. Wormald,et al.  Counting connected graphs inside-out , 2005, J. Comb. Theory, Ser. B.

[3]  Remco van der Hofstad,et al.  Expansion in ${\boldsymbol{n^{-1}}}$ for Percolation Critical Values on the $n$-cube and ${\boldsymbol{{\mathbb Z}^n}}$: the First Three Terms , 2006, Comb. Probab. Comput..

[4]  H. Kesten,et al.  Inequalities with applications to percolation and reliability , 1985 .

[5]  B. Pittel,et al.  The structure of a random graph at the point of the phase transition , 1994 .

[6]  Remco van der Hofstad,et al.  Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time , 2009 .

[7]  Tomasz Luczak,et al.  Component Behavior Near the Critical Point of the Random Graph Process , 1990, Random Struct. Algorithms.

[8]  Michael Aizenman,et al.  On the critical behavior of the magnetization in high-dimensional Ising models , 1986 .

[9]  Gordon Slade,et al.  Self-avoiding walk enumeration via the lace expansion , 2007 .

[10]  Remco van der Hofstad,et al.  Non-backtracking Random Walk , 2012, 1212.6390.

[11]  Michael Aizenman,et al.  Percolation Critical Exponents Under the Triangle Condition , 1991 .

[12]  Svante Janson,et al.  The Birth of the Giant Component , 1993, Random Struct. Algorithms.

[13]  Remco van der Hofstad,et al.  Asymptotic expansions in n−1 for percolation critical values on the n‐Cube and ℤn , 2005, Random Struct. Algorithms.

[14]  Remco van der Hofstad,et al.  Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms , 2004 .

[15]  Joel H. Spencer,et al.  Random subgraphs of finite graphs: I. The scaling window under the triangle condition , 2005, Random Struct. Algorithms.

[16]  Joel H. Spencer,et al.  Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube , 2006, Comb..

[17]  R. Hofstad,et al.  Unlacing the lace expansion: a survey to hypercube percolation , 2012, 1210.8213.

[18]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

[19]  Yoshiharu Kohayakawa,et al.  The Evaluation of Random Subgraphs of the Cube , 1992, Random Struct. Algorithms.

[20]  Roberto H. Schonmann,et al.  Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs , 2001 .

[21]  Y. Peres,et al.  Critical random graphs: Diameter and mixing time , 2007, math/0701316.

[22]  G. Slade,et al.  Mean-field critical behaviour for percolation in high dimensions , 1990 .

[23]  David Reimer,et al.  Proof of the Van den Berg–Kesten Conjecture , 2000, Combinatorics, Probability and Computing.

[24]  G. Slade,et al.  Polygons and the Lace Expansion , 2009 .

[25]  David Aldous,et al.  Brownian excursions, critical random graphs and the multiplicative coalescent , 1997 .

[26]  J. Spencer,et al.  EVOLUTION OF THE n-CUBE , 1979 .

[27]  Remco van der Hofstad,et al.  Hypercube percolation , 2012 .

[28]  Remco van der Hofstad,et al.  Random subgraphs of finite graphs : II. The lace expansion and the triangle condition , 2003 .

[30]  Dana Randall,et al.  The van den Berg-Kesten-Reimer Inequality: A Review , 1999 .

[31]  János Komlós,et al.  Largest random component of ak-cube , 1982, Comb..

[32]  G. Kozma.  Percolation on a product of two trees , 2010, 1003.5240.

[33]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[34]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[35]  Charles M. Newman,et al.  Tree graph inequalities and critical behavior in percolation models , 1984 .

[36]  Asaf Nachmias,et al.  Non-amenable Cayley graphs of high girth have $p_c < p_u$ and mean-field exponents , 2012 .

[37]  Svante Janson.  Multicyclic Components in a Random Graph Process , 1993, Random Struct. Algorithms.

[38]  Remco van der Hofstad,et al.  Random Graph Asymptotics on High-Dimensional Tori , 2005 .

[39]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[40]  Asaf Nachmias,et al.  The Alexander-Orbach conjecture holds in high dimensions , 2008, 0806.1442.

[41]  Heather J. Ruskin,et al.  Bond percolation processes in d dimensions , 1978 .

[42]  P. Erdos,et al.  On the evolution of random graphs , 1984 .