Hypercube percolation

We study bond percolation on the Hamming hypercube {0,1} around the critical probability pc . It is known that if p = pc (1+O(2−m/3)), then with high probability the largest connected component C1 is of size Θ(2 ) and that this quantity is non-concentrated. Here we show that for any sequence εm such that εm = o(1) but εm ≫ 2−m/3 percolation on the hypercube at pc (1+εm ) has |C1| = (2+o(1))εm 2 and |C2| = o(εm 2 ) , with high probability, where C2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].

[1]  M. Aizenman,et al.  Sharpness of the phase transition in percolation models , 1987 .

[2]  Nicholas C. Wormald,et al.  The mixing time of the giant component of a random graph , 2006, Random Struct. Algorithms.

[3]  Remco van der Hofstad,et al.  Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time , 2009, 0903.4279.

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

[5]  Béla Bollobás,et al.  Random Graphs , 1985 .

[6]  Stefan Kuhr,et al.  Department of Mathematics and Computer Science , 2002 .

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

[8]  Van den Berg,et al.  On a Combinatorial Conjecture Concerning Disjoint Occurrences of Events , 1987 .

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

[10]  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..

[11]  Yuval Peres,et al.  Critical percolation on random regular graphs , 2007, Random Struct. Algorithms.

[12]  Béla Bollobás,et al.  Bootstrap Percolation in High Dimensions , 2009, Combinatorics, Probability and Computing.

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

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

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

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

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

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

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

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

[21]  Remco van der Hofstad,et al.  Random subgraphs of the 2D Hamming graph: the supercritical phase , 2008, 0801.1607.

[22]  M. Penrose.  Random minimal spanning tree and percolation on the N -cube , 1998 .

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

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

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

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

[27]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[28]  N. Alon,et al.  Non-backtracking random walks mix faster , 2006, math/0610550.

[29]  Bruce A. Reed,et al.  The evolution of the mixing rate of a simple random walk on the giant component of a random graph , 2008, Random Struct. Algorithms.

[30]  Joel H. Spencer,et al.  The second largest component in the supercritical 2D Hamming graph , 2010, Random Struct. Algorithms.

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

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

[33]  Béla Bollobás,et al.  The scaling window of the 2‐SAT transition , 1999, Random Struct. Algorithms.

[34]  Y. Peres,et al.  Mixing time of near-critical random graphs , 2009, 0908.3870.

[35]  Asaf Nachmias,et al.  A Note About Critical Percolation on Finite Graphs , 2009, 0909.4351.

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

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

[38]  Asaf Nachmias,et al.  Mean-Field Conditions for Percolation on Finite Graphs , 2007, 0709.1719.

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

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

[41]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[42]  Béla Bollobás,et al.  Majority Bootstrap Percolation on the Hypercube , 2007, Combinatorics, Probability and Computing.

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

[44]  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.

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