Expansion of Percolation Critical Points for Hamming Graphs

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on $H(d,n)$. We show that, for $d \in \mathbb N$ fixed and $n \to \infty$, \begin{equation*} p_c^{(d)}= \dfrac{1}{m} + \dfrac{2d^2-1}{2(d-1)^2}\dfrac{1}{m^2} + O(m^{-3}) + O(m^{-1}V^{-1/3}), \end{equation*} which extends the asymptotics found in \cite{BorChaHofSlaSpe05b} by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} = O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In \cite{FedHofHolHul16a} \st{we show that} this formula is a crucial ingredient in the study of critical bond percolation on $H(d,n)$ for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erd\H{o}s-R\'enyi random graph.

[1]  Remco van der Hofstad,et al.  Progress in High-Dimensional Percolation and Random Graphs , 2017 .

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

[3]  H. Poincaré,et al.  Percolation ? , 1982 .

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

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

[6]  Johan van Leeuwaarden,et al.  Scaling limits for critical inhomogeneous random graphs with finite third moments , 2009, 0907.4279.

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

[8]  Svante Janson,et al.  On the critical probability in percolation , 2016, 1611.08549.

[9]  Adrien Joseph,et al.  The component sizes of a critical random graph with given degree sequence , 2010, 1012.2352.

[10]  A. Pakes,et al.  Some limit theorems for the total progeny of a branching process , 1971, Advances in Applied Probability.

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

[12]  L. Russo.  On the critical percolation probabilities , 1981 .

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

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

[15]  Remco van der Hofstad,et al.  Connectivity threshold for random subgraphs of the Hamming graph , 2015 .

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

[17]  Remco van der Hofstad,et al.  Random Graphs and Complex Networks , 2016, Cambridge Series in Statistical and Probabilistic Mathematics.

[18]  Remco van der Hofstad,et al.  Novel scaling limits for critical inhomogeneous random graphs , 2009, 0909.1472.

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

[20]  Asaf Nachmias,et al.  Unlacing hypercube percolation: a survey , 2013 .

[21]  Y. Peres,et al.  Component sizes of the random graph outside the scaling window , 2006, math/0610466.

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

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

[24]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

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

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

[27]  R. Durrett.  Random Graph Dynamics: References , 2006 .

[28]  Balazs Rath.  A moment-generating formula for Erdős-Rényi component sizes , 2018 .

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

[30]  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 .

[31]  Piotr Milos,et al.  Existence of a phase transition of the interchange process on the Hamming graph , 2016, Electronic Journal of Probability.

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

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

[34]  Rick Durrett,et al.  Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics) , 2006 .

[35]  Boris G. Pittel,et al.  On the Largest Component of the Random Graph at a Nearcritical Stage , 2001, J. Comb. Theory, Ser. B.

[36]  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.

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

[38]  Remco van der Hofstad,et al.  Hypercube percolation , 2012, 1201.3953.

[39]  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..

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

[41]  Thomas Spencer,et al.  Self-avoiding walk in 5 or more dimensions , 1985 .

[42]  Geoffrey Grimmett,et al.  Percolation and disordered systems , 1997 .