Critical percolation on random regular graphs

The behavior of the random graph G(n,p) around the critical probability pc = $ {1 \over n} $ is well understood. When p = (1 + O(n1-3))pc the components are roughly of size n2-3 and converge, when scaled by n-2-3, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1 - e(n))pc with e(n)n1-3 →∞ (the subcritical regime) the largest component is concentrated around 2e-2 log(e3n). When p = (1 + e(n))pc with e(n)n1-3 →∞ (the supercritical regime), the largest component is concentrated around 2en and a duality principle holds: other component sizes are distributed as in the subcritical regime. Itai Benjamini asked whether the same phenomenon occurs in a random d-regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359–1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non-critical regimes) extends to random d-regular graphs. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010

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

[2]  Tomasz Łuczak.  Component behavior near the critical point of the random graph process , 1990 .

[3]  Edward M. Wright,et al.  The number of connected sparsely edged graphs , 1977, J. Graph Theory.

[4]  Boris Pittel,et al.  Edge percolation on a random regular graph of low degree , 2008, 0808.3516.

[5]  F. Spitzer.  Principles Of Random Walk , 1966 .

[6]  Anders Martin-Löf,et al.  Symmetric sampling procedures, general epidemic processes and their threshold limit theorems , 1986, Journal of Applied Probability.

[7]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[8]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

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

[10]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

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

[12]  Richard M. Karp,et al.  The Transitive Closure of a Random Digraph , 1990, Random Struct. Algorithms.

[13]  Remco van der Hofstad,et al.  Random subgraphs of finite graphs: I. The scaling window under the triangle condition , 2003 .

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

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

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

[17]  Nikolaos Fountoulakis,et al.  Percolation on Sparse Random Graphs with Given Degree Sequence , 2007, Internet Math..

[18]  Edward A. Bender,et al.  The asymptotic number of non-negative integer matrices with given row and column sums , 1974, Discret. Math..

[19]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[20]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[21]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[22]  Andreas Goerdt.  The giant component threshold for random regular graphs with edge faults H. Prodinger , 2001, Theor. Comput. Sci..

[23]  Brendan D. McKay,et al.  The Asymptotic Number of Labeled Connected Graphs with a Given Number of Vertices and Edges , 1990, Random Struct. Algorithms.

[24]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

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

[26]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[27]  Asaf Nachmias,et al.  The critical random graph, with martingales , 2005 .

[28]  Noga Alon,et al.  Percolation on finite graphs and isoperimetric inequalities , 2004 .

[29]  B. Bollobás.  The evolution of random graphs , 1984 .