Percolation on finite graphs and isoperimetric inequalities

Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p. The method from Ajtai, Komlos and Szemeredi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.

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

[2]  Igor Pak,et al.  Percolation on Finite Cayley Graphs , 2002, RANDOM.

[3]  A. Wigderson,et al.  ENTROPY WAVES, THE ZIG-ZAG GRAPH PRODUCT, AND NEW CONSTANT-DEGREE , 2004, math/0406038.

[4]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[5]  R. Lyons.  Phase transitions on nonamenable graphs , 1999, math/9908177.

[6]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[7]  L BergvandenJ.,et al.  Random sampling for the monomer-dimer model on a lattice. Probabilistic techniques in equilibrium and nonequilibrium statistical physics , 1999 .

[8]  O. Schramm,et al.  Indistinguishability of Percolation Clusters , 1998, math/9811170.

[9]  R. Schonmann,et al.  Domination by product measures , 1997 .

[10]  I. Benjamini,et al.  Percolation Beyond $Z^d$, Many Questions And a Few Answers , 1996 .

[11]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[12]  Eli Upfal.  Tolerating a Linear Number of Faults in Networks of Bounded Degree , 1994, Inf. Comput..

[13]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[14]  Yoshiharu Kohayakawa,et al.  Percolation in High Dimensions , 1994, Eur. J. Comb..

[15]  Eli Upfal.  Tolerating linear number of faults in networks of bounded degree , 1992, PODC '92.

[16]  W. Woess,et al.  The Dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality , 1992 .

[17]  Geoffrey Grimmett,et al.  On the geometry of random Cantor sets and fractal percolation , 1992 .

[18]  László Babai,et al.  Local Expansion of Ssymmetrical Graphs , 1992, Comb. Probab. Comput..

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

[20]  Noga Alon,et al.  A parallel algorithmic version of the local lemma , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

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

[22]  Andrzej Rucinski,et al.  Random Graphs , 2018, Foundations of Data Science.

[23]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

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

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

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

[27]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .