TY - JOUR
T1 - Percolation on finite graphs and isoperimetric inequalities
AU - Alon, Noga
AU - Benjamin, Itai
AU - Stacey, Alan
PY - 2004/7
Y1 - 2004/7
N2 - 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, Komlós and Szemerédi [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.
AB - 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, Komlós and Szemerédi [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.
KW - Expander
KW - Giant component
KW - Percolation
KW - Random graph
UR - http://www.scopus.com/inward/record.url?scp=4544271923&partnerID=8YFLogxK
U2 - 10.1214/009117904000000414
DO - 10.1214/009117904000000414
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AN - SCOPUS:4544271923
VL - 32
SP - 1727
EP - 1745
JO - Annals of Probability
JF - Annals of Probability
SN - 0091-1798
IS - 3 A
ER -