TY - JOUR
T1 - A Note About Critical Percolation on Finite Graphs
AU - Kozma, Gady
AU - Nachmias, Asaf
PY - 2011/12
Y1 - 2011/12
N2 - In this note we study the geometry of the largest component C1of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137-184, 2005). There it is shown that this component is of size n2/3, and here we show that its diameter is n1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886-1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus ℤdn(with d large and n→∞) and the Hamming cube {0,1}n.
AB - In this note we study the geometry of the largest component C1of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137-184, 2005). There it is shown that this component is of size n2/3, and here we show that its diameter is n1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886-1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus ℤdn(with d large and n→∞) and the Hamming cube {0,1}n.
KW - Critical exponents
KW - Critical percolation
KW - Intrinsic metric
KW - Triangle condition
UR - http://www.scopus.com/inward/record.url?scp=80255141923&partnerID=8YFLogxK
U2 - 10.1007/s10959-010-0283-x
DO - 10.1007/s10959-010-0283-x
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AN - SCOPUS:80255141923
SN - 0894-9840
VL - 24
SP - 1087
EP - 1096
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
IS - 4
ER -