TY - JOUR
T1 - Mean-field conditions for percolation on finite graphs
AU - Nachmias, Asaf
N1 - Funding Information:
Keywords and phrases: percolation, random graphs, giant component, scaling window 2000 Mathematics Subject Classification: 60K35, 60C05, 05C80 Research supported in part by NSF grant #DMS-0605166.
PY - 2009/12
Y1 - 2009/12
N2 - Let {Gn} be a sequence of finite transitive graphs with vertex degree d=d(n) and |Gn| =n. Denote by pt(v, v) the return probability after t steps of the non-backtracking random walk on Gn. We show that if pt(v, v) has quasi-random properties, then critical bond-percolation on Gn behaves as it would on a random graph. More precisely, if then the size of the largest component in p-bond-percolation with is roughly n2/3. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n-1/3 around the critical probability. A consequence of our theorems is that if {Gn} is a transitive expander family with girth at least then {Gn} has the above scaling window around. In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [LuPS] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.
AB - Let {Gn} be a sequence of finite transitive graphs with vertex degree d=d(n) and |Gn| =n. Denote by pt(v, v) the return probability after t steps of the non-backtracking random walk on Gn. We show that if pt(v, v) has quasi-random properties, then critical bond-percolation on Gn behaves as it would on a random graph. More precisely, if then the size of the largest component in p-bond-percolation with is roughly n2/3. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n-1/3 around the critical probability. A consequence of our theorems is that if {Gn} is a transitive expander family with girth at least then {Gn} has the above scaling window around. In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [LuPS] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.
KW - Giant component
KW - Percolation
KW - Random graphs
KW - Scaling window
UR - http://www.scopus.com/inward/record.url?scp=71049154050&partnerID=8YFLogxK
U2 - 10.1007/s00039-009-0032-4
DO - 10.1007/s00039-009-0032-4
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AN - SCOPUS:71049154050
SN - 1016-443X
VL - 19
SP - 1171
EP - 1194
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 4
ER -