TY - JOUR

T1 - Universal critical amplitude ratios for percolation

AU - Aharony, Amnon

PY - 1980

Y1 - 1980

N2 - The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes B|t|, AF|t|2, C|t|-, and 0|t|- [where t=(pc-p)pc, p being the concentration of nonzero bonds, and +(-) stands for ppc)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios AF+AF-, C+C-, AF+B-2C+,00-, and AF+(0+)d (d is the dimensionality) are universal. Similar quantities are found for the behavior at p=pc (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in =6-d. The (extrapolated) results are compared with available information in dimensionalities d=2, <3,4, 5, with reasonable agreements. The amplitude relations become exact at d=6, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.

AB - The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes B|t|, AF|t|2, C|t|-, and 0|t|- [where t=(pc-p)pc, p being the concentration of nonzero bonds, and +(-) stands for ppc)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios AF+AF-, C+C-, AF+B-2C+,00-, and AF+(0+)d (d is the dimensionality) are universal. Similar quantities are found for the behavior at p=pc (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in =6-d. The (extrapolated) results are compared with available information in dimensionalities d=2, <3,4, 5, with reasonable agreements. The amplitude relations become exact at d=6, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.

UR - http://www.scopus.com/inward/record.url?scp=0010175628&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.22.400

DO - 10.1103/PhysRevB.22.400

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AN - SCOPUS:0010175628

SN - 0163-1829

VL - 22

SP - 400

EP - 414

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

IS - 1

ER -