Universal critical amplitude ratios for percolation

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95 Scopus citations


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 p<pc (p>pc)] 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.

Original languageEnglish
Pages (from-to)400-414
Number of pages15
JournalPhysical Review B
Issue number1
StatePublished - 1 Jan 1980
Externally publishedYes

ASJC Scopus subject areas

  • Condensed Matter Physics


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