Abstract
Using a direct position-space renormalization-group approach we study percolation clusters in the limits → ∞, wheres is the number of occupied elements in a cluster. We do this by assigning a fugacityK per cluster element; asK approaches a critical valueK c , the conjugate variables → ∞. All exponents along the path (K−K c ) → 0 are then related to a corresponding exponent along the paths → ∞. We calculate the exponent ρ, which describes how the radius of ans-site cluster grows withs at the percolation threshold, in dimensionsd=2, 3. Ind=2 our numerical estimate of ρ=0.52±0.02, obtained from extrapolation and from cell-to-cell transformation procedures, is in agreement with the best known estimates. We combine this result with previous PSRG calculations for the connectedness-length exponent ν, to make an indirect test of cluster-radius scaling by calculating the scaling function exponent σ using the relation σ=ρ/ν. Our result for σ is in agreement with direct Monte-Carlo calculations of σ, and thus supports the cluster-radius scaling assumption. We also calculate ρ ind=3 for both site and bond percolation, using a cell of linear sizeb=2 on the simple-cubic lattice. Although the result of such small-cell calculations are at best only approximate, they nevertheless are consistent with the most recent numerical estimates.
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Supported in part by grants from ARO and ONR
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Family, F., Reynolds, P.J. Radius of clusters at the percolation threshold: A position space renormalization group study. Z. Physik B - Condensed Matter 45, 123–128 (1981). https://doi.org/10.1007/BF01293325
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DOI: https://doi.org/10.1007/BF01293325