Exact results at the 2-D percolation point
P. Kleban (1), R. M. Ziff (2) ((1) Laboratory for Surface Science and, Technology & Department of Physics, Astronomy, University of Maine, Orono,, ME, (2) Department of Chemical Engineering, University of Michigan, Ann, Arbor, MI)
TL;DR
This paper derives exact formulas for cluster counts and cumulants at the 2-D percolation point, confirming predictions with high-precision simulations and highlighting their universality and shape dependence.
Contribution
It provides the first exact expressions for excess cluster numbers and cumulants at the 2-D percolation threshold, validated by simulations.
Findings
Exact formulas match simulation data
Cluster quantities are universal and shape-dependent
Higher-order corrections lack fractional power dependence
Abstract
We derive exact expressions for the excess number of clusters b and the excess cumulants b_n of a related quantity at the 2-D percolation point. High-accuracy computer simulations are in accord with our predictions. b is a finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula for the number of clusters per site n_c in the infinite system limit; the bn correct bulk cumulants. b and b_n are universal, and thus depend only on the system's shape. Higher-order corrections show no apparent dependence on fractional powers of the system size.
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Taxonomy
TopicsTheoretical and Computational Physics · Stochastic processes and statistical mechanics · Random Matrices and Applications