The Largest Cluster in Subcritical Percolation

Details The Largest Cluster in Subcritical Percolation The Largest Cluster in Subcritical Percolation

1999-05-13

arxiv.org/abs/cond-mat/9905191v4

Phys. Rev. E 62, 1660 (2000).

Physics

Condensed Matter

Statistical Mechanics

16 pages, 5 figs, expanded version to appear in Phys Rev E

Scientific paper

10.1103/PhysRevE.62.1660

The statistical behavior of the size (or mass) of the largest cluster in subcritical percolation on a finite lattice of size $N$ is investigated (below the upper critical dimension, presumably $d_c=6$). It is argued that as $N \to \infty$ the cumulative distribution function converges to the Fisher-Tippett (or Gumbel) distribution $e^{-e^{-z}}$ in a certain weak sense (when suitably normalized). The mean grows like $s_\xi^* \log N$, where $s_\xi^*(p)$ is a ``crossover size''. The standard deviation is bounded near $s_\xi^* \pi/\sqrt{6}$ with persistent fluctuations due to discreteness. These predictions are verified by Monte Carlo simulations on $d=2$ square lattices of up to 30 million sites, which also reveal finite-size scaling. The results are explained in terms of a flow in the space of probability distributions as $N \to \infty$. The subcritical segment of the physical manifold ($0 < p < p_c$) approaches a line of limit cycles where the flow is approximately described by a ``renormalization group'' from the classical theory of extreme order statistics.

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