Largest cluster in subcritical percolation

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 $d_c=6).\mathrm{}$ It is argued that as $\overrightarrow{N}\infty$ 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 as $s_{\xi }^{*}\log N,$ where $s_{\xi }^{*}\mathrm{}\left(p\right)\mathrm{}$ 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 $d=2\mathrm{}$ 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 $\overrightarrow{N}\infty .$ The subcritical segment of the physical manifold $(0<p<\mathrm{}{p}_c)$ approaches a line of limit cycles where the flow is approximately described by a “renormalization group” from the classical theory of extreme order statistics.