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). It is argued that as N→∞ 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_ξ^* log N, where s_ξ^*(p) is a “crossover size.” The standard deviation is bounded near s_ξ^*π/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→∞. 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.