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-->infinity, the cumulative distribution function converges to the Fisher-Tippett (or Gumbel) distribution e(-e-xi) in a certain weak sense (when suitably normalized). The mean grows as s(xi)* log N, where s(xi)*(p) is a "crossover size.'' The standard deviation is bounded near s(xi)* pi/root 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 sires, which also reveal finite-size scaling. The results are explained in terms of a flow in the space of probability distributions as N-->infinity The subcritical segment of the physical manifold (0<p<p(c)) approaches a line of limit cycles where the Row is approximately described by a "renormalization group" from the classical theory of extreme order statistics.