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 dc=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*logN, where s_ξ*(p) is a ``crossover size.'' The standard deviation is bounded near s_ξ*π/√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<pc) approaches a line of limit cycles where the flow is approximately described by a ``renormalization group'' from the classical theory of extreme order statistics.