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(-z)) 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/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-->infinity. 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.