The Largest Cluster in Subcritical Percolation

TL;DR

This paper investigates the distribution and size of the largest cluster in subcritical percolation on finite lattices, showing convergence to a Gumbel distribution and verifying results through large-scale simulations.

Contribution

It demonstrates the convergence of the largest cluster size distribution to a Gumbel distribution and introduces a renormalization group perspective for understanding finite-size effects.

Findings

Largest cluster size distribution converges to Gumbel distribution as N increases.

Mean of largest cluster grows logarithmically with lattice size N.

Monte Carlo simulations confirm theoretical predictions and finite-size scaling.

Abstract

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 $N \to \infty$ 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 like $s_\xi^* \log N$, 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 $N \to \infty$. 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.