Zipf’s law in percolation

Abstract

Cluster-size distributions are examined for percolation processes on square and triangular lattices by means of the Monte Carlo simulation. Zipf's law is found to exist in the relation between cluster sizes and their ranks in the size order on finite lattices. The law is predicted to appear just at the thresholds ${\mathit{p}}_{\mathit{c}}$ for infinite lattices. The value of critical exponent \ensuremath{\tau} is derived analytically from the sizes with the law. The value is found to be \ensuremath{\tau}=2, showing the validity of the appearance of the law at ${\mathit{p}}_{\mathit{c}}$ for the lattices. \textcopyright{} 1996 The American Physical Society.