Non-Gaussian Limiting Behavior of the Percolation Threshold in a Large System

Abstract:

We study short-range percolation models. In a finite box we define the percolation threshold as a random variable obtained from a stochastic procedure used in actual numerical calculations, and study the asymptotic behavior of these random variables as the size of the box goes to infinity. We formulate very general conditions under which in two dimensions rescaled threshold variables cannot converge to a Gaussian and determine the asymptotic behavior of their second moments in terms of a widely used definition of correlation length. We also prove that in all dimensions the finite-volume percolation thresholds converge in probability to the percolation threshold of the infinite system. The convergence result is obtained by estimating the rate of decay of the limiting distribution function's tail in terms of the correlation length exponent ν. The proofs use exponential estimates of crossing probabilities. Substantial parts of the proofs apply in all dimensions.