Universality of finite-size corrections to the number of critical percolation clusters

TL;DR

This study demonstrates that the finite-size correction to the number of critical percolation clusters is a universal quantity depending only on system shape, confirmed through extensive Monte Carlo simulations and theoretical predictions.

Contribution

It provides the first verification of theoretical predictions for fluctuations in the number of clusters at criticality, establishing universality across different lattice types and percolation models.

Findings

Finite-size correction is universal and shape-dependent.

Monte Carlo results match theoretical predictions.

Fluctuation predictions are experimentally verified.

Abstract

Monte-Carlo simulations on a variety of 2d percolating systems at criticality suggest that the excess number of clusters in finite systems over the bulk value of nc is a universal quantity, dependent upon the system shape but independent of the lattice and percolation type. Values of nc are found to high accuracy, and for bond percolation confirm the theoretical predictions of Temperley and Lieb, and Baxter, Temperley, and Ashley, which we have evaluated explicitly in terms of simple algebraic numbers. Predictions for the fluctuations are also verified for the first time.