Bond and Site Percolation in Three Dimensions

TL;DR

This paper presents high-precision simulations of bond and site percolation on a cubic lattice, accurately estimating critical thresholds, exponents, and universal amplitudes, and analyzing finite-size corrections.

Contribution

The study provides the most precise estimates to date of percolation thresholds, critical exponents, and universal quantities in three-dimensional percolation models.

Findings

Percolation thresholds: $p_c({\rm bond})=0.24881182(10)$, $p_c({\rm site})=0.3116077(2)$

Critical exponents: $1/\nu=1.1410(15)$, $\beta/\nu=0.47705(15)$, $d_{\rm min}=1.3756(3)$

Finite-size correction behavior differs from initial expectations, with some corrections governed by an exponent of approximately -2.

Abstract

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512, and estimate the percolation thresholds to be $p_c ({\rm bond})=0.248\,811\,82(10)$ and $p_c ({\rm site})=0.311\,607\,7(2)$. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\nu =1.141\,0(15)$, $\beta/\nu=0.477\,05(15)$, the leading correction exponent $y_i =-1.2(2)$, and the shortest-path exponent $d_{\rm min}=1.375\,6(3)$. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\approx -2$, rather than $y_i \approx -1.2$.