Critical polynomials in the nonplanar and continuum percolation models

TL;DR

This paper extends the critical polynomial method to nonplanar and continuum percolation models, providing high-precision thresholds and confirming asymptotic behaviors with minimal finite-size effects.

Contribution

It demonstrates the effectiveness of the critical polynomial $P_B$ in estimating thresholds for nonplanar and continuum percolation models with unprecedented accuracy.

Findings

High-precision thresholds for equivalent-neighbor percolation up to z~10^5

Confirmation of the asymptotic behavior zp_c - 1 ~ 1/√z as z→∞

Minimal finite-size correction in $P_B$ for continuum percolation with L ≥ 3

Abstract

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently the critical polynomial $P_{\rm B}(p,L)$ was introduced for planar-lattice percolation models, where $p$ is the occupation probability and $L$ is the linear system size. The solution of $P_{\rm B} = 0$ can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of $P_{\rm B}$, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, $P_{\rm B}$ suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds $p_c(z)$ as a function of coordination number $z$ for equivalent-neighbor percolation with $z$ up to O$(10^5)$, and clearly confirm the asymptotic behavior $zp_c-1 \sim 1/\sqrt{z}$ for $z \rightarrow \infty$. For the continuum percolation model, we surprisingly observe that the finite-size correction in $P_{\rm B}$ is unobservable within uncertainty O$(10^{-5})$ as long as $L \geq 3$. The estimated threshold number density of disks is $\rho_c = 1.436 325 05(10)$, slightly below the most recent result $\rho_c = 1.436 325 45(8)$ of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.