Critical polynomials in the nonplanar and continuum percolation models

Abstract

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently the critical polynomial $P_{\mathrm{B}}(p,L)$ was introduced for planar-lattice percolation models, where $p$ is the occupation probability and $L$ is the linear system size. The solution of $P_{\mathrm{B}}=0$ 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 $P_{\mathrm{B}}$, 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, $P_{\mathrm{B}}$ 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 $z{p}_c-1\sim 1/\sqrt{z}$ for $z\to \infty$. For the continuum percolation model, we surprisingly observe that the finite-size correction in $P_{\mathrm{B}}$ is unobservable within uncertainty O$(10^{-5})$ as long as $L\ge 3$. The estimated threshold number density of disks is ${\rho }_c=1.43632505(10)$, slightly below the most recent result ${\rho }_c=1.43632545(8)$ 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.

arXiv comment12 pages, 12 figures

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