Critical polynomials in the nonplanar and continuum percolation models
Wenhui Xu, Junfeng Wang, Hao Hu, Youjin Deng
Abstract
Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial ${P}_{B}(p,L)$ was introduced for planar-lattice percolation models, where $p$ is the occupation probability and $L$ is the linear system size. The solution of ${P}_{B}=0$ 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 ${P}_{B}$, 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, ${P}_{B}$ 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 $\mathrm{O}({10}^{5})$ and clearly confirm the asymptotic behavior $z{p}_{c}\ensuremath{-}1\ensuremath{\sim}1/\sqrt{z}$ for $z\ensuremath{\rightarrow}\ensuremath{\infty}$. For the continuum percolation model, we surprisingly observe that the finite-size correction in ${P}_{B}$ is unobservable within uncertainty $\mathrm{O}({10}^{\ensuremath{-}5})$ as long as $L\ensuremath{\ge}3$. The estimated threshold number density of disks is ${\ensuremath{\rho}}_{c}=1.436\phantom{\rule{0.16em}{0ex}}325\phantom{\rule{0.16em}{0ex}}05(10)$, slightly below the most recent result ${\ensuremath{\rho}}_{c}=1.436\phantom{\rule{0.16em}{0ex}}325\phantom{\rule{0.16em}{0ex}}45(8)$ 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.