Critical Polynomials In The Nonplanar And Continuum Percolation Models

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 O(10(5)) and clearly confirm the asymptotic behavior zp(c) - 1 similar to 1/root z for z -> infinity. For the continuum percolation model, we surprisingly observe that the finite-size correction in P-B is unobservable within uncertainty O(10(-5)) as long as L >= 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.