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 (10⁵) and clearly confirm the asymptotic behavior z p_c-1 ∼1 /√{z } for z →∞ . 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 ρ_c=1.436 325 05 (10 ) , slightly below the most recent result ρ_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.