论文标题
非平面和连续渗透模型中的临界多项式
Critical polynomials in the nonplanar and continuum percolation models
论文作者
论文摘要
自引入渗透模型以来，已经对精确或确切的阈值进行了深入的研究。最近，针对平面晶格渗透模型引入了关键的多项式$ p _ {\ rm b}（p，l）$，其中$ p $是职业概率，$ l $是线性系统大小。 $ p _ {\ rm b} = 0 $的解决方案可以重现所有已知的确切阈值，并导致对未解决的平面晶格模型阈值的前所未有的估计。在二维中，假设$ p _ {\ rm b} $的普遍性，我们使用它来研究非平面晶格模型，即，通过Monte Carlo Mimulations Mimulations Mimulations Mimulations和Fialite-Sparite-Spaking分析分析，相同的渗透性磁盘的同等纽约晶格粘结式渗透以及相同的可渗透磁盘的连续渗透。发现与其他数量相比，$ p _ {\ rm b} $在有限尺寸的校正中遭受的损失要少得多。结果，我们获得了一系列高精度阈值$ p_c（z）$，是协调数$ z $的函数，$ z $的等效纽伯特渗透性，$ z $ to $ $ $ $（10^5）$，并清楚地确认了$ zp_c-cp_c-cp_c-1 \ sim 1/\ sim 1/\ sqrt $ $ sqrt $ $ zp_c-ccy $ zp_c-ccrivation $对于连续渗透模型，我们出人意料地观察到$ p _ {\ rm b} $中的有限尺寸校正在不确定性o $ $ $（10^{ - 5}）$中是无法观察到的，只要$ l \ geq 3 $。圆盘的估计阈值数量密度为$ρ_C= 1.436 325 05（10）$，略低于最新结果$ρ_C= 1.436 325 45（8）$的MERTENS和MOORE通过其他方式获得。我们的工作表明，关键的多项式方法可以成为研究统计力学中非平面和连续体系统的强大工具。
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 $ρ_c = 1.436 325 05(10)$, 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.
