Cluster densities at 2D critical points in rectangular geometries
Cluster densities at 2D critical points in rectangular geometries
复制标题
矩形几何中二维临界点的簇密度
DOI：
--
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
R. Ziff
中科院分区：
文献类型：
--
作者：
Jacob J. H. Simmons;P. Kleban;S. M. Flores;R. Ziff
利用手性六点相关函数的完整计算[1]，在任意矩形的四角和内部的点z = x + i y处有四个ϕ 1,2算子，对于任意中心电荷（等效的SLE参数κ > 0），我们计算了渗透（κ = 6）和许多其他二维临界点的各种兴趣量。特别地，我们使用C来指定临界簇在z处的密度，这些临界簇被限制接触矩形的一个或两个垂直边，这些边是“有线的”，即被限制在一个簇中，水平边是自由的。这些量探测各种簇构型的结构，包括那些对交叉概率有贡献的簇构型。我们首先研究了临界O(n)环模型在高密度和低密度相中的边界条件对C的影响，以及临界q态Potts模型中的Fortuin-Kasteleyn （FK）和自旋团簇。然后，库仑气体分析允许我们根据[]中计算的共形块计算具有各种条件的簇密度。并给出了将Cardy水平交叉概率推广到这些模型的显式公式（使用先前已知的结果）。这些解被用来推广先前的结果，证明高阶相关函数分解到所提到的关键系统。给出了在自由边界条件下横向跨越矩形临界渗滤团密度的显式公式。给出了各种模型解中超几何函数的简化。高精度模拟验证了这些预测的渗透和Q = 2和3状态波茨模型，包括FK和自旋簇。本文还验证了自由边界条件下渗流中交叉簇密度的计算公式。
Making use of the complete calculation [1] of the chiral six-point correlation function with the four ϕ1, 2 operators at the corners of an arbitrary rectangle and the point z = x + i y in the interior, for arbitrary central charge (equivalently, SLE parameter κ > 0), we calculate various quantities of interest for percolation (κ = 6) and many other two-dimensional critical points. In particular, we use C to specify the density at z of critical clusters conditioned to touch either or both vertical sides of the rectangle, with these sides ‘wired’, i.e. constrained to be in a single cluster, and the horizontal sides free. These quantities probe the structure of various cluster configurations, including those that contribute to the crossing probability. We first examine the effects of boundary conditions on C for the critical O(n) loop models in both high- and low-density phases and for both Fortuin–Kasteleyn (FK) and spin clusters in the critical Q-state Potts models. A Coulomb gas analysis then allows us to calculate the cluster densities with various conditionings in terms of the conformal blocks calculated in []. Explicit formulas generalizing Cardy’s horizontal crossing probability to these models (using previously known results) are also presented. These solutions are employed to generalize previous results demonstrating factorization of higher order correlation functions to the critical systems mentioned. An explicit formula for the density of critical percolation clusters that cross a rectangle horizontally with free boundary conditions is also given. Simplifications of the hypergeometric functions in our solutions for various models are presented. High-precision simulations verify these predictions for percolation and for the Q = 2- and 3-state Potts models, including both FK and spin clusters. Our formula for the density of crossing clusters in percolation with free boundary conditions is also verified.
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