Hsu Hsiao-Ping, Nadler Walter, Grassberger Peter
John-von-Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jun;71(6 Pt 2):065104. doi: 10.1103/PhysRevE.71.065104. Epub 2005 Jun 24.
Lattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of two-dimensional site animals on square and triangular lattices in nontrivial geometries. The simulations are done with the pruned-enriched Rosenbluth method (PERM) algorithm, which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent theta (Z(N) approximately micro(N)N(-theta)). In particular, we studied animals grafted to the tips of wedges with a wide range of angles alpha, to the tips of cones (wedges with the sides glued together), and to branching points of Riemann surfaces. The latter can either have k sheets and no boundary, generalizing in this way cones to angles alpha>360 degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, theta approximately 1/alpha , only for small angles (alpha << 2pi) , while theta approximately = const-alpha/2pi for alpha << 2pi. These scalings hold both for wedges and cones. A heuristic (nonconformal) argument for the behavior at large alpha is given, and comparison is made with critical percolation.
格点动物是统计力学中少数违反共形不变性的临界模型之一。我们在此展示了在非平凡几何形状的正方形和三角形晶格上对二维格点动物的模拟。模拟是使用修剪富集罗森布鲁斯方法（PERM）算法进行的，该算法能对配分函数给出非常精确的估计，从而得到熵指数θ的精确值（Z(N)近似于μ(N)N^(-θ)）。特别地，我们研究了嫁接到具有各种角度α的楔形尖端、圆锥尖端（侧面胶合在一起的楔形）以及黎曼曲面分支点的格点动物。后者要么有k个叶且无边界，以这种方式将圆锥推广到角度α>360度，要么可以有边界，推广楔形。我们发现，仅对于小角度（α << 2π）才有共形不变性行为，θ近似为1/α，而对于α << 2π，θ近似等于常数 - α/2π。这些标度关系对楔形和圆锥都成立。给出了关于大α时行为的一个启发式（非共形）论证，并与临界渗流进行了比较。
