几何指数、SLE和对数极小模型
Geometric Exponents, SLE and Logarithmic Minimal Models
摘要：
在统计力学中，可观测量通常与局部自由度有关，例如Q状态Potts模型的Q < 4不同状态或受限固体对固体模型的高度。在连续标度极限下，这些模型用有理共形场论描述，即对适当的p，p '，用极小模型M（p，p'）描述。更一般地说，在随机Loewner演化（SLE_kappa）中，可以考虑与非局部自由度相关的可观测量，例如集群的路径或边界。这导致分形维数或几何指数的值的保形尺寸中没有发现的有限集的值所允许的合理的极小模型。工作的情况下，一个环的气体与环逸度β = -2 cos（4 π/κ），我们使用蒙特卡罗模拟测量的分形维数的各种几何对象，如路径和概括的集群质量，集群外壳，外部周长和红色债券。特别是当SLE参数kappa = 4p '/p是有理数且p <p'时，我们认为几何指数与对数极小模型LM（p，p '）的无限扩展Kac表中的共形维数有关.这些理论描述了具有非局部自由度的晶格系统。我们给出了临界致密聚合物LM（1，2），临界渗流LM（2，3），对数伊辛模型LM（3，4），对数三临界伊辛模型LM（4，5）以及LM（3，5）的结果.我们的结果进行了比较严格的结果SLE_kappa，从理论物理和其他数值实验的预测。在整个过程中，我们强调SLE_kappa，几何指数和共形尺寸的基础CFTs之间的关系。
英文摘要：
In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.
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