Probabilistic investigation of phase transitions for lattice spin systems

用一族基本形状“标准壁”来描述三维Ising模型的正负相界面，并给出了具有长程多体相互作用的标准壁系统的Gibbs态概率分布。研究了界面上随机场的一个极限定理和一个大偏差原理。特别地，我们考虑了标准壁w在[o，tL] * [o，sL]中的泛函F（w）的重新标度和以及由外部标准壁封闭的域的体积。为了研究极限定理和大偏差原理，必须得到标准壁的相关函数的性质，如聚类性质。我们通过使用一种簇展开的方法得到了这些相关函数的估计。该方法有望推广到更一般的格点系统。我们正试图把这种方法推广到更一般的情况。

An interface of plus minus phases for three dimensional Ising model is described by a family of elementary shapes called "standard walls", and its probability distribution is given as a Gibbs state for standard wall system with long range, multibody interaction. We investigated a limit theorem and a large deviation principle for random field on the interface. In particular we considered a rescaled sum of functionals F (w) for standard walls w in [o, tL] * [o, sL] and a volume of enclosed domain by outer standard walls. To investigate a limit theovem and a large deviation principle it was neccesary to obtain properties of correlation functions for standard walls such as clustering properties. We obtained these estimates for correlation functions by using a method of cluster expansion. This method is expected to extend to more general lattice system. We are trying to extend this method to more general cases.