Probabilistic investigation of phase transitions for lattice spin systems

对于长程多体相互作用的标准壁系统，用一族基本形状族“标准壁”描述了三维伊辛模型的正负相界面，并给出了它的概率分布。我们研究了界面上随机场的一个极限定理和大偏差原理。特别地，我们考虑了[o，TL]*[o，SL]中标准墙的泛函F(W)的重新标度和和外标准墙封闭区域的体积。为了研究极限定理和大偏差原理，必须获得标准墙的相关函数的性质，如聚集性。我们用簇展开的方法得到了关联函数的这些估计。这种方法有望推广到更一般的晶格系统。我们正试图将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.