Research of fluctuations of phase boundaries in large interacting systems from the probabilistic point of view

本研究的主要目的是从概率的观点来理解出现在许多相变数学模型中的相边界的涨落。本质上，我们只能对二维Widom-Rowlinson模型做这件事，作为相边界的条件中心极限定理。然而，这种类型的现象，现在很好地理解在这四年的作品约费，博迪诺和其他人。还有仍然有待了解有关这个问题，但我们知道，主要的问题是解决了。我们的第二个目标是要了解的过渡机制渗流时，底层图形没有翻译的行为组的自同构的底层图形。一个典型的问题是在图具有无限分歧分形结构的情况下。作为一个例子，在谢尔宾斯基地毯晶格中的渗流还没有得到很好的理解。我们可以证明这个模型的渗流是尖锐的。自1997年以来一直开放。逾渗转变的尖锐性被理解为：1.只有一个临界点2.在临界点以下，连通性函数呈指数衰减3。在临界点以上，无限簇是唯一的，对偶连通性相对于对偶图距离呈指数衰减。

The main aim of this research is to understand the fluctuation of phase boundaries appearing in many mathematical models of phase transitions from the probabilistic view point. Essentially, we could do it only for the Widom-Rowlinson model in two dimensions as a conditional central limit theorem for phase boundaries. However, this type of phenomenon is now well understood during these four years by the works of Ioffe, Bodineau and others. There still remains to be understood related to this problem but we understand that the main problem is solved.Our second aim was to understand the transition mechanism in percolation when the underlying graph has no translations which act as group of automorphisms of the underlying graph. A typical problem is in the case where the graph has infinitely ramified fractal structure. As an example, percolation in the Sierpinski carpet lattice has not been understood well. We could prove that percolation is sharp for this model. This has been open since 1997. The sharpness of the percolation transition is understood as1.there is only one critical point2.below the critical point, the connectivity function decays exponentially3. above the critical point, the infinite cluster is unique and the dual connectivity decays exponentially with respect to the dual graph distance.