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

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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.