方向性のあるパーコレーション問題における組合せ問題

抄録

This paper is aimed at a bridge between a directed percolation problem and combinational problems. Percolation theory was developed to mathematically deal with the degree of connectivity in disordered media. In particular, directed percolation in known as one of the simplest model for strongly anisotropic systems exhibiting a phase transition. A number of approaches are applied to the percolation problems, but, there are many open problems still now. The percolation probability that a given point belongs to an infinite cluster is a basic order parameter and it change drastically near the critical point. The series expansion method gives us the best values for the critical point and critical exponents. Furthermore, the solvability is also considered from the power-series data. Recently, it is found that the percolation probability is obtained by enumerating the paths of the friendly walkers. We focus on the connection between the power-series of the percolation probability and the generating function of the random walkers.