Annals of Combinatorics 3 (1999) 337-356

Baxter--Guttmann--Jensen Conjecture for Power Series in Directed Percolation Problem

M. Katori1, T. Tsuchiya1, N. Inui2, and H. Kakuno2

1Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
{katori, tomoko}

2Department of Mechanical and Intelligent Engineering, Himeji Institute of Technology, 2167 Shosha, Himeji, Hyogo 671-2201, Japan

Received December 11, 1998

AMS Subject Classification: 82C43, 60K35, 30B10, 05A19

Abstract. A probabilistic model of a flow of fluid through a random medium, percolation model, provides a typical example of statistical mechanical problems which are easy to describe but difficult to solve. While the percolation problem on undirected planar lattices is exactly solved as a limit of the Potts models, there still has been no exact solution for the directed lattices. The most reliable method to provide good approximations is a numerical estimation using finite power-series expansion data of the infinite formal power series for percolation probability. In order to calculate higher-order terms in power series, Baxter and Guttmann[6] and Jensen and Guttmann[33] proposed an extrapolation procedure based on an assumption that the correction terms, which show the difference between the exact infinite power series and approximate finite series, are expressed as linear combinations of the Catalan numbers. In this paper, starting from a brief review on the directed percolation problem and the observation by Baxter, Guttmann, and Jensen, we state some theorems in which we explain the reason why the combinatorial numbers appear in the correction terms of power series. In the proof of our theorems, we show several useful combinatorial identities for the ballot numbers, which become the Catalan numbers in a special case. These identities ensure that a summation of products of the ballot numbers with polynomial coefficients can be expanded using the ballot numbers. There is still a gap between our theorems and the Baxter--Guttmann--Jensen observation, and we also give some conjectures. As a generalization of the percolation problem on a directed planar lattice, we present two topics at the end of this paper: The friendly walker problem and the stochastic cellular automata in higher dimensions. We hope that these two topics as well as the directed percolation problem will be of much interest to researchers of combinatorics.

Keywords: directed percolation, formal power series, Catalan numbers and ballot numbers, friendly walker problem, stochastic cellular automata


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