A matrix equation approach to solving recurrence relations in two-dimensional random walks

Abstract

The purpose of this paper is to provide explicit formulas for a variety of probabilistic quantities associated with anasymmetric random walk on a finite rectangular lattice with absorbing barriers.Quantities of interest include probabilities that a walker will exit the lattice onto some particular set of boundary states, the expected duration of the walk, and the expected number of visits to one state given a start in another. These quantities are shown to satisfy two-dimensional recurrence relations that are very similar in structure. In each case, the recurrence relations may be represented by matrix equations of the formX=AX+XB+C, whereAandBare tridiagonal Toeplitz matrices. The spectral properties ofAandBare investigated and used to provide solutions to this matrix equation. The solutions to the matrix equations then lead to solutions for the recurrence relations in very general cases.