Martin Boundary and Elliptic Curves

I.A. Kurkova, V.A. Malyshev

1998, v.4, Issue 2, 203-272


Martin boundary is found for two-dimensional transient random walks on a plane lattice with different jumps in a finite number of other points, a half-plane and a quarter-plane. The random walks are homogeneous outside the boundary and possibly in a finite number of other points. The approach is based on the analysis of the elliptic curve defined by the jump generating function. In most cases the Martin boundary is proved to be homeomorphic to some subset of ``real'' points of this curve. In other cases the minimal Martin boundary consists of one or two points.

Keywords: random walk,Martin boundary,Riemann surface,algebraic function,saddle-point method


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