Random walks and harmonic functions on infinite planar graphs using square tilings. (English) Zbl 0862.60053

Benjamini, Itai; Schramm, Oded

Ann. Probab. 24, No. 3, 1219-1238 (1996).

Summary: We study a wide class of transient planar graphs, through a geometric model given by a square tiling of a cylinder. For many graphs, the geometric boundary of the tiling is a circle and is easy to describe in general. The simple random walk on the graph converges (with probability 1) to a point in the geometric boundary. We obtain information on the harmonic measure and estimates on the rate of convergence. This allows us to extend results we previously proved for triangulations of a disk.

MSC:

60G50	Sums of independent random variables; random walks
60J45	Probabilistic potential theory
52C20	Tilings in \(2\) dimensions (aspects of discrete geometry)

Keywords:

planar graphs; random walks; harmonic measure; Dirichlet problem