Recurrence and transience of the edge graph of a tiling of the euclidean plane

Suppose that Γ is an infinite, connected graph such that every vertex V has finite degree d v . Let #7B-V and #7B-E denote the vertex and edge set of Γ respectively. It is a standard procedure to associate with Γ a Markov chain with state space #7B-V and probability d V −1 of moving from V to any neighbour V'. The graph Γ is called recurrent or transient depending on whether the associated Markov chain is recurrent or transient. The main result of this paper states that the edge graph of a quasi normal tiling of the plane is recurrent. This result can be viewed as an extension of Polya's theorem on the recurrence of the square lattice of the plane