ar X iv : c on d - m at / 9 40 80 97 v 1 3 1 A ug 1 99 4 Lattice Green Function ( at 0 ) for the 4 d Hypercubic Lattice

The generating function for recurrent Polya walks on the four dimensional hypercubic lattice is expressed as a Kampé-de-Fériet function. Various properties of the associated walks are enumerated. Lattice statistics play an important role in many areas of chemistry and statistical physics. Random lattice walks have been investigated extensively, both numerically and analytically, and explicit analytic expressions for recurrent lattice walks (those that end where they began) are available for several lattices in one, two and three dimensions [1,2]. The interest in this problem follows from the numerous connections between random walks and other problems in mathematical physics. They form the basis of certain proofs in the problem of self-avoiding walks [3], they are a key component in the study of lattice dynamics [4], and they have an extensive following in the mathematical literature [1,3]. Furthermore, d-dimensional lattice Green functions have recently [5] been shown to underlie the theory of d-dimensional staircase polygons and to be related to the generating function for d-dimensional multinomial coefficients. The aim of this note is to extend this list to four dimensions by summarizing a study of the lattice Green function