Asymptotic behaviour of the lattice Green function

Abstract

The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long distance behaviour using the Laplace method applied to an integral representation involving modified Bessel functions. Our emphasis is on the decay of the massive lattice Green function in the vicinity of the massless (critical) case, and the recovery of Euclidean isotropy in the massless limit.