Asymptotic behaviour of the lattice Green function

TL;DR

This paper analyzes the asymptotic decay of the lattice Green function, revealing its behavior near criticality and the emergence of Euclidean isotropy, with implications for statistical physics models.

Contribution

It provides a detailed analysis of the long-distance behavior of the lattice Green function, especially near the critical massless limit, connecting to models in statistical mechanics.

Findings

Decay rate of the massive lattice Green function near criticality

Recovery of Euclidean isotropy in the massless limit

Prototype for long-distance behavior of two-point functions in statistical models

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 via a detailed analysis of 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. This provides a prototype for the expected but unproven long-distance behaviour of near-critical two-point functions in statistical mechanical models such as percolation, the Ising model, and the self-avoiding walk above their upper critical dimensions.