Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices

TL;DR

This paper investigates the lattice Green's functions of face-centered cubic lattices in up to six dimensions, deriving differential equations and proving conjectures using advanced computer algebra techniques.

Contribution

It provides rigorous proofs of ODEs for 4- and 5-dimensional fcc lattices and derives the ODE for the 6-dimensional case, surpassing previous computational limitations.

Findings

Rigorous proof of ODEs for 4D and 5D fcc lattices

Derivation of the 6D fcc lattice Green's function ODE

Extension of computational methods to higher dimensions

Abstract

We study the face-centered cubic lattice (fcc) in up to six dimensions. In particular, we are concerned with lattice Green's functions (LGF) and return probabilities. Computer algebra techniques, such as the method of creative telescoping, are used for deriving an ODE for a given LGF. For the four- and five-dimensional fcc lattices, we give rigorous proofs of the ODEs that were conjectured by Guttmann and Broadhurst. Additionally, we find the ODE of the LGF of the six-dimensional fcc lattice, a result that was not believed to be achievable with current computer hardware.