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Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices
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.Comment: 16 pages, final versio