Lattice Green Functions: the seven-dimensional face-centred cubic lattice

TL;DR

This paper develops a recursive method to derive the lattice Green function for a seven-dimensional face-centred cubic lattice, analyzing its differential equation, Galois group, and asymptotic return probability behavior.

Contribution

It introduces a recursive approach to generate Green functions for high-dimensional fcc lattices and determines the minimal differential equation and its algebraic properties.

Findings

Derived the minimal order differential equation for 7D fcc lattice Green function.

Identified the differential Galois group as a subgroup of SO(11,C).

Showed the return probability decreases as d^{-2} with increasing dimension.

Abstract

We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d =7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional face-centred cubic (fcc) lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in $SO(11, \mathbb{C})$. This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a "decomposition" of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as $d^{-2}$ as the dimension d goes to infinity.