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

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.