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

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2015-01, Vol.48 (3), p.35205
• Main Authors: Zenine, N, Hassani, S, Maillard, J M
• Format: Article
• Language: English
• Subjects: apparent singularities; face-centred cubic lattice; Fuchsian ODE; Landau conditions; lattice Green function; long series expansions; Mathematics; Physics; return probability
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/48/3/035205

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 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 . 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.