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.

arXiv comment19 pages

References in corpus (12)

• A Fast Approach to Creative Telescoping
• Lattice Green functions in all dimensions
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• The Fuchsian differential equation of the square lattice Ising model $\chi (3)$ susceptibility
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• Ising model susceptibility: Fuchsian differential equation for ${\chi }^{(4)}$ and its factorization properties
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• High order Fuchsian equations for the square lattice Ising model: ${\chi }^{(6)}$
• Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices
• Landau singularities and singularities of holonomic integrals of the Ising class
• Square lattice Ising model ${\tilde{\chi }}^{(5)}$ ODE in exact arithmetic

Cited by in corpus (6)

• Multiple-scale structures: from Faraday waves to soft-matter quasicrystals
• Impedance responses and size-dependent resonances in topolectrical circuits via the method of images
• Diagonals of rational functions and selected differential Galois groups
• Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12
• Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
• Scaling functions in the square Ising model