Lattice Green functions: the d-dimensional face-centred cubic lattice,
d = 8, 9, 10, 11, 12

Saoud Hassani, Christoph Koutschan, Jean-Marie Maillard, Nadjah Zenine


We previously reported on a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic lattice. The method was used to generate many coefficients for d = 7 and the corresponding linear differential equation has been obtained. We show the strength and the limit of the method by producing the series and the corresponding differential equations for d = 8, 9, 10, 11. The differential Galois groups of these differential equations are shown to be symplectic for d = 8, 10 and orthogonal for d = 9, 11.


The full paper is available here.

Supplementary electronic material

Here we give the minimal-order differential operators for the lattice Green functions of the face-centred cubic lattice, for different dimensions d. The second column displays the order of the operator, the third column the maximal degree of its coefficients, and the fourth column shows the number of terms of the Taylor series that are necessary to guess some non-minimal differential operators which allow to obtain the minimal-order one as their greatest common right divisor. In the sixth column, we give the recurrence equation for the number a(n) of returning walks in the d-dimensional f.c.c. lattice; the n-th Taylor coefficient of the lattice Green function is equal to a(n)/cn, where c = 2d(d-1). Note that the operator for d = 12 is only available modulo the prime p = 231 - 1.

d order degree terms operator recurrence
3 3 5 20 fcc3_mop fcc3_rec
4 4 10 40 fcc4_mop fcc4_rec
5 6 17 88 fcc5_mop fcc5_rec
6 8 43 228 fcc6_mop fcc6_rec
7 11 68 391 fcc7_mop fcc7_rec
8 14 126 714 fcc8_mop fcc8_rec
9 18 169 999 fcc9_mop fcc9_rec
10 22 300 1739 fcc10_mop fcc10_rec
11 27 409 2464 fcc11_mop
12 32 617 3618 fcc12_mop_mod

In the following we provide the ODEs and PDEs (in operator notation) for T2(z,y) and T3(z,y):