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
AbstractWe 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.
PaperThe 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.
In the following we provide the ODEs and PDEs (in operator notation) for T2(z,y) and T3(z,y):
- ODE L5 with respect to z for T2(z,y): T2_ode_z.txt
- ODE N5 with respect to y for T2(z,y): T2_ode_y.txt
- The operator PDE3(1) for T2(z,y): T2_pde_3_1.txt
- The operator PDE3(2) for T2(z,y): T2_pde_3_2.txt
- The operator PDE2 for T2(z,y): T2_pde_2.txt
- Gröbner basis of PDE for ann(T2): T2_pde_gb.txt
- ODE L9 with respect to z for T3(z,y): T3_ode_z.txt
- ODE N9 with respect to y for T3(z,y): T3_ode_y.txt
- The operator PDE4(1) for T3(z,y): T3_pde_4_1.txt
- The operator PDE4(2) for T3(z,y): T3_pde_4_2.txt
- The operator PDE3 for T3(z,y): T3_pde_3.txt
- Gröbner basis of PDE for ann(T3): T3_pde_gb.txt