The Face-Centered Cubic Lattice

Description

A lattice in $\R^d$ is given as an infinite set of points \[ \bigg\{\{\sum_{i=1}^d n_i\a_i: n_1,\dots,n_d\in\Z\bigg\} \subseteq\R^d \] for some linearly independent vectors $\a_1,\dots,\a_d\in\R^d$. The simplest instance of such a lattice is obtained by choosing $\a_i=\e_i$, the $i$-th unit vector; the result is the integer lattice~$\Z^d$. This talk deals with the family of \emph{face-centered cubic (fcc) lattices}, which are obtained from the lattice~$\Z^d$ by adding the center point of each (two-dimensional) face to the set of lattice points. The three-dimensional fcc lattice is regularly encountered in nature, for example in the atomic structure of aluminium, copper, silver, and gold. We want to study random walks on the fcc lattice in several dimensions, namely $d=3,4,5,6$. We consider walks that allow only steps to the nearest neighbors and assume that all steps are taken with the same probability. Let $p_n(\x)$ denote the probability that a random walk which started at the origin~$\0$ ends at point~$\x$ after $n$~steps. The object of interest is the probability generating function \[ P(\x;z) = \sum_{n=0}^\infty p_n(\x)z^n. \] which also is called the \emph{lattice Green's function}. It can be expressed as a $d$-dimensional integral \[ P(\x;z) = \frac{1}{\pi^d}\int_0^\pi\dots\int_0^\pi \frac{e^{i\x\cdot\k}}{1-z\lambda(\k)}\,\mathrm{d} k_1\dots\,\mathrm{d} k_d. \] where \[ \lambda(\k) = \lambda(k_1,\dots,k_d) = \sum_{\x\in\R^d} p_1(\x)e^{i\x\cdot\k} \] is the discrete Fourier transform of the single-step probability function~$p_1(\x)$. We will discuss several computer algebra approaches how to obtain a differential equation for $P(\0;z)$, the probability generating function for excursions. Our work is mainly based on two methodologies: the first is \emph{guessing} of linear recurrences and differential equations, the second is \emph{creative telescoping} in the spirit of Zeilberger's holonomic systems approach.

Period	05 Mar 2012
Event title	Functional Equations in LIMoges (FELIM)
Event type	Conference
Location	FranceShow on map