Brillouin Zones of Integer Lattices and Their Perturbations
Résumé
For a locally finite set, A \subseteq \BbbR d, the kth Brillouin zone of a \in A is the region of points
x \in \BbbR d for which \| x a\| is the kth smallest among the Euclidean distances between x and the points
in A. If A is a lattice, the kth Brillouin zones of the points in A are translates of each other, and
together they tile space. Depending on the value of k, they express medium- or long-range order in
the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing
on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone
under perturbations, a linear upper bound on the number of chambers in a zone for lattices in \BbbR 2,
and the convergence of the maximum volume of a chamber to zero for the integer lattice.
Domaines
Informatique [cs]
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