Brillouin Zones of Integer Lattices and Their Perturbations
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The k-th Brillouin zone of a point in a locally finite set in ℝ^d is the region in which the point is the k-closest in the set. If the set is a lattice, the k-th Brillouin zones of different points are translates of each other, which 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 ℝ^2, and the convergence of the maximum volume of a chamber to zero for the integer lattice.
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