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Brillouin Zones of Integer Lattices and Their Perturbations
Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian, Mathijs Wintraecken
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For a locally finite set, A ⊆ℝᵈ, the k-th Brillouin zone of a ∈A is the region of points x ∈ℝᵈ for which x-a is the k-th smallest among the Euclidean distances between x and the points in A. If A is a lattice, the k-th Brillouin zones of the points in A 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 ℝ², and the convergence of the maximum volume of a chamber to zero for the integer lattice.
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