Brillouin Zones of Integer Lattices and Their Perturbations - Archive ouverte HAL
Article Dans Une Revue SIAM Journal on Discrete Mathematics Année : 2024

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.
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Dates et versions

hal-04628684 , version 1 (01-07-2024)

Identifiants

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Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian, et al.. Brillouin Zones of Integer Lattices and Their Perturbations. SIAM Journal on Discrete Mathematics, 2024, 38 (2), pp.1784-1807. ⟨10.1137/22M1489071⟩. ⟨hal-04628684⟩
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