{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/brillouin-zones-of-integer-lattices-and-their","title":"Brillouin Zones of Integer Lattices and Their Perturbations","arxiv_id":"2204.01077","date":"2022-04-03","proceeding":null,"authors":["Herbert Edelsbrunner","Alexey Garber","Mohadese Ghafari","Teresa Heiss","Morteza Saghafian","Mathijs Wintraecken"],"abstract":"For a locally finite set, $A \\subseteq \\mathbb{R}^d$, the $k$-th Brillouin zone of $a \\in A$ is the region of points $x \\in \\mathbb{R}^d$ 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 $\\mathbb{R}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.","url_abs":"https://arxiv.org/abs/2204.01077v2","url_pdf":"https://arxiv.org/pdf/2204.01077v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"brillouin-zones-of-integer-lattices-and-their","repo_url":"https://github.com/almaho/brillouin_zones_integer_lattices_perturbations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}