{"paper":{"title":"Minimizing Lattice Energy and Hexagonal Crystallization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kaixin Deng, Senping Luo","submitted_at":"2024-11-26T08:12:13Z","abstract_excerpt":"Consider the energy per particle on the lattice given by $\\min_{ \\Lambda }\\sum_{ \\mathbb{P}\\in \\Lambda} \\left|\\mathbb{P}\\right|^4 e^{-\\pi \\alpha \\left|\\mathbb{P}\\right|^2 }$, where $\\alpha >0$ and $\\Lambda$ is a two dimensional lattice. We prove that for $\\alpha\\geq\\frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\\frac{\\pi}{3}}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by B\\'etermin."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17199","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.17199/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}