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Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations
Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order [Formula: see text] . The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer Berlin Heidelberg
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7897325/ https://www.ncbi.nlm.nih.gov/pubmed/33678808 http://dx.doi.org/10.1007/s00220-020-03920-z |
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author | Glazman, Alexander Manolescu, Ioan |
author_facet | Glazman, Alexander Manolescu, Ioan |
author_sort | Glazman, Alexander |
collection | PubMed |
description | Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order [Formula: see text] . The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model. |
format | Online Article Text |
id | pubmed-7897325 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer Berlin Heidelberg |
record_format | MEDLINE/PubMed |
spelling | pubmed-78973252021-03-05 Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations Glazman, Alexander Manolescu, Ioan Commun Math Phys Article Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order [Formula: see text] . The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model. Springer Berlin Heidelberg 2021-01-07 2021 /pmc/articles/PMC7897325/ /pubmed/33678808 http://dx.doi.org/10.1007/s00220-020-03920-z Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
spellingShingle | Article Glazman, Alexander Manolescu, Ioan Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations |
title | Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations |
title_full | Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations |
title_fullStr | Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations |
title_full_unstemmed | Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations |
title_short | Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations |
title_sort | uniform lipschitz functions on the triangular lattice have logarithmic variations |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7897325/ https://www.ncbi.nlm.nih.gov/pubmed/33678808 http://dx.doi.org/10.1007/s00220-020-03920-z |
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