[2211.14365] A dichotomy theory for the height functions of the BKT transition Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2211.14365 (math) [Submitted on 25 Nov 2022 ( v1 ), last revised 25 Jul 2026 (this version, v4)] Title: A dichotomy theory for the height functions of the BKT transition Authors: Piet Lammers View a PDF of the paper titled A dichotomy theory for the height functions of the BKT transition, by Piet Lammers View PDF HTML (experimental) Abstract: This text considers the discrete height functions associated with the Berezinskii--Kosterlitz--Thouless transition (BKT) at slope zero. Our main results are as follows. * Sharpness: If the model is localised, then the two-point function (covariance) decays exponentially fast in the distance between the points. * Effective temperature gap: If the model is delocalised, then the variance grows at least as $c\log n$, where $n$ is the distance to the boundary and $c>0$ a universal constant not depending on the temperature. Thus, the effective temperature must jump from $0$ to at least $c$ at the transition point; values in the interval $(0,c)$ are forbidden. * Delocalisation at the transition point: The delocalised phase includes the transition point, in the sense that it is a closed set in the phase diagram in the appropriate topology. These results contribute to the understanding of the regime at and around the transition point which remained largely unexplored. In a follow-up paper, the sharpness derived here is used to establish that the localisation-delocalisation transition is equivalent to the BKT transition in the dual XY and Villain models. Comments: 55 pages, 26 figures; manuscript rewritten entirely for improved presentation; references added Subjects: Probability (math.PR) ; Mathematical Physics (math-ph) MSC classes: 82B20, 82B41 (Primary) 82B30 (Secondary) Cite as: arXiv:2211.14365 [math.PR] (or arXiv:2211.14365v4 [math.PR] for this version) https://doi.org/10.48550/arXiv.2211.14365 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Piet Lammers [ view email ] [v1] Fri, 25 Nov 2022 20:17:39 UTC (1,199 KB) [v2] Tue, 18 Apr 2023 17:18:06 UTC (950 KB) [v3] Thu, 30 Apr 2026 10:26:46 UTC (587 KB) [v4] Sat, 25 Jul 2026 09:40:38 UTC (606 KB) Full-text links: Access Paper: View a PDF of the paper titled A dichotomy theory for the height functions of the BKT transition, by Piet Lammers View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2022-11 Change to browse by: math math-ph math.MP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from