The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality

[2608.15120] The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Condensed Matter > Statistical Mechanics arXiv:2608.15120 (cond-mat) [Submitted on 15 Aug 2026] Title: The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality Authors: Zhiyi Li , Kun Chen , Zhijie Fan , Youjin Deng View a PDF of the paper titled The $6-\epsilon$ Expansion for Long-Range Lee--Yang and Percolation Criticality, by Zhiyi Li and 2 other authors View PDF HTML (experimental) Abstract: The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $\epsilon'=3\sigma-d$ expansion fixes the anomalous dimension at $\eta=2-\sigma$, whereas SR percolation has $\eta_{\rm SR}<0$ near $d=6$. Sak's matching condition then places the crossover above $\sigma=2$, outside the regime in which the LR interaction dominates. In spatial dimension $d=6-\epsilon$, we formulate a perturbative expansion for the LR $\phi^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<\delta<\epsilon/3$, where $\delta = 2-\sigma$. We derive the one-loop corrections to the critical exponents $\eta$ and $\nu$, which acquire nontrivial dependence on $\epsilon$ and $\delta$. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-\epsilon$ results as $\sigma\to2$. These results support a crossover threshold $\sigma_*=2$ and remove the apparent discontinuity of $\eta$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$-state Potts universality classes with $q<2$. Comments: 7 pages, 2 figures Subjects: Statistical Mechanics (cond-mat.stat-mech) Cite as: arXiv:2608.15120 [cond-mat.stat-mech] (or arXiv:2608.15120v1 [cond-mat.stat-mech] for this version) https://doi.org/10.48550/arXiv.2608.15120 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Zhiyi Li [ view email ] [v1] Sat, 15 Aug 2026 08:40:49 UTC (45 KB) Full-text links: Access Paper: View a PDF of the paper titled The $6-\epsilon$ Expansion for Long-Range Lee--Yang and Percolation Criticality, by Zhiyi Li and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.stat-mech < prev | next > new | recent | 2026-08 Change to browse by: cond-mat 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? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) 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

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