The four-dimensional Anderson model: a case study for critical SPDEs

[2607.10105] The four-dimensional Anderson model: a case study for critical SPDEs Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2607.10105 (math) [Submitted on 11 Jul 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: The four-dimensional Anderson model: a case study for critical SPDEs Authors: Yu Deng , Hao Shen View PDF HTML (experimental) Abstract: We study the weakly coupled elliptic Anderson model with spatial white noise on the four-dimensional torus, which provides a basic example of a critical SPDE requiring renormalization at arbitrarily high orders. With coupling $\lambda |\log\varepsilon|^{-\frac12}$ where $\lambda>0$ is sufficiently small, we prove that the Green's function of the corresponding random Schrödinger operator, suitably centered and rescaled, converges to a centered Gaussian random field with explicit covariance. The main difficulty is that, for such critical models, one must expand up to order $|\log\varepsilon|$, while the perturbative expansion contains factorially many pairings and a growing number of renormalization terms. To overcome this, we construct a truncated renormalized parametrix and prove sharp high-order bounds for its remainder. A central ingredient is a multiscale analysis based on a new version of Hepp trees, combined with new estimates for summations over permutations. These estimates reveal a precise balance between logarithmic losses from scale summation and factorial gains from the structure of primitive pairings. The methods developed here are intended as a first step toward a general theory for critical SPDEs with weak couplings. Comments: 45 pages. Fixed some typos. Added link to Lean 4 autoformalization Subjects: Probability (math.PR) ; Mathematical Physics (math-ph); Analysis of PDEs (math.AP) Cite as: arXiv:2607.10105 [math.PR] (or arXiv:2607.10105v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2607.10105 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hao Shen [ view email ] [v1] Sat, 11 Jul 2026 03:45:48 UTC (55 KB) [v2] Tue, 25 Aug 2026 01:47:42 UTC (55 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-07 Change to browse by: math math-ph math.AP math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

Related documents

• The four-dimensional Anderson model: a case study for critical SPDEs#670681
• Review cuprates -- Anderson's unhappy electrons and their fate#971646
• The conductivity matrix at the topological phase transition#926712
• Reevaluating Quantum Geometric Criteria for Itinerant Magnetic Instabilities#1052130
• Non-standard quantum algebra Uh(sl(2, R)) and h-Dicke states#233282