{"paper":{"title":"The four-dimensional Anderson model: a case study for critical SPDEs","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.PR","authors_text":"Hao Shen, Yu Deng","submitted_at":"2026-07-11T03:45:48Z","abstract_excerpt":"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 Schr\\\"odinger operator, suitably centered and rescaled, converges to a centered Gaussian random field with explicit covariance.\n  The main difficulty is that, for such critical models, one must expand up to order $|\\log\\varepsilon|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10105","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/2607.10105/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"}