{"paper":{"title":"Fractal geometry of the PAM in 2D and 3D with white noise potential","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.AP","math.FA","math.MP"],"primary_cat":"math.PR","authors_text":"Jaeyun Yi, Promit Ghosal","submitted_at":"2023-03-28T15:43:10Z","abstract_excerpt":"We study the parabolic Anderson model (PAM) \\begin{equation}\n  {\\partial \\over \\partial t}u(t,x) =\\frac{1}{2}\\Delta u(t,x) + u(t,x)\\xi(x), \\quad t>0, x\\in \\mathbb{R}^d, \\quad \\text{and} \\quad\n  u(0,x) \\equiv 1, \\quad \\forall x\\in \\mathbb{R}^d,\n  \\end{equation} where $\\xi$ is spatial white noise on $\\mathbb{R}^d$ with $d \\in\\{2,3\\}$. We show that the peaks of the PAM are macroscopically multifractal. More precisely, we prove that the spatial peaks of the PAM have infinitely many distinct values and we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) of those peaks. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.16063","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/2303.16063/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"}