{"paper":{"title":"Thick points of the Gaussian free field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jason Miller, Xiaoyu Hu, Yuval Peres","submitted_at":"2009-02-23T03:34:33Z","abstract_excerpt":"Let $U\\subseteq\\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\\int_U\\nabla f(x)\\cdot \\nabla g(x)\\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\\in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\\sqrt{a/\\pi}\\log\\frac{1}{r}$ as $r\\to 0$. We show that for each $0\\leq a\\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $\\nu_{2-a}(T(a;U))=\\infty$ when $0<a\\leq2$ and $\\nu_2(T(0;U))=\\nu_2(U)$ almost "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.3842","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}