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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0902.3842","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-23T03:34:33Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d516c74719e3d8dbc04de791c66f985a2f383f35e203b94e218e9c7fd28a89ce","abstract_canon_sha256":"906143bdc538bef9b3c097e3fe41e29b57e0984136c674cf3d0f863ee97202a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:59.641375Z","signature_b64":"fZMclulp43id/gKM3NyIewGQrXLaxuSKOI8ziTXMZ8YYgEd0ZbnVx/dSZoUOpOUpyii1/V7DVbklqmhsxQkKAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bddc18483dc6e932dc717fcdcabf35f26e0a1ff48b23a627aea941dbaf32ebf","last_reissued_at":"2026-05-18T04:39:59.640739Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:59.640739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\int_U\\nabla f(x)\\cdot \\nabla g(x)\\,dx$. 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