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In the case $E[\\nu\\ln(\\nu)]<\\infty$, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to $\\ln(m)$ (\\cite{hawkes}, \\cite{lpp95}). When $E[\\nu \\ln(\\nu)]=\\infty$, we show that the dimension drops to $0$. This answers a question of Lyons, Pemantle and Peres \\cite{LyPemPer97}."},"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":"1101.1816","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-10T13:54:38Z","cross_cats_sorted":[],"title_canon_sha256":"727cb751c7cf0302aefa029761d668c99de491962ba7717206629c5e553668bf","abstract_canon_sha256":"772ca0ce818dc9c754201596d7748901f7d28a8178aa780cfca812efac861c69"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:51.738341Z","signature_b64":"ERkU3/pK4QZcDCcrziAAtw85m4S+UnrTwn3/B2mvAoyws+VvoRQaRWxUx1msKmPIivxqWJXZE49tbDfTe3npCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4bb4e84c453398b1813591f17eca17b57fc933cdf43a95d8a77c072d1cec3bd","last_reissued_at":"2026-05-18T04:31:51.737717Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:51.737717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The uniform measure on a Galton-Watson tree without the XlogX condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"elie aidekon","submitted_at":"2011-01-10T13:54:38Z","abstract_excerpt":"We consider a Galton--Watson tree with offspring distribution $\\nu$ of finite mean. 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