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If, for fixed $\\alpha\\in(0,\\sqrt{2})$ and $\\kappa>0$, one has $\\mathbb{E}\\max_i X_i\\geq\\alpha\\sqrt{\\log N}$ and $\\mathbb{E}\\max_i X_i+\\kappa\\sqrt{\\log N}\\leq\\sqrt{2\\log N}$, then \\[ \\mathbb{P}\\left(\\max_i X_i\\geq \\mathbb{E}\\max_i X_i+\\kappa\\sqrt{\\log N}\\right) \\leq N^{-\\kappa^2/(2-\\alpha^2)+o(1)}. \\] This answers a question of Ding, Eldan and Zhai, and the exponent is attained by an equicorrelated Gaussian field. Second, for the Sherrington--Kirkpatrick m"},"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":"2607.21392","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-23T14:55:06Z","cross_cats_sorted":[],"title_canon_sha256":"bc5f0b9817477d6bbcde67c35a6ce92e6c575b001a687387b1a1ee0a98e5874e","abstract_canon_sha256":"50d66ae33601c0a1f42c5cbc259fc62ebc1d1d8d055b8b5a27c58ab732710221"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T01:24:30.030512Z","signature_b64":"7TyniiYtAhrC98SOoDlYNJRgKA7h9hLiog0sWr/ywaGcHWJ6UbynwFC3iwiHsljaUi9Z4lH0xwF3GnGZrQ5BBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3a3258860457928815cd79b15c5e8fe2d73a00ce6c6cccdbe2b860edbd22cff","last_reissued_at":"2026-07-24T01:24:30.029609Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T01:24:30.029609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moderate Deviations for Gaussian Maxima and an Entropy Proof of Critical SK Free Energy Fluctuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yiming Chen","submitted_at":"2026-07-23T14:55:06Z","abstract_excerpt":"We study two problems for Gaussian maxima. First, let $(X_1,\\ldots,X_N)$ be a centered Gaussian vector with $\\operatorname{Var}(X_i)\\leq 1$. If, for fixed $\\alpha\\in(0,\\sqrt{2})$ and $\\kappa>0$, one has $\\mathbb{E}\\max_i X_i\\geq\\alpha\\sqrt{\\log N}$ and $\\mathbb{E}\\max_i X_i+\\kappa\\sqrt{\\log N}\\leq\\sqrt{2\\log N}$, then \\[ \\mathbb{P}\\left(\\max_i X_i\\geq \\mathbb{E}\\max_i X_i+\\kappa\\sqrt{\\log N}\\right) \\leq N^{-\\kappa^2/(2-\\alpha^2)+o(1)}. \\] This answers a question of Ding, Eldan and Zhai, and the exponent is attained by an equicorrelated Gaussian field. 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