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By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [2], we obtain some explicit formulas for $\\mathbb{E}[u(t,x)^3]$. One application of these formulas is given to show the exact phase transition for the intermittency fro"},"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":"1609.01005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-05T00:22:05Z","cross_cats_sorted":[],"title_canon_sha256":"920e0050ae994a66b7085cd5de050fe3c16bb43d598d3f054a6906b84f3b26f9","abstract_canon_sha256":"3781819c7ce0296547096be3bef91f864c3c123df93544fdb39f0ee544f4c8ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:17.473015Z","signature_b64":"PNOWXz33bh6doMV6e0mSlvlbl7Opmu2Fw1GggSxchMt9kIkF/4WaBPavllvCJsy5MnLTk7ENcRxEo7WGex87Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c91314648e3e9a75f31320214d3c89a796ab6079a1907bebaeef54ccb44a594","last_reissued_at":"2026-05-18T01:04:17.472508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:17.472508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The third moment for the parabolic Anderson model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Le Chen","submitted_at":"2016-09-05T00:22:05Z","abstract_excerpt":"In this paper, we study the {\\it parabolic Anderson model} starting from the Dirac delta initial data: \\[ \\left(\\frac{\\partial}{\\partial t} -\\frac{\\nu}{2}\\frac{\\partial^2}{\\partial x^2} \\right) u(t,x) = \\lambda u(t,x) \\dot{W}(t,x), \\qquad u(0,x)=\\delta_0(x), \\quad x\\in\\mathbb{R}, \\] where $\\dot{W}$ denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [2], we obtain some explicit formulas for $\\mathbb{E}[u(t,x)^3]$. 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