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For N \\to \\infty, this jpdf is peaked around M = \\sqrt{2N} and \\tau_M = 1/2, while the typical fluctuations behave for large N like M - \\sqrt{2N} \\propto s N^{-1/6} and \\tau_M - 1/2 \\propto w N^{-1/3} where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE)"},"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":"1203.1658","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-03-07T23:27:36Z","cross_cats_sorted":["cond-mat.dis-nn","cond-mat.stat-mech","math.MP","math.PR"],"title_canon_sha256":"5d28214e1996d7183013a14dcc23d77f21539bc3cc8de29a51671e3c20a36fae","abstract_canon_sha256":"c148496af5b4865d5cce582fa937abfc8a1c8abedcc5edaa679791f5d406e544"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:13.317984Z","signature_b64":"E0TgyHbGSkkg02QFnZueXPg2tKHqeTq8BQTYX68ELnKpeoCIrO/gQcMqBikRvrWW4pLtnujxbKuoTyWUcXodDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d842ddf67ad7ec3532c3bc5cde779df595e8ea6140cdb4d0a0b8aa0b3a182ed","last_reissued_at":"2026-05-18T01:58:13.317109Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:13.317109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","cond-mat.stat-mech","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Gregory Schehr","submitted_at":"2012-03-07T23:27:36Z","abstract_excerpt":"We compute the joint probability density function (jpdf) P_N(M, \\tau_M) of the maximum M and its position \\tau_M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N \\to \\infty, this jpdf is peaked around M = \\sqrt{2N} and \\tau_M = 1/2, while the typical fluctuations behave for large N like M - \\sqrt{2N} \\propto s N^{-1/6} and \\tau_M - 1/2 \\propto w N^{-1/3} where s and w are correlated random variables. 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