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We compute the distribution P_B(A,n) of the area A = \\sum_{m=1}^n x_B(m) under such a L\\'evy bridge and show that, for large n, it has the scaling form P_B(A,n) \\sim n^{-1-1/\\alpha} F_\\alpha(A/n^{1+1/\\alpha}), with the asymptotic behavior F_\\alpha(Y) \\sim Y^{-2(1+\\alpha)} for large Y. For \\alpha=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions.  We also compute the average "},"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":"1004.5046","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-04-28T14:39:17Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"172a9753e45beb8ed39ded5cf26ef6b11cfc4fe07991a02da688e05a772c5998","abstract_canon_sha256":"ce9a4b969b221979f1e4d10f31dcfae5e3a43479f47a78de16debc130a071e91"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:26.104481Z","signature_b64":"vxeeldzlt1KdY467lHLFiXLIbAsboTPuELp9p8wmL82pVKZBm/w92997r4WfZl+S8ZhH7DP/5B2GdZWmx+uMCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99e607698e5867db2318a5e9da30054c6f2086cea1f4a42de6dacd70e02219c1","last_reissued_at":"2026-05-18T04:41:26.104055Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:26.104055Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Area distribution and the average shape of a L\\'evy bridge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Gregory Schehr, Satya N. Majumdar","submitted_at":"2010-04-28T14:39:17Z","abstract_excerpt":"We consider a one dimensional L\\'evy bridge x_B of length n and index 0 < \\alpha < 2, i.e. a L\\'evy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \\sum_{m=1}^n x_B(m) under such a L\\'evy bridge and show that, for large n, it has the scaling form P_B(A,n) \\sim n^{-1-1/\\alpha} F_\\alpha(A/n^{1+1/\\alpha}), with the asymptotic behavior F_\\alpha(Y) \\sim Y^{-2(1+\\alpha)} for large Y. For \\alpha=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions.  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