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This is motivated by Feynman's path representation of the quantum Bose gas; the choice $X:=\\mathbb{Z}$ and $V(x):=\\alpha x^2$ is of principal interest. Under suitable regularity conditions on the set $X$ and the potential $V$, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permuta"},"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":"1310.0248","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-01T11:32:46Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"2629e250d183fc9d145c7d7adae21924f1870282c0017bf4329e90b2cb601ee4","abstract_canon_sha256":"4930f50ee2a857182dd3fa56867971df3343864d936903662df64dc5994b4c9b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:22:30.093617Z","signature_b64":"SexpsneKW3HavhiZnoKMuMGYTF2460ykN6W9WxPvK+JcR1dvDxc+f4ZfP124aruxCp/vEIf44ez64qXxm9sYBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3fae0592dabdc92eb77d293b08415f00e63850e34171fb5fdd69bac930b1eab","last_reissued_at":"2026-05-18T02:22:30.093009Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:22:30.093009Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gibbs measures on permutations over one-dimensional discrete point sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.PR","authors_text":"Marek Biskup, Thomas Richthammer","submitted_at":"2013-10-01T11:32:46Z","abstract_excerpt":"We consider Gibbs distributions on permutations of a locally finite infinite set $X\\subset\\mathbb{R}$, where a permutation $\\sigma$ of $X$ is assigned (formal) energy $\\sum_{x\\in X}V(\\sigma(x)-x)$. 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