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In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of the distance from the identity at time $cn/2$ has a phase transition at $c=1$. Here we investigate some consequences of this result for the geometry of $G_n$. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occu"},"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":"math/0411011","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-10-31T15:59:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"92a21dbc4637614801aeb071726817948ca91a5dd166cbc8ffa12b8e2e45ed93","abstract_canon_sha256":"9c6903aa10c46d47549fb34b81fbc952055b461d815d22f6ea45582f5b235886"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:51.502922Z","signature_b64":"Job6RjtgJE64naoaBknmJDYM19kvNY4B+3bDRAUhkFGPF4IpadfO/QBsiLDhzZokaJ2PreQo4XGz4hH6K3ObBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84a488b95947ba03894f7bc31ce739fcdd921476cb0bc6e0d3ea103dc53633cc","last_reissued_at":"2026-05-18T01:08:51.502330Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:51.502330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The hyperbolic geometry of random transpositions","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Nathana\\\"el Berestycki","submitted_at":"2004-10-31T15:59:46Z","abstract_excerpt":"Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\\sigma_t$ be the simple random walk on this graph. 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