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We quantify the overlap between this single vector with the eigenvectors of the initial matrix and identify precisely a \"Cauchy-flight\" regime. In particular, we compute the local density of this vector in the eigenvalues space of the initial matrix. Our results are obtained in a non perturbative setting and are derived using the ideas of [O. Ledoit and S. P\\'ech\\'e, Prob. Th. Rel. Fields, {\\bf 151} 233 (2011)]. Finally, we give"},"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":"1301.4939","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-01-21T17:48:04Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"66e90c0d8a1e688b1fb95e97cc49577bee056a94a5495aeb46ce4c83f23e4d7d","abstract_canon_sha256":"b9b2a996f2a8f03e6f0bf1e4aaaed5d8b17edee81a81d8d354255aa2b647a58a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:01.381363Z","signature_b64":"933KdNFQ6n0gYNCQyjNEHdGQCQ4iWXI0D2LSMK842TdlY506adzZndc/NjcOqMQurNbG9NCcpEIxJozMxn7lBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b4204ab187a0eec6d4115e1b56cff4e0b581ccd32a5dcf49e387c9a11011292","last_reissued_at":"2026-05-18T02:30:01.380954Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:01.380954Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvector dynamics under free addition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"math.PR","authors_text":"Jean-Philippe Bouchaud, Romain Allez","submitted_at":"2013-01-21T17:48:04Z","abstract_excerpt":"We investigate the evolution of a given eigenvector of a symmetric (deterministic or random) matrix under the addition of a matrix in the Gaussian orthogonal ensemble. 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