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It is well-known that $F_N$ itself is spectrally rigid; it also follows from the result of Smillie and Vogtmann that there does not exist a finite spectrally rigid subset of $F_N$. We prove that if $A$ is a free basis of $F_N$ (where $N\\ge 2$) then almost every trajectory of a non-backtracking simple random walk on $F_N$ with respect to $A$ is a spectrally rigid subset of "},"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":"1001.1729","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-01-12T00:38:18Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"a377a048b06eeb3d27f46da5db8bf3ecbe3f6fc2324c9959aa9404be481a733f","abstract_canon_sha256":"769c9d6f683a381a496533fbac9ec93268a320d4ba2b0caa152641974e93edc4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:03.845163Z","signature_b64":"HJNyxTNC228YpL57/EBttOy0OWvRSsSvBvw1/3X8wiFzk1c6MIx32aK0noqv/ctCJbav4IR3qXnbMzyuO7LjCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"209e77bae978d1e38fd3a326074f5e282e01cefce2422bcd6374d1380cb4fece","last_reissued_at":"2026-05-18T04:30:03.844750Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:03.844750Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random length-spectrum rigidity for free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Ilya Kapovich","submitted_at":"2010-01-12T00:38:18Z","abstract_excerpt":"We say that a subset $S\\subseteq F_N$ is \\emph{spectrally rigid} if whenever $T_1, T_2\\in cv_N$ are points of the (unprojectivized) Outer space such that $||g||_{T_1}=||g||_{T_2}$ for every $g\\in S$ then $T_1=T_2$ in $\\cvn$. It is well-known that $F_N$ itself is spectrally rigid; it also follows from the result of Smillie and Vogtmann that there does not exist a finite spectrally rigid subset of $F_N$. 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