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The transition matrix of walk $i$ takes the form $Q^i(x, p_i) = p_i \\Pi^i(x) + (1-p_i)I$, where $\\pi^i(x)$ is the unique invariant measure, independently of $p_i$. Consequently, the limiting points of the occupation measure $X(n)$ coincide with those of the simultaneous-transition model ($p_i = 1$): the solutions of $x = \\pi(x)$. Verifying almost sure convergence to these points is non-trivial, since the stochastic input $U("},"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":"2606.05386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-03T19:43:02Z","cross_cats_sorted":[],"title_canon_sha256":"107ad987168643abbcb83cc9f10239542cc6c543ad2ca648c697e464ecadc96d","abstract_canon_sha256":"6ec3ba45ad4afa9336bb3a6c4da9d534800ba198f99d7f09a40e3123c1a1f01d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T00:13:56.061465Z","signature_b64":"mzV1cdJColl8kbLPyb7BmRQDayeybCJP3PcU7wWBJ/pczgGj+4S0cvkZ3z7BTDb8pFRaP827TG8oMLhKYE/9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eaf8f3f6f402a0bf928e217e82e8ceae5b890dcdda8f9b36737cc4f77172aea0","last_reissued_at":"2026-06-05T00:13:56.060930Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T00:13:56.060930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reinforced random walks with geometric inter-transition times","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fernando P. A. Prado, Mirela G. Coelho","submitted_at":"2026-06-03T19:43:02Z","abstract_excerpt":"We consider interacting vertex-reinforced random walks on a finite graph, where each walk transitions at independent geometric random times with parameter $p_i \\in (0,1]$. The transition matrix of walk $i$ takes the form $Q^i(x, p_i) = p_i \\Pi^i(x) + (1-p_i)I$, where $\\pi^i(x)$ is the unique invariant measure, independently of $p_i$. Consequently, the limiting points of the occupation measure $X(n)$ coincide with those of the simultaneous-transition model ($p_i = 1$): the solutions of $x = \\pi(x)$. Verifying almost sure convergence to these points is non-trivial, since the stochastic input $U("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05386","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.05386/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.05386","created_at":"2026-06-05T00:13:56.060995+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.05386v1","created_at":"2026-06-05T00:13:56.060995+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05386","created_at":"2026-06-05T00:13:56.060995+00:00"},{"alias_kind":"pith_short_12","alias_value":"5L4PH5XUAKQL","created_at":"2026-06-05T00:13:56.060995+00:00"},{"alias_kind":"pith_short_16","alias_value":"5L4PH5XUAKQL7EUO","created_at":"2026-06-05T00:13:56.060995+00:00"},{"alias_kind":"pith_short_8","alias_value":"5L4PH5XU","created_at":"2026-06-05T00:13:56.060995+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ","json":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ.json","graph_json":"https://pith.science/api/pith-number/5L4PH5XUAKQL7EUOEF7IF2GOVZ/graph.json","events_json":"https://pith.science/api/pith-number/5L4PH5XUAKQL7EUOEF7IF2GOVZ/events.json","paper":"https://pith.science/paper/5L4PH5XU"},"agent_actions":{"view_html":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ","download_json":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ.json","view_paper":"https://pith.science/paper/5L4PH5XU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.05386&json=true","fetch_graph":"https://pith.science/api/pith-number/5L4PH5XUAKQL7EUOEF7IF2GOVZ/graph.json","fetch_events":"https://pith.science/api/pith-number/5L4PH5XUAKQL7EUOEF7IF2GOVZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ/action/storage_attestation","attest_author":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ/action/author_attestation","sign_citation":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ/action/citation_signature","submit_replication":"https://pith.science/pith/5L4PH5XUAKQL7EUOEF7IF2GOVZ/action/replication_record"}},"created_at":"2026-06-05T00:13:56.060995+00:00","updated_at":"2026-06-05T00:13:56.060995+00:00"}