{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:5APXCOHAFF3CASBPDYB2J6ZNTO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c8fc0217a425b62f31729d8af523a571bf17c53a7b0149e1f35a915c46ca2057","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-11-14T18:04:16Z","title_canon_sha256":"19a0fcb9235099d94b03a2a9eae52615ad4f075c19afb4ce492a92c9dd21f447"},"schema_version":"1.0","source":{"id":"2411.09640","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.09640","created_at":"2026-07-05T09:35:30Z"},{"alias_kind":"arxiv_version","alias_value":"2411.09640v1","created_at":"2026-07-05T09:35:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.09640","created_at":"2026-07-05T09:35:30Z"},{"alias_kind":"pith_short_12","alias_value":"5APXCOHAFF3C","created_at":"2026-07-05T09:35:30Z"},{"alias_kind":"pith_short_16","alias_value":"5APXCOHAFF3CASBP","created_at":"2026-07-05T09:35:30Z"},{"alias_kind":"pith_short_8","alias_value":"5APXCOHA","created_at":"2026-07-05T09:35:30Z"}],"graph_snapshots":[{"event_id":"sha256:bebaa91118c635c09e23d4417aaeb3e39498eaa9d99b15ad808c2f6ab52df91f","target":"graph","created_at":"2026-07-05T09:35:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.09640/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Benjamini, Yadin, and Yehudayoff (2007) showed that if the maximum degree of a graph $G$ is 'sub-logarithmic,' then the typical range of random $\\mathbb Z$-homomorphisms is super-constant. Furthermore, they showed that there is a sharp transition on the range of random $\\mathbb Z$-homomorphisms on the graph $C_{n,k}$, the tensor product of the $n$-cycle and the complete graph on $k$ vertices with self-loops, around $k=2\\log n$. We extend (to some extent) their results to random $M$-Lipschitz functions and random real-valued Lipschitz functions.","authors_text":"Jinyoung Park, Senem I\\c{s}{\\i}k","cross_cats":["math-ph","math.CO","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-11-14T18:04:16Z","title":"Random Lipschitz functions on graphs with weak expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.09640","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:55011749ad8c18305b5264dacb03494263c1ba00c853cf41a734a3a423f31a28","target":"record","created_at":"2026-07-05T09:35:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c8fc0217a425b62f31729d8af523a571bf17c53a7b0149e1f35a915c46ca2057","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-11-14T18:04:16Z","title_canon_sha256":"19a0fcb9235099d94b03a2a9eae52615ad4f075c19afb4ce492a92c9dd21f447"},"schema_version":"1.0","source":{"id":"2411.09640","kind":"arxiv","version":1}},"canonical_sha256":"e81f7138e0297620482f1e03a4fb2d9ba5e54c5afae32800a3b4b1977619aa17","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e81f7138e0297620482f1e03a4fb2d9ba5e54c5afae32800a3b4b1977619aa17","first_computed_at":"2026-07-05T09:35:30.242215Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:35:30.242215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MIK7quVQnEVBMJnzNUVop9wB6oJq2Q2nJOU2g7V3WhsAzaNSbOrKzX0p0vz8oqAMsl2YQKpa1QEs/r9pCGP2BA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:35:30.243155Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.09640","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:55011749ad8c18305b5264dacb03494263c1ba00c853cf41a734a3a423f31a28","sha256:bebaa91118c635c09e23d4417aaeb3e39498eaa9d99b15ad808c2f6ab52df91f"],"state_sha256":"533698ba4120ac92a9f6ce5c645366eea9ca4dd51a6d0a470b564ee718b63138"}