{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JN4SARS5COAEMXHLDSBCDMQBXK","short_pith_number":"pith:JN4SARS5","canonical_record":{"source":{"id":"1408.6895","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-29T01:28:53Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0b3dc9c2d285fe98fa40d98d0da2eb2ccdb37e7423642b408df699428abafad0","abstract_canon_sha256":"d00b695bc89bb179d2fb28161d02f0f7d95a43d9fa68987a1bb118734dec1927"},"schema_version":"1.0"},"canonical_sha256":"4b7920465d1380465ceb1c8221b201baa4d1b1eb6eada46eddb517574710d41e","source":{"kind":"arxiv","id":"1408.6895","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.6895","created_at":"2026-05-18T00:26:16Z"},{"alias_kind":"arxiv_version","alias_value":"1408.6895v2","created_at":"2026-05-18T00:26:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6895","created_at":"2026-05-18T00:26:16Z"},{"alias_kind":"pith_short_12","alias_value":"JN4SARS5COAE","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JN4SARS5COAEMXHL","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JN4SARS5","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JN4SARS5COAEMXHLDSBCDMQBXK","target":"record","payload":{"canonical_record":{"source":{"id":"1408.6895","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-29T01:28:53Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0b3dc9c2d285fe98fa40d98d0da2eb2ccdb37e7423642b408df699428abafad0","abstract_canon_sha256":"d00b695bc89bb179d2fb28161d02f0f7d95a43d9fa68987a1bb118734dec1927"},"schema_version":"1.0"},"canonical_sha256":"4b7920465d1380465ceb1c8221b201baa4d1b1eb6eada46eddb517574710d41e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:16.894267Z","signature_b64":"pt88rvFVViE4LXaSkfVlXPnYOh9QdVQV65oRu3KKPh7u7jt35Mw62gPE/2lsJsk24d915ZXl23kfpRbpAxdlAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b7920465d1380465ceb1c8221b201baa4d1b1eb6eada46eddb517574710d41e","last_reissued_at":"2026-05-18T00:26:16.893695Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:16.893695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.6895","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0MPL6Z+V2IpPeCcuGvdU+gDl3khXKsJLkwRxP9iQiY0cajt4G0yWZdvPbe/0FANHYeLxKIu4V3As1ru6c8t4BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T12:20:44.054224Z"},"content_sha256":"50651355b481a3074d4aedb0827883e1009f62d00b985e0f8bde59491fbaccd9","schema_version":"1.0","event_id":"sha256:50651355b481a3074d4aedb0827883e1009f62d00b985e0f8bde59491fbaccd9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JN4SARS5COAEMXHLDSBCDMQBXK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-Liouville groups with return probability exponent at most 1/2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.GR","authors_text":"B\\'alint Vir\\'ag, Micha{\\l} Kotowski","submitted_at":"2014-08-29T01:28:53Z","abstract_excerpt":"We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \\gtrsim e^{-n^{1/2 + o(1)}}$. Recent results suggest that $1/2$ is indeed the smallest possible return probability exponent for non-Liouville groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6895","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"96mf/mnC/FoxoxhZZ06m4mfPzFP5h0iu2ReJ//LSAnvou6d5qcdw5lvWqXS2V4DnLgFJOBCbIbEdyPyNmCuAAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T12:20:44.054587Z"},"content_sha256":"08b554c2ab75486dcd54db908390a879d45349a1f21244be1c79ece45c37e566","schema_version":"1.0","event_id":"sha256:08b554c2ab75486dcd54db908390a879d45349a1f21244be1c79ece45c37e566"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JN4SARS5COAEMXHLDSBCDMQBXK/bundle.json","state_url":"https://pith.science/pith/JN4SARS5COAEMXHLDSBCDMQBXK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JN4SARS5COAEMXHLDSBCDMQBXK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-28T12:20:44Z","links":{"resolver":"https://pith.science/pith/JN4SARS5COAEMXHLDSBCDMQBXK","bundle":"https://pith.science/pith/JN4SARS5COAEMXHLDSBCDMQBXK/bundle.json","state":"https://pith.science/pith/JN4SARS5COAEMXHLDSBCDMQBXK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JN4SARS5COAEMXHLDSBCDMQBXK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JN4SARS5COAEMXHLDSBCDMQBXK","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":"d00b695bc89bb179d2fb28161d02f0f7d95a43d9fa68987a1bb118734dec1927","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-29T01:28:53Z","title_canon_sha256":"0b3dc9c2d285fe98fa40d98d0da2eb2ccdb37e7423642b408df699428abafad0"},"schema_version":"1.0","source":{"id":"1408.6895","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.6895","created_at":"2026-05-18T00:26:16Z"},{"alias_kind":"arxiv_version","alias_value":"1408.6895v2","created_at":"2026-05-18T00:26:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6895","created_at":"2026-05-18T00:26:16Z"},{"alias_kind":"pith_short_12","alias_value":"JN4SARS5COAE","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JN4SARS5COAEMXHL","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JN4SARS5","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:08b554c2ab75486dcd54db908390a879d45349a1f21244be1c79ece45c37e566","target":"graph","created_at":"2026-05-18T00:26:16Z","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"},"paper":{"abstract_excerpt":"We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \\gtrsim e^{-n^{1/2 + o(1)}}$. Recent results suggest that $1/2$ is indeed the smallest possible return probability exponent for non-Liouville groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.","authors_text":"B\\'alint Vir\\'ag, Micha{\\l} Kotowski","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-29T01:28:53Z","title":"Non-Liouville groups with return probability exponent at most 1/2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6895","kind":"arxiv","version":2},"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":"sha256:50651355b481a3074d4aedb0827883e1009f62d00b985e0f8bde59491fbaccd9","target":"record","created_at":"2026-05-18T00:26:16Z","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":"d00b695bc89bb179d2fb28161d02f0f7d95a43d9fa68987a1bb118734dec1927","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-29T01:28:53Z","title_canon_sha256":"0b3dc9c2d285fe98fa40d98d0da2eb2ccdb37e7423642b408df699428abafad0"},"schema_version":"1.0","source":{"id":"1408.6895","kind":"arxiv","version":2}},"canonical_sha256":"4b7920465d1380465ceb1c8221b201baa4d1b1eb6eada46eddb517574710d41e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b7920465d1380465ceb1c8221b201baa4d1b1eb6eada46eddb517574710d41e","first_computed_at":"2026-05-18T00:26:16.893695Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:16.893695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pt88rvFVViE4LXaSkfVlXPnYOh9QdVQV65oRu3KKPh7u7jt35Mw62gPE/2lsJsk24d915ZXl23kfpRbpAxdlAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:16.894267Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.6895","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50651355b481a3074d4aedb0827883e1009f62d00b985e0f8bde59491fbaccd9","sha256:08b554c2ab75486dcd54db908390a879d45349a1f21244be1c79ece45c37e566"],"state_sha256":"142434063f1c5d7851040f1dde84f3dc38d01111c35853a63498e4d6dccbfa72"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"68O/8HSdtuaTfsJ+9VGumrfjFwBtQ1H9RA4Ub8PoYrK4Wp4NJ84B5H5XhjXueUf5Fm2be/uDSEg4n4CUbrSOBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T12:20:44.056437Z","bundle_sha256":"8a88ba252c51dd60dec60fcebeb5f99abf6a7b5d29bff6a26089fca00bbe9057"}}