{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:D3BDNFUDDHPWZCGXF2XCGRLRN2","short_pith_number":"pith:D3BDNFUD","canonical_record":{"source":{"id":"2309.03192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-06T17:55:03Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"04019208112885e9ebc21851670cba58c1772601664b2c63b565f39c7abc610c","abstract_canon_sha256":"28d38c58b8441afc9705dc5a1b9c752d3fb91730b17e817c8ee7c7f91d72d24c"},"schema_version":"1.0"},"canonical_sha256":"1ec236968319df6c88d72eae2345716eb90d5930f2c33612ada789037d7f5a8e","source":{"kind":"arxiv","id":"2309.03192","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.03192","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"arxiv_version","alias_value":"2309.03192v1","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.03192","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"pith_short_12","alias_value":"D3BDNFUDDHPW","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"pith_short_16","alias_value":"D3BDNFUDDHPWZCGX","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"pith_short_8","alias_value":"D3BDNFUD","created_at":"2026-07-05T06:48:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:D3BDNFUDDHPWZCGXF2XCGRLRN2","target":"record","payload":{"canonical_record":{"source":{"id":"2309.03192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-06T17:55:03Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"04019208112885e9ebc21851670cba58c1772601664b2c63b565f39c7abc610c","abstract_canon_sha256":"28d38c58b8441afc9705dc5a1b9c752d3fb91730b17e817c8ee7c7f91d72d24c"},"schema_version":"1.0"},"canonical_sha256":"1ec236968319df6c88d72eae2345716eb90d5930f2c33612ada789037d7f5a8e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:48:21.953043Z","signature_b64":"XVd+pAZUTISbNRHan1nrqs3cG0mVfX4GhLflbvlqJLp+UQJ3kkitwuHwCNFeqg3SvMHRlqiebNZb7YnIJOguDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ec236968319df6c88d72eae2345716eb90d5930f2c33612ada789037d7f5a8e","last_reissued_at":"2026-07-05T06:48:21.952610Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:48:21.952610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2309.03192","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:48:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aE9TVMzgP4WoH6L65njV63+Qe8gIIFLc/E6Ii8keBH1QzrDs6ZXIMYJ6GL/s73bz+njgQGBulyZe3kApQesaBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T06:13:38.671413Z"},"content_sha256":"fed4a4b29a0db256699a361e8b6ee9c630e0b10046bf1b5bbb9718759a7bc0a8","schema_version":"1.0","event_id":"sha256:fed4a4b29a0db256699a361e8b6ee9c630e0b10046bf1b5bbb9718759a7bc0a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:D3BDNFUDDHPWZCGXF2XCGRLRN2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Phase transition for the late points of random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alexis Pr\\'evost, Perla Sousi, Pierre-Fran\\c{c}ois Rodriguez","submitted_at":"2023-09-06T17:55:03Z","abstract_excerpt":"Let $X$ be a random walk on the torus of side length $N$ in dimension $d\\geq 3$ with uniform starting point, and $t_{\\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $\\alpha > 0$, the set $\\mathcal{L}^{\\alpha}$ of $\\alpha$-late points consists of those points not visited by $X$ at time $\\alpha t_{\\text{cov}}$. We prove the existence of a value $\\alpha_* \\in (\\frac12,1)$ across which $\\mathcal{L}^{\\alpha}$ trivialises as follows: for all $\\alpha > \\alpha_*$ and $\\epsilon\\geq N^{-c}$ there exists a cou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.03192","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/2309.03192/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:48:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wl6j14Reb3zLfYFCRja2CiGaG5FncYT6RkwGOh4TyMdsFFLyueZAaZJ9idtyHJs+ySmsSJq+Q9KH5Z4+weQ7AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T06:13:38.672355Z"},"content_sha256":"9b48d7c6b81beb771db3d766eb3b0bb01039253d43ab3323e1e8a7f8c36f1bf7","schema_version":"1.0","event_id":"sha256:9b48d7c6b81beb771db3d766eb3b0bb01039253d43ab3323e1e8a7f8c36f1bf7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2/bundle.json","state_url":"https://pith.science/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2/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":"2026-08-13T06:13:38Z","links":{"resolver":"https://pith.science/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2","bundle":"https://pith.science/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2/bundle.json","state":"https://pith.science/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D3BDNFUDDHPWZCGXF2XCGRLRN2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:D3BDNFUDDHPWZCGXF2XCGRLRN2","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":"28d38c58b8441afc9705dc5a1b9c752d3fb91730b17e817c8ee7c7f91d72d24c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-06T17:55:03Z","title_canon_sha256":"04019208112885e9ebc21851670cba58c1772601664b2c63b565f39c7abc610c"},"schema_version":"1.0","source":{"id":"2309.03192","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.03192","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"arxiv_version","alias_value":"2309.03192v1","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.03192","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"pith_short_12","alias_value":"D3BDNFUDDHPW","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"pith_short_16","alias_value":"D3BDNFUDDHPWZCGX","created_at":"2026-07-05T06:48:21Z"},{"alias_kind":"pith_short_8","alias_value":"D3BDNFUD","created_at":"2026-07-05T06:48:21Z"}],"graph_snapshots":[{"event_id":"sha256:9b48d7c6b81beb771db3d766eb3b0bb01039253d43ab3323e1e8a7f8c36f1bf7","target":"graph","created_at":"2026-07-05T06:48:21Z","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/2309.03192/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X$ be a random walk on the torus of side length $N$ in dimension $d\\geq 3$ with uniform starting point, and $t_{\\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $\\alpha > 0$, the set $\\mathcal{L}^{\\alpha}$ of $\\alpha$-late points consists of those points not visited by $X$ at time $\\alpha t_{\\text{cov}}$. We prove the existence of a value $\\alpha_* \\in (\\frac12,1)$ across which $\\mathcal{L}^{\\alpha}$ trivialises as follows: for all $\\alpha > \\alpha_*$ and $\\epsilon\\geq N^{-c}$ there exists a cou","authors_text":"Alexis Pr\\'evost, Perla Sousi, Pierre-Fran\\c{c}ois Rodriguez","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-06T17:55:03Z","title":"Phase transition for the late points of random walk"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.03192","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:fed4a4b29a0db256699a361e8b6ee9c630e0b10046bf1b5bbb9718759a7bc0a8","target":"record","created_at":"2026-07-05T06:48:21Z","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":"28d38c58b8441afc9705dc5a1b9c752d3fb91730b17e817c8ee7c7f91d72d24c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-06T17:55:03Z","title_canon_sha256":"04019208112885e9ebc21851670cba58c1772601664b2c63b565f39c7abc610c"},"schema_version":"1.0","source":{"id":"2309.03192","kind":"arxiv","version":1}},"canonical_sha256":"1ec236968319df6c88d72eae2345716eb90d5930f2c33612ada789037d7f5a8e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ec236968319df6c88d72eae2345716eb90d5930f2c33612ada789037d7f5a8e","first_computed_at":"2026-07-05T06:48:21.952610Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:48:21.952610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XVd+pAZUTISbNRHan1nrqs3cG0mVfX4GhLflbvlqJLp+UQJ3kkitwuHwCNFeqg3SvMHRlqiebNZb7YnIJOguDA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:48:21.953043Z","signed_message":"canonical_sha256_bytes"},"source_id":"2309.03192","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fed4a4b29a0db256699a361e8b6ee9c630e0b10046bf1b5bbb9718759a7bc0a8","sha256:9b48d7c6b81beb771db3d766eb3b0bb01039253d43ab3323e1e8a7f8c36f1bf7"],"state_sha256":"824f89ff3d7979bbe4a9872448d591101c4eb212e6533ae339c191e14b887d85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ci2uI3IS9BxKPvgiGeBU5DaUK6HSaxAl3vZI66SmTFfqq3QafCITf3XwEUWgHWUbuGsgzzQz6XjfskMxI9JrCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T06:13:38.679374Z","bundle_sha256":"1c8c5d0212513a1ceaf78f42b9bb3b594895b5f65a56ca11bbb89738d3163002"}}