{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:Z3ZEMRBBCEUYFCSXUFBBUWM6H7","short_pith_number":"pith:Z3ZEMRBB","canonical_record":{"source":{"id":"1904.11183","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-04-25T07:34:48Z","cross_cats_sorted":[],"title_canon_sha256":"119df1ecfc6062e5bbea829ba48882a4b05e6eb5ad746779a98a8023f98a50dd","abstract_canon_sha256":"fbc2b41c5df03b0ded2a3d9283975c34f1e61d2baaebba1659a296c0cb8ac27f"},"schema_version":"1.0"},"canonical_sha256":"cef24644211129828a57a1421a599e3fee86a693f395cfc74adf2b982dff255e","source":{"kind":"arxiv","id":"1904.11183","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.11183","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"arxiv_version","alias_value":"1904.11183v2","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.11183","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"pith_short_12","alias_value":"Z3ZEMRBBCEUY","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"pith_short_16","alias_value":"Z3ZEMRBBCEUYFCSX","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"pith_short_8","alias_value":"Z3ZEMRBB","created_at":"2026-07-05T01:13:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:Z3ZEMRBBCEUYFCSXUFBBUWM6H7","target":"record","payload":{"canonical_record":{"source":{"id":"1904.11183","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-04-25T07:34:48Z","cross_cats_sorted":[],"title_canon_sha256":"119df1ecfc6062e5bbea829ba48882a4b05e6eb5ad746779a98a8023f98a50dd","abstract_canon_sha256":"fbc2b41c5df03b0ded2a3d9283975c34f1e61d2baaebba1659a296c0cb8ac27f"},"schema_version":"1.0"},"canonical_sha256":"cef24644211129828a57a1421a599e3fee86a693f395cfc74adf2b982dff255e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:13:41.480859Z","signature_b64":"tj9feaaPom0FWUAUz5Po1GwSbL2ddrXy6Y1RQIMx+YCVhxZBe+dfQ+PBKVOe0ltI3YC6HgcVqZ7szBQvapWpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cef24644211129828a57a1421a599e3fee86a693f395cfc74adf2b982dff255e","last_reissued_at":"2026-07-05T01:13:41.480510Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:13:41.480510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.11183","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":"2026-07-05T01:13:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MadG7oUjTJshcaT7UUw+gkOJTh197AHOWAkY61GkJ865PtDXGmMTWzB6IGrC+5VwcaQBd47E/UMTxrYkCVaLAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:50:57.466621Z"},"content_sha256":"d3e7e2fd42dab57b1ff4558c8f5ebcd3489259d1a0b94be5a5bda69df623dcb2","schema_version":"1.0","event_id":"sha256:d3e7e2fd42dab57b1ff4558c8f5ebcd3489259d1a0b94be5a5bda69df623dcb2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:Z3ZEMRBBCEUYFCSXUFBBUWM6H7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Capacity of the range in dimension 5","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bruno Schapira","submitted_at":"2019-04-25T07:34:48Z","abstract_excerpt":"We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\\ge 5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covaria"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.11183","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1904.11183/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":"2026-07-05T01:13:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wqb8BWLKvDqP2SE9Vhx5tw6MINLn6eGHrO/DyyX3yU9peO+ZBZQ9Tt64BwhbVMlIftLNAdi4zG6aGmA85Q0eDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:50:57.467308Z"},"content_sha256":"38ea76f4f399f831b9b9a28e1344ae92d6048e696facaa095aa817fd0c110c2d","schema_version":"1.0","event_id":"sha256:38ea76f4f399f831b9b9a28e1344ae92d6048e696facaa095aa817fd0c110c2d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7/bundle.json","state_url":"https://pith.science/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7/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":"2026-08-16T20:50:57Z","links":{"resolver":"https://pith.science/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7","bundle":"https://pith.science/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7/bundle.json","state":"https://pith.science/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z3ZEMRBBCEUYFCSXUFBBUWM6H7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:Z3ZEMRBBCEUYFCSXUFBBUWM6H7","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":"fbc2b41c5df03b0ded2a3d9283975c34f1e61d2baaebba1659a296c0cb8ac27f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-04-25T07:34:48Z","title_canon_sha256":"119df1ecfc6062e5bbea829ba48882a4b05e6eb5ad746779a98a8023f98a50dd"},"schema_version":"1.0","source":{"id":"1904.11183","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.11183","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"arxiv_version","alias_value":"1904.11183v2","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.11183","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"pith_short_12","alias_value":"Z3ZEMRBBCEUY","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"pith_short_16","alias_value":"Z3ZEMRBBCEUYFCSX","created_at":"2026-07-05T01:13:41Z"},{"alias_kind":"pith_short_8","alias_value":"Z3ZEMRBB","created_at":"2026-07-05T01:13:41Z"}],"graph_snapshots":[{"event_id":"sha256:38ea76f4f399f831b9b9a28e1344ae92d6048e696facaa095aa817fd0c110c2d","target":"graph","created_at":"2026-07-05T01:13:41Z","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/1904.11183/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\\ge 5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covaria","authors_text":"Bruno Schapira","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-04-25T07:34:48Z","title":"Capacity of the range in dimension 5"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.11183","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:d3e7e2fd42dab57b1ff4558c8f5ebcd3489259d1a0b94be5a5bda69df623dcb2","target":"record","created_at":"2026-07-05T01:13:41Z","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":"fbc2b41c5df03b0ded2a3d9283975c34f1e61d2baaebba1659a296c0cb8ac27f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-04-25T07:34:48Z","title_canon_sha256":"119df1ecfc6062e5bbea829ba48882a4b05e6eb5ad746779a98a8023f98a50dd"},"schema_version":"1.0","source":{"id":"1904.11183","kind":"arxiv","version":2}},"canonical_sha256":"cef24644211129828a57a1421a599e3fee86a693f395cfc74adf2b982dff255e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cef24644211129828a57a1421a599e3fee86a693f395cfc74adf2b982dff255e","first_computed_at":"2026-07-05T01:13:41.480510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:13:41.480510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tj9feaaPom0FWUAUz5Po1GwSbL2ddrXy6Y1RQIMx+YCVhxZBe+dfQ+PBKVOe0ltI3YC6HgcVqZ7szBQvapWpAg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:13:41.480859Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.11183","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3e7e2fd42dab57b1ff4558c8f5ebcd3489259d1a0b94be5a5bda69df623dcb2","sha256:38ea76f4f399f831b9b9a28e1344ae92d6048e696facaa095aa817fd0c110c2d"],"state_sha256":"45fe54cde8f12ca88cf0e5992b0ec3778e36987947e444b7efe7e56aa46acb69"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kLK1/v7cDvu5M6qwexYm2bh5cVXqPf3O4z0NB+F7Uhl5RwIx9mV3P+DDMYPzaRlPln3iYSEVIdl6qXNLj6jiCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T20:50:57.473046Z","bundle_sha256":"2e862b80d2980908b98f38a5406e45ef36a3799e4ab9dbd16d67cb5b9f8de16d"}}