{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:HBIFA75ZKQ5IVTL3DM4XZJFRKZ","short_pith_number":"pith:HBIFA75Z","canonical_record":{"source":{"id":"0903.3179","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-18T15:11:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d43f1ab206eb4a3c4c11f6a64b0d016eb88e1be473815581c7946f38543f28f7","abstract_canon_sha256":"fa104b2e780981e05ff4b1c2880c5d9ef3c27ec00ee599666582fe5d822ada5d"},"schema_version":"1.0"},"canonical_sha256":"3850507fb9543a8acd7b1b397ca4b1567959d4d3641e6eb8daff146dbf3260c9","source":{"kind":"arxiv","id":"0903.3179","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.3179","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"0903.3179v1","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.3179","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"HBIFA75ZKQ5I","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"HBIFA75ZKQ5IVTL3","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"HBIFA75Z","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:HBIFA75ZKQ5IVTL3DM4XZJFRKZ","target":"record","payload":{"canonical_record":{"source":{"id":"0903.3179","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-18T15:11:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d43f1ab206eb4a3c4c11f6a64b0d016eb88e1be473815581c7946f38543f28f7","abstract_canon_sha256":"fa104b2e780981e05ff4b1c2880c5d9ef3c27ec00ee599666582fe5d822ada5d"},"schema_version":"1.0"},"canonical_sha256":"3850507fb9543a8acd7b1b397ca4b1567959d4d3641e6eb8daff146dbf3260c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:14:27.502662Z","signature_b64":"kCcVNMGTuSr2YcLAqwTeklny92yzmtowq2KD8Dd8RKkg1pmVZgJ6rgQqiQTTeKIt4nQMvLvItXPgMNcr0wrcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3850507fb9543a8acd7b1b397ca4b1567959d4d3641e6eb8daff146dbf3260c9","last_reissued_at":"2026-05-18T02:14:27.502066Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:14:27.502066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.3179","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":"2026-05-18T02:14:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZY/EcCm2o6T1q9fnty45SE8bkq/CZY/rFHa9oKJxSOfIXbLomt/tGJuJQjcMpEoUP8VlsoiH64gs4dM3n3SvAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T18:19:32.704307Z"},"content_sha256":"cb2e1558aab08e33948a59ac46acb662284345cf1d3b26b6330a08a9d662a4e7","schema_version":"1.0","event_id":"sha256:cb2e1558aab08e33948a59ac46acb662284345cf1d3b26b6330a08a9d662a4e7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:HBIFA75ZKQ5IVTL3DM4XZJFRKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropy of Random Walk Range","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Amir Yehudayoff, Ariel Yadin, Gady Kozma, Itai Benjamini","submitted_at":"2009-03-18T15:11:04Z","abstract_excerpt":"We study the entropy of the set traced by an $n$-step random walk on $\\Z^d$. We show that for $d \\geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\\log^2 n$. These values are essentially governed by the size of the boundary of the trace."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.3179","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":""},"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":"2026-05-18T02:14:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CPnP9pWZtkWcVxik8TeIiaTpwDg+k0g29V8r3kXDRumNAx6dWXaauL1vCyyiSMvafSv9TVQDgw1vt+oXObjrAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T18:19:32.705074Z"},"content_sha256":"0e96216a4beb105eef3e18be6b01759d351d0dbb7623fc70256a532a3c4961fd","schema_version":"1.0","event_id":"sha256:0e96216a4beb105eef3e18be6b01759d351d0dbb7623fc70256a532a3c4961fd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ/bundle.json","state_url":"https://pith.science/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ/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":"2026-05-25T18:19:32Z","links":{"resolver":"https://pith.science/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ","bundle":"https://pith.science/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ/bundle.json","state":"https://pith.science/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HBIFA75ZKQ5IVTL3DM4XZJFRKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:HBIFA75ZKQ5IVTL3DM4XZJFRKZ","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":"fa104b2e780981e05ff4b1c2880c5d9ef3c27ec00ee599666582fe5d822ada5d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-18T15:11:04Z","title_canon_sha256":"d43f1ab206eb4a3c4c11f6a64b0d016eb88e1be473815581c7946f38543f28f7"},"schema_version":"1.0","source":{"id":"0903.3179","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.3179","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"0903.3179v1","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.3179","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"HBIFA75ZKQ5I","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"HBIFA75ZKQ5IVTL3","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"HBIFA75Z","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:0e96216a4beb105eef3e18be6b01759d351d0dbb7623fc70256a532a3c4961fd","target":"graph","created_at":"2026-05-18T02:14:27Z","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 study the entropy of the set traced by an $n$-step random walk on $\\Z^d$. We show that for $d \\geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\\log^2 n$. These values are essentially governed by the size of the boundary of the trace.","authors_text":"Amir Yehudayoff, Ariel Yadin, Gady Kozma, Itai Benjamini","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-18T15:11:04Z","title":"Entropy of Random Walk Range"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.3179","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:cb2e1558aab08e33948a59ac46acb662284345cf1d3b26b6330a08a9d662a4e7","target":"record","created_at":"2026-05-18T02:14:27Z","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":"fa104b2e780981e05ff4b1c2880c5d9ef3c27ec00ee599666582fe5d822ada5d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-18T15:11:04Z","title_canon_sha256":"d43f1ab206eb4a3c4c11f6a64b0d016eb88e1be473815581c7946f38543f28f7"},"schema_version":"1.0","source":{"id":"0903.3179","kind":"arxiv","version":1}},"canonical_sha256":"3850507fb9543a8acd7b1b397ca4b1567959d4d3641e6eb8daff146dbf3260c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3850507fb9543a8acd7b1b397ca4b1567959d4d3641e6eb8daff146dbf3260c9","first_computed_at":"2026-05-18T02:14:27.502066Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:14:27.502066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kCcVNMGTuSr2YcLAqwTeklny92yzmtowq2KD8Dd8RKkg1pmVZgJ6rgQqiQTTeKIt4nQMvLvItXPgMNcr0wrcBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:14:27.502662Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.3179","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb2e1558aab08e33948a59ac46acb662284345cf1d3b26b6330a08a9d662a4e7","sha256:0e96216a4beb105eef3e18be6b01759d351d0dbb7623fc70256a532a3c4961fd"],"state_sha256":"7f0f0bd220dad9877900b7913876158af59511c2e9c953e71ab74fd42164d074"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yGWDOKUf/v5/UGpyP3BWdmVa7Qiugamvso+51pxys7RHO58YR7T9D33LrMFq9ztRZZCztSQ4bX9qWgFXrLpNAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T18:19:32.708971Z","bundle_sha256":"b5ed9d79796e4a1c2e1ab43f0117f997b52c60ec8eb592f0993e2f53390e0ff1"}}