{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:UBE6COBEWVYVBMJNO577GP44IN","short_pith_number":"pith:UBE6COBE","canonical_record":{"source":{"id":"1005.0376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-05-03T19:33:14Z","cross_cats_sorted":[],"title_canon_sha256":"747bf42491b57c6df33830ed6c5bb6f662eabaa60721bd08f62ea392a18133af","abstract_canon_sha256":"aed4785818020a9150bfabc024de2a64d528cb12e813d9794d50bf8d4ed2d1f2"},"schema_version":"1.0"},"canonical_sha256":"a049e13824b57150b12d777ff33f9c4362a4b62a1146dd3e91dd22c09bacf1c9","source":{"kind":"arxiv","id":"1005.0376","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0376","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0376v2","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0376","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"pith_short_12","alias_value":"UBE6COBEWVYV","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UBE6COBEWVYVBMJN","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UBE6COBE","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:UBE6COBEWVYVBMJNO577GP44IN","target":"record","payload":{"canonical_record":{"source":{"id":"1005.0376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-05-03T19:33:14Z","cross_cats_sorted":[],"title_canon_sha256":"747bf42491b57c6df33830ed6c5bb6f662eabaa60721bd08f62ea392a18133af","abstract_canon_sha256":"aed4785818020a9150bfabc024de2a64d528cb12e813d9794d50bf8d4ed2d1f2"},"schema_version":"1.0"},"canonical_sha256":"a049e13824b57150b12d777ff33f9c4362a4b62a1146dd3e91dd22c09bacf1c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:48.894635Z","signature_b64":"hhUUthJqRulnYgrA9k7FnAV/Mt5JnfxIjSFTVEIr/KPjfIeDQvkXdnd89eb4UTFMQXrDCwGwmz3giiAx61NnDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a049e13824b57150b12d777ff33f9c4362a4b62a1146dd3e91dd22c09bacf1c9","last_reissued_at":"2026-05-18T03:58:48.893873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:48.893873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.0376","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":"2026-05-18T03:58:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rlEkl1J5jz177Azncc0ibGKQfbLSs55dBZ43RitZhjy3kgueCiEQqEx1hB2CGkDgGHovIDqijZjXK3CwXZOLAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:39:15.452416Z"},"content_sha256":"9b157be4e98f81fe1d105b2d1d89269e67cee7bbdd1c884a58297634073025a8","schema_version":"1.0","event_id":"sha256:9b157be4e98f81fe1d105b2d1d89269e67cee7bbdd1c884a58297634073025a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:UBE6COBEWVYVBMJNO577GP44IN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alejandro F. Ram\\'irez, Alexander Drewitz","submitted_at":"2010-05-03T19:33:14Z","abstract_excerpt":"Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each $\\gamma\\in(0,1)$ the ballisticity condition $(T)_{\\gamma}$ and the condition $(T')$ defined as the fulfillment of $(T)_{\\gamma}$ for each $\\gamma\\in(0,1)$. Sznitman proved that $(T')$ implies a ballistic law of large numbers. Furthermore, he showed that for all $\\gamma\\in (0.5,1)$, $(T)_{\\gamma}$ is equivalent to $(T')$. Recently, Berger has proved that in dimensions larger than three, for each $\\gamma\\in (0,1)$, condition $(T)_{\\gamma}$ implies a ballistic la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0376","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":"2026-05-18T03:58:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9rg31TXtkE7AgAxFbfdfGWogPJAv4SG3VehR0oQ3+qE7nQehWL1p7wf3cOctvCG9a087zEFzUL8ArWeYb1aSBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:39:15.452949Z"},"content_sha256":"764b6369b62063f2e95f6e3feb670153dafebfa1b941ca3b2135c919b54b7e44","schema_version":"1.0","event_id":"sha256:764b6369b62063f2e95f6e3feb670153dafebfa1b941ca3b2135c919b54b7e44"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UBE6COBEWVYVBMJNO577GP44IN/bundle.json","state_url":"https://pith.science/pith/UBE6COBEWVYVBMJNO577GP44IN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UBE6COBEWVYVBMJNO577GP44IN/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":"2026-05-27T11:39:15Z","links":{"resolver":"https://pith.science/pith/UBE6COBEWVYVBMJNO577GP44IN","bundle":"https://pith.science/pith/UBE6COBEWVYVBMJNO577GP44IN/bundle.json","state":"https://pith.science/pith/UBE6COBEWVYVBMJNO577GP44IN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UBE6COBEWVYVBMJNO577GP44IN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:UBE6COBEWVYVBMJNO577GP44IN","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":"aed4785818020a9150bfabc024de2a64d528cb12e813d9794d50bf8d4ed2d1f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-05-03T19:33:14Z","title_canon_sha256":"747bf42491b57c6df33830ed6c5bb6f662eabaa60721bd08f62ea392a18133af"},"schema_version":"1.0","source":{"id":"1005.0376","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0376","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0376v2","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0376","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"pith_short_12","alias_value":"UBE6COBEWVYV","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UBE6COBEWVYVBMJN","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UBE6COBE","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:764b6369b62063f2e95f6e3feb670153dafebfa1b941ca3b2135c919b54b7e44","target":"graph","created_at":"2026-05-18T03:58:48Z","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":"Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each $\\gamma\\in(0,1)$ the ballisticity condition $(T)_{\\gamma}$ and the condition $(T')$ defined as the fulfillment of $(T)_{\\gamma}$ for each $\\gamma\\in(0,1)$. Sznitman proved that $(T')$ implies a ballistic law of large numbers. Furthermore, he showed that for all $\\gamma\\in (0.5,1)$, $(T)_{\\gamma}$ is equivalent to $(T')$. Recently, Berger has proved that in dimensions larger than three, for each $\\gamma\\in (0,1)$, condition $(T)_{\\gamma}$ implies a ballistic la","authors_text":"Alejandro F. Ram\\'irez, Alexander Drewitz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-05-03T19:33:14Z","title":"Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0376","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":"sha256:9b157be4e98f81fe1d105b2d1d89269e67cee7bbdd1c884a58297634073025a8","target":"record","created_at":"2026-05-18T03:58:48Z","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":"aed4785818020a9150bfabc024de2a64d528cb12e813d9794d50bf8d4ed2d1f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-05-03T19:33:14Z","title_canon_sha256":"747bf42491b57c6df33830ed6c5bb6f662eabaa60721bd08f62ea392a18133af"},"schema_version":"1.0","source":{"id":"1005.0376","kind":"arxiv","version":2}},"canonical_sha256":"a049e13824b57150b12d777ff33f9c4362a4b62a1146dd3e91dd22c09bacf1c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a049e13824b57150b12d777ff33f9c4362a4b62a1146dd3e91dd22c09bacf1c9","first_computed_at":"2026-05-18T03:58:48.893873Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:48.893873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hhUUthJqRulnYgrA9k7FnAV/Mt5JnfxIjSFTVEIr/KPjfIeDQvkXdnd89eb4UTFMQXrDCwGwmz3giiAx61NnDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:48.894635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.0376","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b157be4e98f81fe1d105b2d1d89269e67cee7bbdd1c884a58297634073025a8","sha256:764b6369b62063f2e95f6e3feb670153dafebfa1b941ca3b2135c919b54b7e44"],"state_sha256":"4f5a72a8f71f3d7c4530f1cbf616c33b69e25264da1fbd4ce6cd6df9a1221fd8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RjnaIwQ5GSzkWgY3DuB4eyeRI2k3ET9GNCJNXmBsq+BNQzZxKuv7ZUZsHBdYlgJQ0Sk+v7UaDZD719p/KYayDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T11:39:15.456463Z","bundle_sha256":"fe2ee75c5aebbc4691983645c7de26ea0e5e2b90d33fddb852545b7f7dddbfa5"}}