{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ZBB2FNFPUINEGAWW2DRQNALONX","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":"f825db36c7ea766429388f45e4c4f3ae29221e2f17b741f0314bd678d946b14d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-24T08:07:19Z","title_canon_sha256":"87b6ca230e0aced35dfde843abce068076235f6a2a0e5e86762eb2a825e4a4da"},"schema_version":"1.0","source":{"id":"1102.4921","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.4921","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"arxiv_version","alias_value":"1102.4921v1","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4921","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"pith_short_12","alias_value":"ZBB2FNFPUINE","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZBB2FNFPUINEGAWW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZBB2FNFP","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:89208e91a9e48cfd941f274008038c5e410ae67cb51e1b839f45f683457fcc05","target":"graph","created_at":"2026-05-18T04:27:58Z","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":"The parabolic Anderson problem is the Cauchy problem for the heat equation $\\partial_tu(t,z)=\\Delta u(t,z)+\\xi(z)u(t,z)$ on $(0,\\infty)\\times {\\mathbb{Z}}^d$ with random potential $(\\xi(z):z\\in{\\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\\xi(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also i","authors_text":"Hubert Lacoin, Nadia Sidorova, Peter M\\\"orters, Wolfgang K\\\"onig","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-24T08:07:19Z","title":"A two cities theorem for the parabolic Anderson model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4921","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:a7ecb1e5ce7119be67196e7de6125b2119928efd2c1eaa0e001c99de70aaf748","target":"record","created_at":"2026-05-18T04:27:58Z","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":"f825db36c7ea766429388f45e4c4f3ae29221e2f17b741f0314bd678d946b14d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-24T08:07:19Z","title_canon_sha256":"87b6ca230e0aced35dfde843abce068076235f6a2a0e5e86762eb2a825e4a4da"},"schema_version":"1.0","source":{"id":"1102.4921","kind":"arxiv","version":1}},"canonical_sha256":"c843a2b4afa21a4302d6d0e306816e6de53c083e7dab8daaa28ff38ec8e279ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c843a2b4afa21a4302d6d0e306816e6de53c083e7dab8daaa28ff38ec8e279ee","first_computed_at":"2026-05-18T04:27:58.463488Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:58.463488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cZKFiv5AlPpJUwQQjQK6wGcEE5aiU81PlR2jhS58qtrlLukAGofDy1uzeXapYT7sCqn9CsnNep4VkhmyCk2FBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:58.464122Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.4921","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7ecb1e5ce7119be67196e7de6125b2119928efd2c1eaa0e001c99de70aaf748","sha256:89208e91a9e48cfd941f274008038c5e410ae67cb51e1b839f45f683457fcc05"],"state_sha256":"35eff97d8822335d467d8dd6b921d6c6f6f1e3624bd007f45cc55b7024da4e62"}