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The obstacles appear at each site with probability $p\\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical.\n  Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\\mathbb P_p$-measure, as $n\\to\\infty$,\n  (i) Critical case:\nP"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.09731","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-28T18:04:34Z","cross_cats_sorted":[],"title_canon_sha256":"94b196a7b12a72879272ffb1a9c0fd454c3fc75c52c88e26b241c50e07517afd","abstract_canon_sha256":"56a86059fabcfb44cadaeea530c9805cfb692fb90bf10ac5ad20fbdc5a0835a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:39.663396Z","signature_b64":"axTMc8JdTUmm//dATohEnAtMj0MGSM5tvP+DYIdfivCrvkpuzRp0dxN1JlFVBn3byeEQVcDhhTcmA03f//FOCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ee3feff2ee303f67953c8b27a0bdd48a278e76f6d95f905da4160f7e4db8d88","last_reissued_at":"2026-05-18T00:47:39.662666Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:39.662666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Survival asymptotics for branching random walks in IID environments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Janos Englander, Yuval Peres","submitted_at":"2017-03-28T18:04:34Z","abstract_excerpt":"We first study a model, introduced recently in \\cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. 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We show that on a set of full $\\mathbb P_p$-measure, as $n\\to\\infty$,\n  (i) Critical case:\nP"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09731","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1703.09731","created_at":"2026-05-18T00:47:39.662778+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.09731v1","created_at":"2026-05-18T00:47:39.662778+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.09731","created_at":"2026-05-18T00:47:39.662778+00:00"},{"alias_kind":"pith_short_12","alias_value":"F3R757ZO4MB7","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_16","alias_value":"F3R757ZO4MB7M6KT","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_8","alias_value":"F3R757ZO","created_at":"2026-05-18T12:31:15.632608+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC","json":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC.json","graph_json":"https://pith.science/api/pith-number/F3R757ZO4MB7M6KTZCZHUC65JC/graph.json","events_json":"https://pith.science/api/pith-number/F3R757ZO4MB7M6KTZCZHUC65JC/events.json","paper":"https://pith.science/paper/F3R757ZO"},"agent_actions":{"view_html":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC","download_json":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC.json","view_paper":"https://pith.science/paper/F3R757ZO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.09731&json=true","fetch_graph":"https://pith.science/api/pith-number/F3R757ZO4MB7M6KTZCZHUC65JC/graph.json","fetch_events":"https://pith.science/api/pith-number/F3R757ZO4MB7M6KTZCZHUC65JC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC/action/storage_attestation","attest_author":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC/action/author_attestation","sign_citation":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC/action/citation_signature","submit_replication":"https://pith.science/pith/F3R757ZO4MB7M6KTZCZHUC65JC/action/replication_record"}},"created_at":"2026-05-18T00:47:39.662778+00:00","updated_at":"2026-05-18T00:47:39.662778+00:00"}