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The random walk moves along open edges in $\\mathbb{Z}^d$ (resp., open sites in $\\mathcal{T}$) at rate $1$. For the critical regime $p=p_c$, we prove the following two results: on $\\mathcal{T}$, the mean squared displacement of the random walk from $0$ to $t$ is at most $O(t\\mu^{5/132-\\epsilon})$ for any $\\epsilon>0$; on $\\mathbb{Z}^d$ with $d\\geq 11$, th"},"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":"2407.15162","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-07-21T13:33:53Z","cross_cats_sorted":[],"title_canon_sha256":"d81cbc7cf7fcfcece6423ec9efc24ca15d34a59e0fc356345e5871d0489b98fa","abstract_canon_sha256":"7c44068b187112129a23717ef23cab7eeae97fe898975d4ec725e27b75bb8241"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:28:53.469464Z","signature_b64":"7MLrse587G+4ZmVuNfoIR9l0dtHLnGa64i3zbU10BzQRSUlLhd3iQegxhy8WtosU99BKyBdkeyYTgUgfUnmnDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b8971d30f977424903f58e6ad7946944eaa345910417add2766e6657f2ec943","last_reissued_at":"2026-07-05T09:28:53.468985Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:28:53.468985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random walk on dynamical percolation in Euclidean lattices: separating critical and supercritical regimes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chenlin Gu, Fan Yang, Hao Wu, Jianping Jiang, Yuval Peres, Zhan Shi","submitted_at":"2024-07-21T13:33:53Z","abstract_excerpt":"We study the random walk on dynamical percolation of $\\mathbb{Z}^d$ (resp., the two-dimensional triangular lattice $\\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $\\mu\\in (0,1/e]$. 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For the critical regime $p=p_c$, we prove the following two results: on $\\mathcal{T}$, the mean squared displacement of the random walk from $0$ to $t$ is at most $O(t\\mu^{5/132-\\epsilon})$ for any $\\epsilon>0$; on $\\mathbb{Z}^d$ with $d\\geq 11$, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.15162","kind":"arxiv","version":3},"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/2407.15162/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2407.15162","created_at":"2026-07-05T09:28:53.469040+00:00"},{"alias_kind":"arxiv_version","alias_value":"2407.15162v3","created_at":"2026-07-05T09:28:53.469040+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.15162","created_at":"2026-07-05T09:28:53.469040+00:00"},{"alias_kind":"pith_short_12","alias_value":"DOEXDUYPS52C","created_at":"2026-07-05T09:28:53.469040+00:00"},{"alias_kind":"pith_short_16","alias_value":"DOEXDUYPS52CJEB7","created_at":"2026-07-05T09:28:53.469040+00:00"},{"alias_kind":"pith_short_8","alias_value":"DOEXDUYP","created_at":"2026-07-05T09:28:53.469040+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2607.02477","citing_title":"Mixing times of spin systems on dynamical percolation","ref_index":10,"is_internal_anchor":false},{"citing_arxiv_id":"2604.07207","citing_title":"Mixing times of step-reinforced random walks","ref_index":16,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR","json":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR.json","graph_json":"https://pith.science/api/pith-number/DOEXDUYPS52CJEB7LDTK26KGSR/graph.json","events_json":"https://pith.science/api/pith-number/DOEXDUYPS52CJEB7LDTK26KGSR/events.json","paper":"https://pith.science/paper/DOEXDUYP"},"agent_actions":{"view_html":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR","download_json":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR.json","view_paper":"https://pith.science/paper/DOEXDUYP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2407.15162&json=true","fetch_graph":"https://pith.science/api/pith-number/DOEXDUYPS52CJEB7LDTK26KGSR/graph.json","fetch_events":"https://pith.science/api/pith-number/DOEXDUYPS52CJEB7LDTK26KGSR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR/action/storage_attestation","attest_author":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR/action/author_attestation","sign_citation":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR/action/citation_signature","submit_replication":"https://pith.science/pith/DOEXDUYPS52CJEB7LDTK26KGSR/action/replication_record"}},"created_at":"2026-07-05T09:28:53.469040+00:00","updated_at":"2026-07-05T09:28:53.469040+00:00"}