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Let $\\Upsilon$ be the carpet of $\\Gamma$, i.e., the set of points in $D$ not surrounded by a loop of $\\Gamma$. We prove that certain approximations to the chemical distance metric in $\\Upsilon$ are tight. More precisely, for each path $\\omega \\colon [0,1] \\to \\Upsilon$ and $\\epsilon > 0$ we let ${\\mathfrak N}_\\epsilon(\\omega)$ be the Lebesgue measure of the $\\epsilon$-neighborhood "},"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":"2112.08335","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2021-12-15T18:32:01Z","cross_cats_sorted":["math-ph","math.CV","math.MP"],"title_canon_sha256":"33e7a6cf8d030c678d1a45c49e77ae7702596e55282128cda8f54ddd48ee5162","abstract_canon_sha256":"076867fbcfc6d4479fd2b2ab9dc5247aa927b40714f5d029da91f0a3d27d5209"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:41:21.005500Z","signature_b64":"Kbj+qn94mTvcV/YvvVocY/uJpHEAjjx+i8PzRuZgjdFxh3uf0ESxnOs1nA0Her1rV5o668XOrr3AETRE3n83CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd176b34a339567bed7b65aa9dc4de252176d2cbb92118c661308f39fa4af9f3","last_reissued_at":"2026-07-05T03:41:21.005071Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:41:21.005071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tightness of approximations to the chemical distance metric for simple conformal loop ensembles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP"],"primary_cat":"math.PR","authors_text":"Jason Miller","submitted_at":"2021-12-15T18:32:01Z","abstract_excerpt":"Suppose that $\\Gamma$ is a conformal loop ensemble (CLE$_\\kappa$) with simple loops ($\\kappa \\in (8/3,4)$) in a simply connected domain $D \\subseteq {\\mathbf C}$ whose boundary is itself a type of CLE$_\\kappa$ loop. Let $\\Upsilon$ be the carpet of $\\Gamma$, i.e., the set of points in $D$ not surrounded by a loop of $\\Gamma$. We prove that certain approximations to the chemical distance metric in $\\Upsilon$ are tight. More precisely, for each path $\\omega \\colon [0,1] \\to \\Upsilon$ and $\\epsilon > 0$ we let ${\\mathfrak N}_\\epsilon(\\omega)$ be the Lebesgue measure of the $\\epsilon$-neighborhood "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.08335","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.08335/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":"2112.08335","created_at":"2026-07-05T03:41:21.005120+00:00"},{"alias_kind":"arxiv_version","alias_value":"2112.08335v1","created_at":"2026-07-05T03:41:21.005120+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.08335","created_at":"2026-07-05T03:41:21.005120+00:00"},{"alias_kind":"pith_short_12","alias_value":"3ULWWNFDHFLH","created_at":"2026-07-05T03:41:21.005120+00:00"},{"alias_kind":"pith_short_16","alias_value":"3ULWWNFDHFLHX3L3","created_at":"2026-07-05T03:41:21.005120+00:00"},{"alias_kind":"pith_short_8","alias_value":"3ULWWNFD","created_at":"2026-07-05T03:41:21.005120+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.15589","citing_title":"Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets","ref_index":57,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU","json":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU.json","graph_json":"https://pith.science/api/pith-number/3ULWWNFDHFLHX3L3MWVJ3RG6EU/graph.json","events_json":"https://pith.science/api/pith-number/3ULWWNFDHFLHX3L3MWVJ3RG6EU/events.json","paper":"https://pith.science/paper/3ULWWNFD"},"agent_actions":{"view_html":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU","download_json":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU.json","view_paper":"https://pith.science/paper/3ULWWNFD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2112.08335&json=true","fetch_graph":"https://pith.science/api/pith-number/3ULWWNFDHFLHX3L3MWVJ3RG6EU/graph.json","fetch_events":"https://pith.science/api/pith-number/3ULWWNFDHFLHX3L3MWVJ3RG6EU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU/action/storage_attestation","attest_author":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU/action/author_attestation","sign_citation":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU/action/citation_signature","submit_replication":"https://pith.science/pith/3ULWWNFDHFLHX3L3MWVJ3RG6EU/action/replication_record"}},"created_at":"2026-07-05T03:41:21.005120+00:00","updated_at":"2026-07-05T03:41:21.005120+00:00"}