{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KDYJM5363FOXEG5BX2ODEATOHD","short_pith_number":"pith:KDYJM536","canonical_record":{"source":{"id":"1604.05440","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-19T06:18:44Z","cross_cats_sorted":[],"title_canon_sha256":"45d24be17bdb9050a9168bbe09ac5b3216734a99237a17304526e846f8145cc1","abstract_canon_sha256":"3abaa22940a9257127002e67583f5478d682ce33d923fb7554bdaff1cfb0bda1"},"schema_version":"1.0"},"canonical_sha256":"50f096777ed95d721ba1be9c32026e38cfa317477e60e4ed35ffecb8c9b71ac3","source":{"kind":"arxiv","id":"1604.05440","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05440","created_at":"2026-05-18T00:32:26Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05440v2","created_at":"2026-05-18T00:32:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05440","created_at":"2026-05-18T00:32:26Z"},{"alias_kind":"pith_short_12","alias_value":"KDYJM5363FOX","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KDYJM5363FOXEG5B","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KDYJM536","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KDYJM5363FOXEG5BX2ODEATOHD","target":"record","payload":{"canonical_record":{"source":{"id":"1604.05440","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-19T06:18:44Z","cross_cats_sorted":[],"title_canon_sha256":"45d24be17bdb9050a9168bbe09ac5b3216734a99237a17304526e846f8145cc1","abstract_canon_sha256":"3abaa22940a9257127002e67583f5478d682ce33d923fb7554bdaff1cfb0bda1"},"schema_version":"1.0"},"canonical_sha256":"50f096777ed95d721ba1be9c32026e38cfa317477e60e4ed35ffecb8c9b71ac3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:26.134531Z","signature_b64":"qzJha2WXzLXnaOrFobtKwyW5ImAG4IyoPEIUy5qZZ/yFyXVynfj1janjwDjxltqMryd875+MSys8gisIV69sBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50f096777ed95d721ba1be9c32026e38cfa317477e60e4ed35ffecb8c9b71ac3","last_reissued_at":"2026-05-18T00:32:26.133719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:26.133719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.05440","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":"2026-05-18T00:32:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0nF/mkvf4OzHssJW1V+85sZthAZj4EKm1lX19GTKjaW9gGZTRngA3BxxJ4oadnOKOTz0s5mNjXoMTUWdHRdBAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T01:26:48.578758Z"},"content_sha256":"a59b970684908f0b7723c6ee03d6d31191ec3c6187189a5abbda61e05a0dc64c","schema_version":"1.0","event_id":"sha256:a59b970684908f0b7723c6ee03d6d31191ec3c6187189a5abbda61e05a0dc64c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KDYJM5363FOXEG5BX2ODEATOHD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random walks and induced Dirichlet forms on self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ka-Sing Lau, Shi-Lei Kong, Ting-Kam Leonard Wong","submitted_at":"2016-04-19T06:18:44Z","abstract_excerpt":"Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\\partial_HX$ with the Gromov metric is H\\\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \\lambda <1$ on $(X, {\\mathfrak E})$. We show that the Martin boundary ${\\mathcal M}$ can be identified with $\\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05440","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":"2026-05-18T00:32:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YgL9BrQziM81HrzAVdlJ1rxE/2vwO7UAfjl+gwXFstBz/w9unuCZY3AztCTdh9GkK7DgEkmYuJexlS2Q8wOxBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T01:26:48.579507Z"},"content_sha256":"a6f1b47f4f9c314e37570c44632fa7b915551fa0b75aa97a08f560318ad25a76","schema_version":"1.0","event_id":"sha256:a6f1b47f4f9c314e37570c44632fa7b915551fa0b75aa97a08f560318ad25a76"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KDYJM5363FOXEG5BX2ODEATOHD/bundle.json","state_url":"https://pith.science/pith/KDYJM5363FOXEG5BX2ODEATOHD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KDYJM5363FOXEG5BX2ODEATOHD/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":"2026-06-01T01:26:48Z","links":{"resolver":"https://pith.science/pith/KDYJM5363FOXEG5BX2ODEATOHD","bundle":"https://pith.science/pith/KDYJM5363FOXEG5BX2ODEATOHD/bundle.json","state":"https://pith.science/pith/KDYJM5363FOXEG5BX2ODEATOHD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KDYJM5363FOXEG5BX2ODEATOHD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KDYJM5363FOXEG5BX2ODEATOHD","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":"3abaa22940a9257127002e67583f5478d682ce33d923fb7554bdaff1cfb0bda1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-19T06:18:44Z","title_canon_sha256":"45d24be17bdb9050a9168bbe09ac5b3216734a99237a17304526e846f8145cc1"},"schema_version":"1.0","source":{"id":"1604.05440","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05440","created_at":"2026-05-18T00:32:26Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05440v2","created_at":"2026-05-18T00:32:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05440","created_at":"2026-05-18T00:32:26Z"},{"alias_kind":"pith_short_12","alias_value":"KDYJM5363FOX","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KDYJM5363FOXEG5B","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KDYJM536","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:a6f1b47f4f9c314e37570c44632fa7b915551fa0b75aa97a08f560318ad25a76","target":"graph","created_at":"2026-05-18T00:32:26Z","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":"Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\\partial_HX$ with the Gromov metric is H\\\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \\lambda <1$ on $(X, {\\mathfrak E})$. We show that the Martin boundary ${\\mathcal M}$ can be identified with $\\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain ","authors_text":"Ka-Sing Lau, Shi-Lei Kong, Ting-Kam Leonard Wong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-19T06:18:44Z","title":"Random walks and induced Dirichlet forms on self-similar sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05440","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:a59b970684908f0b7723c6ee03d6d31191ec3c6187189a5abbda61e05a0dc64c","target":"record","created_at":"2026-05-18T00:32:26Z","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":"3abaa22940a9257127002e67583f5478d682ce33d923fb7554bdaff1cfb0bda1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-19T06:18:44Z","title_canon_sha256":"45d24be17bdb9050a9168bbe09ac5b3216734a99237a17304526e846f8145cc1"},"schema_version":"1.0","source":{"id":"1604.05440","kind":"arxiv","version":2}},"canonical_sha256":"50f096777ed95d721ba1be9c32026e38cfa317477e60e4ed35ffecb8c9b71ac3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50f096777ed95d721ba1be9c32026e38cfa317477e60e4ed35ffecb8c9b71ac3","first_computed_at":"2026-05-18T00:32:26.133719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:26.133719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qzJha2WXzLXnaOrFobtKwyW5ImAG4IyoPEIUy5qZZ/yFyXVynfj1janjwDjxltqMryd875+MSys8gisIV69sBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:26.134531Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.05440","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a59b970684908f0b7723c6ee03d6d31191ec3c6187189a5abbda61e05a0dc64c","sha256:a6f1b47f4f9c314e37570c44632fa7b915551fa0b75aa97a08f560318ad25a76"],"state_sha256":"9303a353bef708d0d051879b4fa86b1878f3cd2b3c8b228f8a712541bd65d690"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r+8lj242jZWmZRUs0fT+VFs+syfscsGHYWCD8KJ37DqkARVU8gJdUECGILSNKR7FxK3EEYQFbsIcvKJD+PplBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T01:26:48.583580Z","bundle_sha256":"78dfdc54885a068a117a77d69762ff63dabedc7e74bb640abe6414905d169385"}}