{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:OEQI2GMDWYJ63VEL6M3YV6MROT","short_pith_number":"pith:OEQI2GMD","canonical_record":{"source":{"id":"1910.11733","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-10-25T14:00:19Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"caa92574a98e3fad49d25a0a5393c1fe744c1d35b08c7f93e2f24a8351c700e0","abstract_canon_sha256":"2caa95708a00ad3088d48da3f4bf7e19c5050674ddc720e746b033dc3bac8528"},"schema_version":"1.0"},"canonical_sha256":"71208d1983b613edd48bf3378af99174c914311b44c1112d96bb12c835cdb892","source":{"kind":"arxiv","id":"1910.11733","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.11733","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"arxiv_version","alias_value":"1910.11733v1","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.11733","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"pith_short_12","alias_value":"OEQI2GMDWYJ6","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"pith_short_16","alias_value":"OEQI2GMDWYJ63VEL","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"pith_short_8","alias_value":"OEQI2GMD","created_at":"2026-07-05T00:14:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:OEQI2GMDWYJ63VEL6M3YV6MROT","target":"record","payload":{"canonical_record":{"source":{"id":"1910.11733","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-10-25T14:00:19Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"caa92574a98e3fad49d25a0a5393c1fe744c1d35b08c7f93e2f24a8351c700e0","abstract_canon_sha256":"2caa95708a00ad3088d48da3f4bf7e19c5050674ddc720e746b033dc3bac8528"},"schema_version":"1.0"},"canonical_sha256":"71208d1983b613edd48bf3378af99174c914311b44c1112d96bb12c835cdb892","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:14:56.446067Z","signature_b64":"nFc5OaO6RSFXDTtB9ZNXz+Yh/BGpSzSnoNkUqL4KGRteE0KHECsewUDrSNcF2ouUiInXS3pBHawtQ0aH4CVEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71208d1983b613edd48bf3378af99174c914311b44c1112d96bb12c835cdb892","last_reissued_at":"2026-07-05T00:14:56.445651Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:14:56.445651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1910.11733","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:14:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EPqN25x+UMh0UV5mPOkBa4TqycXTImQLg/J3ZNGEQq3Eytix7z8KXBjMEOrm98isaSuTssVNOW5ZsHCLi3pdBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T08:05:10.805170Z"},"content_sha256":"6f29874c2dee5b1b8cb412a35f5d65a22f1c26e27ace2285413600e31a2f95d2","schema_version":"1.0","event_id":"sha256:6f29874c2dee5b1b8cb412a35f5d65a22f1c26e27ace2285413600e31a2f95d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:OEQI2GMDWYJ63VEL6M3YV6MROT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Separation profiles, isoperimetry, growth and compression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.GR","authors_text":"Antoine Gournay, Corentin Le Coz","submitted_at":"2019-10-25T14:00:19Z","abstract_excerpt":"We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Tim\\'ar) for various graphs using the isoperimetric profile, growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile $\\mathrm{Sep}(n)$ is also bounded by powers of $n$. For many amenable groups, we show a lower bound in $n/ \\log(n)^a$ and, for any group which has a non-trivial compression exponent in an $L^p$-space, an upper bound in $n/ \\log(n)^b$. We show that solvable groups of exponential growth cannot have a separation profi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.11733","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/1910.11733/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:14:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kkbqdHs2O7dZFaO2fckTw2HqWXImrnY0m5XHYLkzdBeKSZ7nrvURFyyUYhMfv+2BrFzVvqspR/SCLkbtMmnJAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T08:05:10.805677Z"},"content_sha256":"1ea2b33ed6ef6465f2fcfdafaeba8d8470d88675a9e87076d665b4c0e464b6c6","schema_version":"1.0","event_id":"sha256:1ea2b33ed6ef6465f2fcfdafaeba8d8470d88675a9e87076d665b4c0e464b6c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OEQI2GMDWYJ63VEL6M3YV6MROT/bundle.json","state_url":"https://pith.science/pith/OEQI2GMDWYJ63VEL6M3YV6MROT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OEQI2GMDWYJ63VEL6M3YV6MROT/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":"2026-08-14T08:05:10Z","links":{"resolver":"https://pith.science/pith/OEQI2GMDWYJ63VEL6M3YV6MROT","bundle":"https://pith.science/pith/OEQI2GMDWYJ63VEL6M3YV6MROT/bundle.json","state":"https://pith.science/pith/OEQI2GMDWYJ63VEL6M3YV6MROT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OEQI2GMDWYJ63VEL6M3YV6MROT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:OEQI2GMDWYJ63VEL6M3YV6MROT","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":"2caa95708a00ad3088d48da3f4bf7e19c5050674ddc720e746b033dc3bac8528","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-10-25T14:00:19Z","title_canon_sha256":"caa92574a98e3fad49d25a0a5393c1fe744c1d35b08c7f93e2f24a8351c700e0"},"schema_version":"1.0","source":{"id":"1910.11733","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.11733","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"arxiv_version","alias_value":"1910.11733v1","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.11733","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"pith_short_12","alias_value":"OEQI2GMDWYJ6","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"pith_short_16","alias_value":"OEQI2GMDWYJ63VEL","created_at":"2026-07-05T00:14:56Z"},{"alias_kind":"pith_short_8","alias_value":"OEQI2GMD","created_at":"2026-07-05T00:14:56Z"}],"graph_snapshots":[{"event_id":"sha256:1ea2b33ed6ef6465f2fcfdafaeba8d8470d88675a9e87076d665b4c0e464b6c6","target":"graph","created_at":"2026-07-05T00:14:56Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1910.11733/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Tim\\'ar) for various graphs using the isoperimetric profile, growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile $\\mathrm{Sep}(n)$ is also bounded by powers of $n$. For many amenable groups, we show a lower bound in $n/ \\log(n)^a$ and, for any group which has a non-trivial compression exponent in an $L^p$-space, an upper bound in $n/ \\log(n)^b$. We show that solvable groups of exponential growth cannot have a separation profi","authors_text":"Antoine Gournay, Corentin Le Coz","cross_cats":["math.MG","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-10-25T14:00:19Z","title":"Separation profiles, isoperimetry, growth and compression"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.11733","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":"sha256:6f29874c2dee5b1b8cb412a35f5d65a22f1c26e27ace2285413600e31a2f95d2","target":"record","created_at":"2026-07-05T00:14:56Z","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":"2caa95708a00ad3088d48da3f4bf7e19c5050674ddc720e746b033dc3bac8528","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-10-25T14:00:19Z","title_canon_sha256":"caa92574a98e3fad49d25a0a5393c1fe744c1d35b08c7f93e2f24a8351c700e0"},"schema_version":"1.0","source":{"id":"1910.11733","kind":"arxiv","version":1}},"canonical_sha256":"71208d1983b613edd48bf3378af99174c914311b44c1112d96bb12c835cdb892","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71208d1983b613edd48bf3378af99174c914311b44c1112d96bb12c835cdb892","first_computed_at":"2026-07-05T00:14:56.445651Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:14:56.445651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nFc5OaO6RSFXDTtB9ZNXz+Yh/BGpSzSnoNkUqL4KGRteE0KHECsewUDrSNcF2ouUiInXS3pBHawtQ0aH4CVEBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:14:56.446067Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.11733","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f29874c2dee5b1b8cb412a35f5d65a22f1c26e27ace2285413600e31a2f95d2","sha256:1ea2b33ed6ef6465f2fcfdafaeba8d8470d88675a9e87076d665b4c0e464b6c6"],"state_sha256":"d8364716abadc298985297913540fba8aa58e0daf9a28bd1c49ee808ce8f6b64"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cVwcAWDwLht57e/ernyoEMQArerHrgTzpHqxYegTFY2g5F7fPLzThIQypV13qoCTMYoBp94FAtlHV3ZRXVLsDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T08:05:10.815616Z","bundle_sha256":"d9c2efa964d70f9c172f47b12560313a09f39381696f851a13b0f93f536ae56e"}}