{"paper":{"title":"Sharp Phase Transition for the Formation of Infinite Tubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Tube percolation exhibits sharp thresholds at criticality for infinite tube formation, proven via OSSS inequality and adapted exploration algorithms.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Omer Bobrowski, Primoz Skraba, Shu Kanazawa","submitted_at":"2026-05-14T14:44:28Z","abstract_excerpt":"Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or \"escape\" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization of this question, asking whether a fixed loop (or, more generally, a topological sphere) can escape to infinity via a tube formed by random plaquettes. We refer to this phenomenon as tube percolation. We first compare tube percolation with previously studied higher-dimensional percolation phenomena, including face and cycle percolation. For tubes of codimens"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"the tubular one-arm event exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. [...] the existence of a box-crossing tube also exhibits a sharp threshold.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The OSSS inequality applies to the Boolean function defined by the tubular one-arm event under the adapted exploration algorithm that respects tube topology, without hidden obstructions from non-transitivity of tube connectedness.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Tube percolation exhibits sharp thresholds at criticality for infinite tube formation, proven via OSSS inequality and adapted exploration algorithms.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"db32d9ec963b5807de53705c20866ef2d1f40560f7e5543371d79f3b6fc22bd2"},"source":{"id":"2605.14910","kind":"arxiv","version":1},"verdict":{"id":"a53a002f-b1b7-471b-a0ba-713b33b1521d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:19:24.162217Z","strongest_claim":"the tubular one-arm event exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. [...] the existence of a box-crossing tube also exhibits a sharp threshold.","one_line_summary":"Tube percolation exhibits sharp thresholds at criticality for infinite tube formation, proven via OSSS inequality and adapted exploration algorithms.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The OSSS inequality applies to the Boolean function defined by the tubular one-arm event under the adapted exploration algorithm that respects tube topology, without hidden obstructions from non-transitivity of tube connectedness.","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"3350558aa6d6e99a53e6c4c985bd4930d4b268a96d1c94c0eb1edc20c40435b0"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}