{"paper":{"title":"A Phase Transition for the Metric Distortion of Percolation on the Hypercube","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Itai Benjamini, Omer Angel","submitted_at":"2003-06-25T07:22:29Z","abstract_excerpt":"Let H_n be the hypercube {0,1}^n, and let H_{n,p} denote the same graph with Bernoulli bond percolation with parameter p=n^-\\alpha. It is shown that at \\alpha=1/2 there is a phase transition for the metric distortion between H_n and H_{n,p}. For \\alpha<1/2, asymptotically there is a map from H_n to H_{n,p} with constant distortion (depending only on \\alpha). For \\alpha>1/2 the distortion tends to infinity as a power of n. We indicate the similarity to the existence of a non-uniqueness phase in the context of infinite nonamenable graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0306355","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/math/0306355/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"}