{"paper":{"title":"Instability of set recurrence and Green's function on groups with the Liouville property","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.PR","authors_text":"David Revelle, Itai Benjamini","submitted_at":"2003-10-17T07:22:06Z","abstract_excerpt":"Let $\\mu$ and $\\nu$ be probability measures on a group \\Gamma and let G_\\mu and G_\\nu denote Green's function with respect to \\mu and \\nu . The group \\Gamma is said to admit instability of Green's function if there are symmetric, finitely supported measures $\\mu$ and \\nu and a sequence \\{x_n\\} such that G_\\mu(e, x_n)/G_\\nu(e,x_n) \\to 0, and \\Gamma admits instability of recurrence if there is a set S that is recurrent with respect to \\nu but transient with respect to \\mu . We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously kn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0310258","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/0310258/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"}