{"paper":{"title":"Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ewain Gwynne, Jinwoo Sung","submitted_at":"2025-06-23T16:41:47Z","abstract_excerpt":"Let $\\mathcal G$ be an infinite graph -- not necessarily one-ended -- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\\mathcal G$ which reaches $\\infty$ in finite time and \"reflects off of $\\infty$\" infinitely many times.\n  We show that the Aldous-Broder algorithm for the random walk reflected off of $\\infty$ gives the free uniform spanning forest (FUSF) on $\\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\\infty$ gives the FUSF on $\\mathcal G$ on the event that the FUSF is connected, but not in general."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.18827","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.18827/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"}