{"paper":{"title":"Exceptional graphs for the random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alex Roberts, Alex Scott, Bhargav Narayanan, Carla Groenland, Juhan Aru, Tom Johnston","submitted_at":"2018-05-16T12:47:42Z","abstract_excerpt":"If $\\mathcal{W}$ is the simple random walk on the square lattice $\\mathbb{Z}^2$, then $\\mathcal{W}$ induces a random walk $\\mathcal{W}_G$ on any spanning subgraph $G\\subset \\mathbb{Z}^2$ of the lattice as follows: viewing $\\mathcal{W}$ as a uniformly random infinite word on the alphabet $\\{\\mathbf{x}, -\\mathbf{x}, \\mathbf{y}, -\\mathbf{y} \\}$, the walk $\\mathcal{W}_G$ starts at the origin and follows the directions specified by $\\mathcal{W}$, only accepting steps of $\\mathcal{W}$ along which the walk $\\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \\subset \\mathbb{Z}^2$, the walk $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06277","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":""},"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"}