{"paper":{"title":"A random Schr\\\"odinger operator associated with the Vertex Reinforced Jump Process on infinite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Christophe Sabot, Xiaolin Zeng","submitted_at":"2015-07-28T20:10:35Z","abstract_excerpt":"This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schr\\\"odinger operator. On infinite graphs, we define a 1-dependent random potential $\\beta$ extending that defined in [20] on finite graphs, and consider its associated random Schr\\\"odinger operator $H_\\beta$. We construct a random function $\\psi$ as a limit of martingales, such that $\\psi=0$ when the VRJP is recurrent, and $\\psi$ is a positive generalized eigenfunction of the random Schr\\\"odinger operator with eigenvalue $0$, when the VRJP is transient. Then we p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07944","kind":"arxiv","version":4},"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"}