{"paper":{"title":"Heat kernel for Liouville Brownian motion and Liouville graph distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fuxi Zhang, Jian Ding, Ofer Zeitouni","submitted_at":"2018-07-02T00:26:16Z","abstract_excerpt":"We show the existence of the scaling exponent $\\chi\\in (0,4[(1+\\gamma^2/4)- \\sqrt{1+\\gamma^4/16}]/\\gamma^2]$ of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $\\gamma<2$ on $\\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel satisfies, for any fixed $u,v\\in \\mathbb V^o$, the short time estimates $$ \\lim_{ t \\to 0} \\frac{\\log |\\log {\\mathsf p}_t^\\gamma(u,v)| }{|\\log t|}=\\frac{\\chi}{2-\\chi}, \\ \\mbox{\\rm a.s.} $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.00422","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":""},"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"}