{"paper":{"title":"KPZ formulas for the Liouville quantum gravity metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Ewain Gwynne, Joshua Pfeffer","submitted_at":"2019-05-28T13:10:55Z","abstract_excerpt":"Let $\\gamma\\in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $\\gamma$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \\subset \\mathbb{C}$ which is independent from $h$, the Hausdorff dimensions of $X$ with respect to the Euclidean metric and with respect to the $\\gamma$-LQG metric $D_h$ are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum $\\gamma$-LQG metric is equal to the exponent $d_\\gamma > 2$ studied by Ding and Gwynne (2018), which describes distances"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.11790","kind":"arxiv","version":3},"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/1905.11790/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"}