{"paper":{"title":"Equivalence of Liouville measure and Gaussian free field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nathana\\\"el Berestycki, Scott Sheffield, Xin Sun","submitted_at":"2014-10-20T19:47:41Z","abstract_excerpt":"Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\\gamma \\in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity area measure $\\mu := e^{\\gamma h(z)} dz.$ It is known that the field $h$ a.s. determines the measure $\\mu_h$. We show that the converse is true: namely, $h$ is measurably determined by $\\mu_h$. More generally, given a random closed fractal subset $\\mathcal A$ endowed with a Frostman measure $\\sigma$ whose support is $\\mathcal A$ (independent of $h$), a Gaussian multiplicative chaos measure $\\mu_{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5407","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1410.5407/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"}