{"paper":{"title":"Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Bertrand Duplantier (IPHT), R\\'emi Rhodes (CEREMADE), Scott Sheffield (MIT), Vincent Vargas (CEREMADE)","submitted_at":"2012-12-03T20:30:17Z","abstract_excerpt":"Gaussian Multiplicative Chaos is a way to produce a measure on $\\R^d$ (or subdomain of $\\R^d$) of the form $e^{\\gamma X(x)} dx$, where $X$ is a log-correlated Gaussian field and $\\gamma \\in [0,\\sqrt{2d})$ is a fixed constant. A renormalization procedure is needed to make this precise, since $X$ oscillates between $-\\infty$ and $\\infty$ and is not a function in the usual sense. This procedure yields the zero measure when $\\gamma=\\sqrt{2d}$.\n  Two methods have been proposed to produce a non-trivial measure when $\\gamma=\\sqrt{2d}$. The first involves taking a derivative at $\\gamma=\\sqrt{2d}$ (and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0529","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":""},"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"}