Liouville quantum gravity and KPZ
Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy ... and a constant 0[less than or equal to]γ<2. The Liouville quantum gravity measure on D is the weak limit as epsilon-->0 of the measures ... where dz is Lebesgue measure on D and h epsil...
| Main Authors: | , |
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2012
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| Online Access: | http://hdl.handle.net/1721.1/71590 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1826202654687100928 |
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| author | Duplantier, Bertrand Sheffield, Scott Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Duplantier, Bertrand Sheffield, Scott Roger |
| author_sort | Duplantier, Bertrand |
| collection | MIT |
| description | Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy ... and a constant 0[less than or equal to]γ<2. The Liouville quantum gravity measure on D is the weak limit as epsilon-->0 of the measures ... where dz is Lebesgue measure on D and h epsilon (z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity. |
| first_indexed | 2024-09-23T12:12:38Z |
| format | Article |
| id | mit-1721.1/71590 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:12:38Z |
| publishDate | 2012 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/715902022-09-28T00:43:15Z Liouville quantum gravity and KPZ Duplantier, Bertrand Sheffield, Scott Roger Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger Sheffield, Scott Roger Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy ... and a constant 0[less than or equal to]γ<2. The Liouville quantum gravity measure on D is the weak limit as epsilon-->0 of the measures ... where dz is Lebesgue measure on D and h epsilon (z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity. French National Research Agency (ANR-08-BLAN-0311-CSD5) Centre National de la Recherche Scientifique (CNRS grant PEPS-PTI 2010) National Science Foundation (U.S.) (NSF grant DMS 0403182) National Science Foundation (U.S.) (grant DMS 064558) National Science Foundation (U.S.) (grant OISE 0730136) 2012-07-12T14:46:44Z 2012-07-12T14:46:44Z 2010-12 2010-01 Article http://purl.org/eprint/type/JournalArticle 0020-9910 1432-1297 http://hdl.handle.net/1721.1/71590 Duplantier, Bertrand, and Scott Sheffield. “Liouville Quantum Gravity and KPZ.” Inventiones mathematicae 185.2 (2010): 333–393. https://orcid.org/0000-0002-5951-4933 en_US http://dx.doi.org/10.1007/s00222-010-0308-1 Inventiones Mathematicae Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Springer-Verlag arXiv |
| spellingShingle | Duplantier, Bertrand Sheffield, Scott Roger Liouville quantum gravity and KPZ |
| title | Liouville quantum gravity and KPZ |
| title_full | Liouville quantum gravity and KPZ |
| title_fullStr | Liouville quantum gravity and KPZ |
| title_full_unstemmed | Liouville quantum gravity and KPZ |
| title_short | Liouville quantum gravity and KPZ |
| title_sort | liouville quantum gravity and kpz |
| url | http://hdl.handle.net/1721.1/71590 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT duplantierbertrand liouvillequantumgravityandkpz AT sheffieldscottroger liouvillequantumgravityandkpz |