## Duality and KPZ in Liouville Quantum Gravity

### Lundi 1er février 2010 14:00-15:00 - Duplantier Bertrand - CEA

Résumé : This is a joint work with Scott Sheffield, MIT. Liouville quantum gravity in two dimensions is described by the « random Riemannian manifold » obtained by changing the Lebesgue measure $dz$ in the plane by a random conformal factor $\exp [\gamma h(z)]$, where $h(z)$ is a random function called the Gaussian Free Field, and $\gamma$ a parameter.
This « random surface » is believed to be the continuum scaling limit of certain discretized random surfaces that can be studied with combinatorics and random matrix theory.
A famous formula, due to Knizhnik, Polyakov and Zamolodchikov in ’88, relates standard critical exponents in the Euclidean plane to their counterparts on the random surfaces mentioned above. We describe a recent proof of the KPZ formula in the probabilistic setting given above.

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