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Modèles de matrices et problèmes de bord dans la gravité de Liouville

Abstract : The present thesis concerns the study of boundary 2d gravity using both the methods of Liouville gravity and the matrix models. It hinges on two main topics: the $O(n)$ matrix model and the 2D non-critical string theory. In a first part, we explain the new method developed to study boundary conditions of lattice statistical models. It consists in using a matrix model formulation of the random lattice model in order to derive loop equations, of which we take the continuum limit. We concentrate on the anisotropic boundary conditions recently introduced in the $O(n)$ model. This method allows to find out the boundary phase diagram, the spectrum of boundary operators and the behavior under bulk and boundary RG flows. These results can be extended to other statistical models, such as the ADE models. In a second part, we consider the Lorentzian Liouville gravity coupled to a free boson, which can be re-interpreted as a string theory with a two-dimensional target space. The discrete version of the model is given by a matrix quantum mechanics (MQM). We computed the scattering amplitude of two long strings at the leading order using the chiral formalism of the MQM, and compared the result with a continuum computation. We also conjectured a formula for the scattering of an arbitrary number of long strings.
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https://tel.archives-ouvertes.fr/tel-00541162
Contributor : Jean-Emile Bourgine <>
Submitted on : Tuesday, November 30, 2010 - 3:39:53 AM
Last modification on : Monday, February 10, 2020 - 6:13:39 PM
Long-term archiving on: : Friday, October 26, 2012 - 5:01:28 PM

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• HAL Id : tel-00541162, version 1

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Jean-Emile Bourgine. Modèles de matrices et problèmes de bord dans la gravité de Liouville. Physique mathématique [math-ph]. Université Paris Sud - Paris XI, 2010. Français. ⟨tel-00541162⟩

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