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Résultats exacts sur les modèles de boucles en deux dimensions

Abstract : Using analytical and numerical methods of two-dimensional Statistical Mechanics (transfer matrix, conformal invariance, Coulomb gas, Yang-Baxter equations, Bethe Ansatz, Monte-Carlo), we tackle some problems which do not belong to the scope of the compact Gaussian model : critical antiferromagnetic Potts model, Brauer loop model. These models exhibit original critical properties, such as the appearance of non-compact degrees of freedom. These properties arise when one introduces loop intersections or staggered Boltzmann weights in the lattice loop model. In the case of the antiferromagnetic Potts model, we develop the study of the Yang-Baxter structure, and identify a family of Bethe states corresponding to the non-compact degrees of freedom. Numerical calculations for large system sizes allow us to state a conjecture about the scaling law for the effective compactification radius. In the case of the Brauer model with loop fugacity n = 0, we propose a model for a scale-invariant escape path, and we determine its critical properties by numerical methods. As a non-local observable, the escape path gives a characterization of similarities with and differences from random walks.
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Contributor : Yacine Ikhlef <>
Submitted on : Tuesday, April 1, 2008 - 12:28:13 PM
Last modification on : Wednesday, October 14, 2020 - 3:41:45 AM
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  • HAL Id : tel-00268765, version 1



Yacine Ikhlef. Résultats exacts sur les modèles de boucles en deux dimensions. Physique mathématique [math-ph]. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00268765⟩



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