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Cartes aléatoires hyperboliques

Abstract : This thesis falls into the theory of random planar maps, which has been active in the last fifteen years, and more precisely into the study of hyperbolic models.We are first interested in a model of dynamical random triangulations based on edge-flips, where we prove a lower bound on the mixing time.In the rest of this thesis, the main objects that we study are the random hyperbolic triangulations called PSHT. These are hyperbolic variants of the Uniform Infinite Planar Triangulation (UIPT), and were introduced by Nicolas Curien in 2014. We first establish a near-critical scaling limit result: if we let the hyperbolicity parameter go to its critical value at the same time as the distances are renormalized, the PSHT converge to a random metric space that we call the hyperbolic Brownian plane. We also study precise metric properties of the PSHT and of the hyperbolic Brownian plane, such as the structure of their infinite geodesics. We obtain as well new properties of the Poisson boundary of the PSHT.Finally, we are interested in another natural model of hyperbolic random maps: supercritical causal maps, which are obtained from supercritical Galton--Watson trees by adding edges between vertices at the same height. We establish metric hyperbolicity results about these maps, as well as properties of the simple random walk (including a positive speed result). Some of the properties we obtain are robust, and may be generalized to any planar map containing a supercritical Galton--Watson tree.
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Submitted on : Friday, November 16, 2018 - 5:25:06 PM
Last modification on : Wednesday, September 16, 2020 - 4:05:35 PM
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Thomas Budzinski. Cartes aléatoires hyperboliques. Probabilités [math.PR]. Université Paris-Saclay, 2018. Français. ⟨NNT : 2018SACLS426⟩. ⟨tel-01925486⟩



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