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Cycles séparants, isopérimétrie et modifications de distances dans les grandes cartes planaires aléatoires

Abstract : Planar maps are planar graphs drawn on the sphere and seen up to deformation. Many properties of maps are conjectured to be universal, in the sense that they do not depend on the details of the model.We begin by establishing an isoperimetric inequality in the infinite quadrangulation of the plane. We also confirm a conjecture by Krikun concerning the length of the shortest cycles separating the ball of radius r from infinity. We then consider the effect of modifications of distances on the large-scale geometry of uniform quadrangulations, extending the universality class of the Brownian map. We also show that the Tutte bijection, between quadrangulations and planar maps, is asymptotically an isometry. Finally, we establish an upper bound on the mixing time of the random walk in random maps.
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https://tel.archives-ouvertes.fr/tel-02412978
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Submitted on : Monday, December 16, 2019 - 9:13:06 AM
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Thomas Lehéricy. Cycles séparants, isopérimétrie et modifications de distances dans les grandes cartes planaires aléatoires. Probabilités [math.PR]. Université Paris Saclay (COmUE), 2019. Français. ⟨NNT : 2019SACLS476⟩. ⟨tel-02412978⟩

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