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Abstract
We study large uniform random bipartite quadrangulations whose genus grows linearly with the number of faces. Their local convergence was recently established by Budzinski and the author [9, 10]. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar with high probability up to logarithmic radius, and we prove that there exist non-contractible cycles of constant length with positive probability.
Highlights
Planar maps Maps are surfaces formed by gluing polygons together
Before going to the proofs of the main results, we want to compare our model with pre-existing models of hyperbolic geometry, and present a few problems that would be a natural extension of this work
Comparison with hyperbolic geometry High genus maps can be seen as discrete models of two dimensional hyperbolic geometry
Summary
Planar maps Maps are surfaces formed by gluing polygons together. They have been given a lot of attention in the last decades, especially in the case of planar maps, i.e. maps of the sphere. They exhibit hyperbolic features, as their average degree (which is directly linked to the average curvature of the map) is asymptotically higher than in planar (or fixed genus) maps Some of their geometric properties have been studied, starting with uniform unicellular maps, i.e. maps with only one face. The general case (i.e. uniform maps with various kinds of constraints on the face degrees) has been studied more recently, starting with uniform high genus triangulations, which converge locally in distribution towards a random hyperbolic triangulation of the plane [10]. Our result In [9], it is proven that the local limit of q(n) is a random infinite quadrangulation of the plane It implies that for every fixed r , the ball of radius r around the root is planar with probability 1 − o(1) as n → ∞. In the third section we prove Proposition 1, and the last two sections are devoted to the proofs of Theorems 1 and 2
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