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SCALING LIMITS OF RANDOM PLANAR MAPS WITH A UNIQUE LARGE FACE
, 2012
"... We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We proof that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown tha ..."
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We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We proof that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n −1/2 is described by a Brownian excursion. The planar maps, with the graph metric rescaled by n −1/2, are then shown to converge in distribution towards Aldous’ Brownian tree in the Gromov–Hausdorff topology. In the proofs we rely on the Bouttier–di Francesco–Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.
THE PEELING PROCESS OF INFINITE BOLTZMANN PLANAR MAPS
"... Abstract. We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a biased random walk, in terms of which the admissib ..."
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Abstract. We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a biased random walk, in terms of which the admissibility criterion has a very simple interpretation. The finite random planar maps under consideration were recently proved to possess a welldefined local limit known as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien and Le Gall, we show that the peeling process on the IBPM can be obtained from the peeling process of finite random maps by conditioning the perimeter process to stay positive. The simplicity of the resulting description of the peeling process allows us to obtain the scaling limit of the associated perimeter and volume process for arbitrary regular critical weight sequences. 1.