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Article Dans Une Revue Probability Theory and Related Fields Année : 2017

Compact Brownian surfaces I. Brownian disks

Résumé

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BD_L, 0 < L < infinity) of random metric spaces homeomorphic to the closed unit disk of R^2, the space BD_L being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where L = 0. Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random.
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Dates et versions

hal-01182224 , version 1 (30-07-2015)
hal-01182224 , version 2 (20-10-2015)
hal-01182224 , version 3 (12-02-2016)

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Jérémie Bettinelli, Gregory Miermont. Compact Brownian surfaces I. Brownian disks. Probability Theory and Related Fields, 2017, 167 (3-4), pp.555--614. ⟨10.1007/s00440-016-0752-y⟩. ⟨hal-01182224v3⟩
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