Results 1 - 10
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73
The objective method: Probabilistic combinatorial optimization and local weak convergence
, 2003
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Processes on unimodular random networks
, 2007
"... We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability fr ..."
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Cited by 124 (6 self)
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We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications
The topological structure of scaling limits of large planar maps
- Invent. Math
"... We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space con ..."
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Cited by 73 (13 self)
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We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n −1/4, converges in distribution as n → ∞ towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4. 1
Limits of normalized quadrangulations. The Brownian map
- Ann. Probab
, 2004
"... Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name t ..."
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Cited by 32 (1 self)
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Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of Brownian map is defined. The weak convergences hold in these metric spaces. 1
Invariance principles for random bipartite planar maps
- ANN. PROBAB
, 2007
"... Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling fa ..."
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Cited by 30 (10 self)
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Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with n faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in distribution once rescaled by n 1/4 to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, Di Francesco and Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution putting weight qk on faces of degree 2k: the radius of such maps, conditioned to have n faces (or n vertices) and under a criticality assumption, converges in distribution once rescaled by n 1/4 to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided. The convergence of rescaled bipartite maps to the Brownian map, in the sense introduced by Marckert and Mokkadem, is also shown. The proofs of these results rely on a new invariance principle for two-type spatial Galton–Watson trees.
Local structure of random quadrangulations
, 2005
"... This paper is an adaptation of a method used in [1] to the model of random quadrangulations. We prove local weak convergence of uniform measures on quadrangulations and show that local growth of quadrangulation is governed by certain critical time-reversed branching process. As an intermediate resul ..."
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Cited by 30 (0 self)
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This paper is an adaptation of a method used in [1] to the model of random quadrangulations. We prove local weak convergence of uniform measures on quadrangulations and show that local growth of quadrangulation is governed by certain critical time-reversed branching process. As an intermediate result we calculate a biparametric generating function for certain class of quadrangulations with boundary. 1
Local limit of labelled trees and expected volume growth in a random quadrangulation
, 2005
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Geodesics in large planar maps and in the brownian map
, 2009
"... We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the ..."
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Cited by 21 (6 self)
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We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.
