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20
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 20 (0 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
Inhomogeneous Continuum Random Trees and the Entrance Boundary of the Additive Coalescent
 PROBAB. TH. REL. FIELDS
, 1998
"... Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; ..."
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Cited by 18 (13 self)
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Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time \Gamma1 with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees.
Random mappings, forests, and subsets associated with AbelCayleyHurwitz multinomial expansions
, 2001
"... Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal's b ..."
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Cited by 13 (9 self)
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Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal's bijection between mappings and marked rooted trees, have interesting probabilistic interpretations, and applications to the asymptotic structure of large random trees and mappings. An extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests. Research supported in part by N.S.F. Grants DMS 9703961 and DMS0071448 1 Contents 1
Brownian Bridge Asymptotics for Random pMappings
 Electonic J. Probab
, 2002
"... The Joyal bijection between doublyrooted trees and mappings can be lifted to a transformation on function space which takes treewalks to mappingwalks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 ..."
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Cited by 13 (8 self)
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The Joyal bijection between doublyrooted trees and mappings can be lifted to a transformation on function space which takes treewalks to mappingwalks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 AldousPitman result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random pmappings.
A family of random trees with random edge lengths
, 1999
"... We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpr ..."
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Cited by 11 (9 self)
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We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpretation is given in terms of sampling from the inhomogeneous continuum random tree of Aldous and Pitman (1998). Key words and phrases. Continuum tree, enumeration, random tree, spanning tree, weighted tree, Cayley's multinomial expansion.
Invariance principles for nonuniform random mappings and trees
 ASYMPTOTIC COMBINATORICS WITH APPLICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases ..."
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Cited by 11 (9 self)
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In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases, in this paper we formulate a sampling invariance principle in terms of iterates of a fixed number of random elements. We show that the sampling invariance principle implies many, but not all, of the distributional limits implied by the functional invariance principle. We give direct verifications of the sampling invariance principle in two successive generalizations of the uniform case, to pmappings (where elements are mapped to i.i.d. nonuniform elements) and Pmappings (where elements are mapped according to a Markov matrix). We compare with parallel results in the simpler setting of random trees.
Scaling limits of the uniform spanning tree and looperased random walk on finite graphs
, 2004
"... Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a looperased random walk from x to y on a large class of graphs that include the torus Z d n for d ≥ 5. Moreover, on this family of graphs we show that a suitably normalized finitedimensional scaling ..."
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Cited by 8 (0 self)
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Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a looperased random walk from x to y on a large class of graphs that include the torus Z d n for d ≥ 5. Moreover, on this family of graphs we show that a suitably normalized finitedimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.
Exchangeable pairs and Poisson approximation
 Probab. Surv
, 2005
"... This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poissonbinomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While ..."
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Cited by 7 (4 self)
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This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poissonbinomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered. 1
The matching, birthday and the strong birthday problem: a contemporary review
 JOURNAL OF STATISTICAL PLANNING AND INFERENCE
, 2005
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Weak convergence of random pmappings and the exploration process of inhomogeneous continuum random trees
 PROBAB. THEORY RELAT. FIELDS
, 2005
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