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TreeValued Markov Chains Derived From GaltonWatson Processes.
 Ann. Inst. Henri Poincar'e
, 1997
"... Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditio ..."
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Cited by 56 (9 self)
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Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson() offspring distribution. Running head. Treevalued Markov chains. Key words. Borel distribution, branching process, conditioning, GaltonWatson process, generalized Poisson distribution, htransform, pruning, random tree, sizebiasing, spinal decomposition, thinning. AMS Subject classifications 05C80, 60C05, 60J27, 60J80 Research supported in part by N.S.F. Grants DMS9404345 and 9622859 1 Contents 1 Introduction 2 1.1 Related topics : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Background and technical setup 5 2.1 Notation and terminology for trees : : : : : : : : : : : : : : :...
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : ..."
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Cited by 53 (14 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
Bootstrap percolation on infinite trees and nonamenable groups
 Combinatorics, Probability and Computing
"... Abstract. Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time ..."
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Cited by 49 (6 self)
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Abstract. Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time step, then it becomes occupied in the next step. This process is wellstudied on Z d; here we investigate it on regular and general infinite trees and on nonamenable Cayley graphs. The critical probability is the infimum of those values of p for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than k, then the critical probability is 1, while it is 1 − 1/k on the kary tree. A related result is that in any rooted tree T there is a way of erasing k children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree T ′ has branching number br(T ′ ) ≤ max{br(T) − k, 0}. We also prove that on any 2kregular nonamenable graph, the critical probability for the krule is strictly positive. 1. Introduction and
The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest
, 1997
"... Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solu ..."
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Cited by 30 (8 self)
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Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solution X of the Ito SDE dXv = n 4 \Gamma X 2 v \Gamma t \Gamma R v 0 Xudu \Delta \Gamma1 o dv + 2 p XvdBv on the interval [0; V t (X)), where V t (X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v V t (X). This conditioned form of the RayKnight description of Brownian local times arises from study of the asymptotic distribution as n !1 and 2k= p n ! ` of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a GaltonWatson branching process conditioned on its total progeny....
A phase transition in the random transposition random walk
 Pages 1726 in Banderier and Krattenthaler (2003) Bollobás, B
, 2003
"... Our work is motivated by Bourque and Pevzner’s (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time ..."
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Cited by 22 (9 self)
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Our work is motivated by Bourque and Pevzner’s (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time starting at the identity and let Dt be the minimum number of transpositions needed to go back to the identity from the location at time t. Dt undergoes a phase transition: the distance D cn/2 ∼ u(c)n, where u is an explicit function satisfying u(c) =c/2 for c ≤ 1 and u(c) <c/2 for c>1. In addition, we describe the fluctuations of D cn/2 about its mean in each of the three regimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulationfragmentation process and relating the behavior to the ErdősRenyi random graph model.
Gibbs distributions for random partitions generated by a fragmentation process
, 2006
"... process ..."
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Ranked fragmentations
 ESAIM P&S
"... distributions for random partitions generated by a ..."
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Cited by 20 (3 self)
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distributions for random partitions generated by a
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 19 (6 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).