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35
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 79 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p ..."
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Cited by 38 (15 self)
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In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...
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 37 (18 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...
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 35 (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 : : : : : : : : : : : : : : :...
Random cutting and records in deterministic and random trees
 Alg
, 2006
"... Abstract. We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned ..."
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Cited by 30 (9 self)
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Abstract. We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random, and the case when we condition on the tree. The proofs are based on Aldous ’ theory of the continuum random tree. 1.
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 25 (7 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....
Random matrix theory over finite fields
 Bull. Amer. Math. Soc. (N.S
"... Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with sym ..."
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Cited by 22 (6 self)
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Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, RogersRamanujan type identities, potential theory, and various measures on partitions.
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... ..."
Critical random graphs: diameter and mixing time
"... Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Worm ..."
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Cited by 13 (4 self)
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Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald [5]. These results extend to clusters of size n 2/3 of pbond percolation on any dregular nvertex graph where such clusters exist, provided that p(d − 1) ≤ 1 + O(n −1/3). 1.
AbelCayleyHurwitz multinomial expansions associated with random mappings, forests, and subsets
, 1998
"... Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the r ..."
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Cited by 13 (12 self)
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Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, 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 which generalizes Cayley's expansion over unrooted labeled trees. Contents 1 Introduction 2 Research supported in part by N.S.F. Grant DMS9703961 2 Probabilistic Interpretations 5 3 Cayley's multinomial expansion 11 4 Random Mappings 14 4.1 Mappings from S to S : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4.2 The random set of cyclic points : : : : : : : : : : : : : : : : : : : : : : : 18 5 Random Forests 19 5.1 Distribution of the roots of a pforest : : : : : : : : : : : : : : : : : : : : 19 5.2 Conditioning on the set...