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77
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 Surveys in Combinatorics (Cambridge
, 2005
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A random walk construction of uniform spanning trees and uniform labelled trees
 SIAM Journal on Discrete Mathematics
, 1990
"... Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter. ..."
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Cited by 82 (3 self)
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Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter.
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 39 (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...
The matrixforest theorem and measuring relations in small social groups
 Automation and Remote Control
, 1997
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Parking Functions and Noncrossing Partitions
 Electronic J. Combinatorics
, 1997
"... this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions ..."
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Cited by 28 (4 self)
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this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions
On the Analysis of Linear Probing Hashing
, 1998
"... This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, ..."
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Cited by 23 (9 self)
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This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n3/2), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 2 3, 3.) For sparse tables, the construction cost has expectation O(n), standard deviation O ( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.
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).
Recursive selfsimilarity for random trees, random triangulations and Brownian excursion
 Ann. Probab
, 1994
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 15 (3 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
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&apo ..."
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Cited by 14 (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
Uniform boundedness of critical crossing probabilities implies hyperscaling
 In preparation
, 1999
"... Abstract. We consider bond percolation on the ddimensional hypercubic lattice. Assuming the existence of a single critical exponent, the exponent ρ describing the decay rate of pointtoplane crossings at the critical point, we prove that hyperscaling holds whenever critical rectangle crossing prob ..."
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Cited by 13 (3 self)
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Abstract. We consider bond percolation on the ddimensional hypercubic lattice. Assuming the existence of a single critical exponent, the exponent ρ describing the decay rate of pointtoplane crossings at the critical point, we prove that hyperscaling holds whenever critical rectangle crossing probabilities are uniformly bounded away from 1. 1.