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14
The twosided infinite extension of the Mallows model for random permutations
 ADVANCES IN APPLIED MATHEMATICS
, 2011
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From trees to seeds: on the inference of the seed from large trees in the uniform attachment model
"... We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics ..."
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We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain welldefined sense, global “balancedness ” properties of such trees. Our paper follows recent results on the same question for the preferential attachment model. 1
A refined Quicksort asymptotic
, 2012
"... The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Yn is known to converge almost surely to a nondegenerate random limit Y ..."
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The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Yn is known to converge almost surely to a nondegenerate random limit Y. This assumes a natural embedding of all Yn on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown: n d
A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees,” http://arxiv.org/pdf/1410.0469.pdf
, 2014
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Random Recursive Trees: A boundary theory approach. Preprint at http://arxiv.org/abs/1406.7614
, 2014
"... ABSTRACT. We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated DoobMartin compactification; it also gives a representation of the limit in terms of the input sequence of the algorithm. We furth ..."
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ABSTRACT. We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated DoobMartin compactification; it also gives a representation of the limit in terms of the input sequence of the algorithm. We further show that this approach can be used to obtain strong limit theorems for various tree functionals, such as path length or the Wiener index. 1.
PERSISTING RANDOMNESS IN RANDOMLY GROWING DISCRETE STRUCTURES: GRAPHS AND SEARCH TREES
, 2014
"... The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be dete ..."
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The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be detected and quantified with techniques from discrete potential theory. We also show that this approach can be used to obtain strong limit theorems in cases where previously only distributional convergence was known.
Growing random 3connected maps, or comment s’enfuir de l’hexagone. arXiv:1402.2632 [math.PR
, 2014
"... Abstract. We use a growth procedure for binary trees [10], a bijection between binary trees and irreducible quadrangulations of the hexagon [6], and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every m ..."
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Abstract. We use a growth procedure for binary trees [10], a bijection between binary trees and irreducible quadrangulations of the hexagon [6], and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. As n → ∞, the probability that the n’th map in the sequence is 3connected tends to 28/36. The sequence of maps has an almost sure limit G∞, and we show that G ∞ is the distributional local limit of large, uniformly random 3connected graphs. 1.
PRUNED DISCRETE RANDOM SAMPLES
"... Abstract. Let Xi, i ∈ N, be independent and identically distributed random variables with values in N0. We transform (‘prune’) the sequence {X1,..., Xn}, n ∈ N, of discrete random samples into a sequence {0, 1, 2,..., Yn}, n ∈ N, of contiguous random sets by replacing Xn+1 with Yn + 1 if Xn+1> Yn ..."
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Abstract. Let Xi, i ∈ N, be independent and identically distributed random variables with values in N0. We transform (‘prune’) the sequence {X1,..., Xn}, n ∈ N, of discrete random samples into a sequence {0, 1, 2,..., Yn}, n ∈ N, of contiguous random sets by replacing Xn+1 with Yn + 1 if Xn+1> Yn. We consider the asymptotic behaviour of Yn as n → ∞. Applications include path growth in digital search trees and the number of tables in Pitman’s Chinese restaurant process if the latter is conditioned on its limit value. 1.