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Even Walks and Estimates of High Moments of Large Wigner Random Matrices, preprint: arXiv:0806.0157
"... We revisit the problem of estimates of moments E{Tr(An) 2s} of random n ×n matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given by the structure of closed even walks w2s and their graphs g(w2s) th ..."
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Cited by 7 (5 self)
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We revisit the problem of estimates of moments E{Tr(An) 2s} of random n ×n matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given by the structure of closed even walks w2s and their graphs g(w2s) that arise in these studies. We show that the degree of a vertex α of g(w2s) depends not only on the selfintersections degree of α but also on the total number of all nonclosed instants of selfintersections of w2s or more precisely, on the number of instants of broken tree structure. This result is used to fill the gaps of earlier considerations. 1
The Wiener Index of Random Digital Trees
, 2012
"... The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An impo ..."
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Cited by 3 (2 self)
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The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An important class of random trees for which the Wiener index was not studied so far are random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in affirmative two questions posed by Neininger.
Tail estimates for the Brownian excursion area and other Brownian areas
, 2007
"... Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested ..."
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Cited by 3 (3 self)
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Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the twodimensional saddle point method.
The integral of the supremum process of Brownian motion
 J. APPL. PROBAB
, 2007
"... In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0, T]. The Laplace transform of A(T) follows as a consequence. The main proof involves ..."
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Cited by 2 (2 self)
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In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0, T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion.
Patterns in random permutations avoiding the pattern 132
, 2014
"... We consider a random permutation drawn from the set of 132avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by nλ(σ)/2 where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be ..."
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Cited by 2 (0 self)
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We consider a random permutation drawn from the set of 132avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by nλ(σ)/2 where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.
Probabilistic Analysis of Additive Shape Parameters in Random Digital Trees
, 2014
"... Established by D. E. Knuth in 1963, analysis of algorithms is an important area which lies in the overlap of mathematics and theoretical computer science. The main aim of this area is to understand the stochastic behavior of algorithms. Moreover, another important issue is to obtain asymptotic prop ..."
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Cited by 1 (1 self)
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Established by D. E. Knuth in 1963, analysis of algorithms is an important area which lies in the overlap of mathematics and theoretical computer science. The main aim of this area is to understand the stochastic behavior of algorithms. Moreover, another important issue is to obtain asymptotic properties of data structures. In this thesis, we study one of the most often used data structure, the digital tree family, through additive shape parameters. The first part of this thesis is an introduction to the digital tree family, including the construction, the random model we are going to use and a survey of known results and related mathematical techniques. We also give some examples to illustrate how these mathematical techniques have been utilized in past researches of random digital trees. The second part contains the new results we derived, including new applications of the PoissonLaplaceMellin method and general frameworks for
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices
, 2007
"... Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability an ..."
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Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.