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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 ..."
Abstract

Cited by 12 (7 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.
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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Susceptibility in subcritical random graphs
 125207. OF RANDOM GRAPHS WITH GIVEN VERTEX DEGREES 25
"... Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its ..."
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Cited by 5 (4 self)
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Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. 1.
Individual displacements for linear probing hashing with different insertion policies
 ACM Transactions on Algorithms
, 2005
"... Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occ ..."
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Cited by 4 (1 self)
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Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.
SUSCEPTIBILITY IN INHOMOGENEOUS RANDOM GRAPHS
"... Abstract. We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives ..."
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Cited by 2 (2 self)
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Abstract. We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives a way of determining the critical point by solving certain linear equations. Here we relate the susceptibility of suitable random graphs to a quantity associated to the corresponding branching process, and study both quantities in various natural examples. 1.
The ForegoundBackground Processor Sharing Queue: an overview
, 2004
"... We give an overview of the results in the literature on singleserver queues with the FB discipline. The FB discipline gives service to the customer that has received the least amount of service. This not so wellknown discipline has some appealing features, and performs well for heavytailed servic ..."
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We give an overview of the results in the literature on singleserver queues with the FB discipline. The FB discipline gives service to the customer that has received the least amount of service. This not so wellknown discipline has some appealing features, and performs well for heavytailed service times. We describe results on the queue length, sojourn time, and the influence of variability in the service times.