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Ergodicity of multiplicative statistics
, 2008
"... For a subfamily of multiplicative measures on integer partitions we give conditions for associated Young diagrams to converge in probability after a proper rescaling to a certain curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape cove ..."
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For a subfamily of multiplicative measures on integer partitions we give conditions for associated Young diagrams to converge in probability after a proper rescaling to a certain curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.
On time dynamics of coagulationfragmentation processes
, 2008
"... 1 transient 2 We establish a characterization of coagulationfragmentation processes, such that the induced birth and death processes depicting the total number of groups at time t ≥ 0 are time homogeneous. Based on this, we provide a characterization of meanfield Gibbs coagulationfragmentation mod ..."
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1 transient 2 We establish a characterization of coagulationfragmentation processes, such that the induced birth and death processes depicting the total number of groups at time t ≥ 0 are time homogeneous. Based on this, we provide a characterization of meanfield Gibbs coagulationfragmentation models, which extends the one derived by Hendriks et al. As a by product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to meanfield models. transient 3 1 Introduction, objective and the context The time dynamics of a time homogeneous Markov process X(t), t ≥ 0 on a space Ω = {η} of states η is described by the set of transition probabilities p ˜ ζ (η; t): = P(X(t) = η X(0) = ˜ ζ), ˜ ζ, η ∈ Ω, t ≥ 0.
The limit shape of random permutations with polynomially growing cycle weights
, 2014
"... In this work we are considering the behaviour of the limit shape of Young diagrams associated to random permutations on the set {1,..., n} under a particular class of multiplicative measures with polynomial growing cycle weights. Our method is based on generating functions and complex analysis (s ..."
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In this work we are considering the behaviour of the limit shape of Young diagrams associated to random permutations on the set {1,..., n} under a particular class of multiplicative measures with polynomial growing cycle weights. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the socalled randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.
Limiting shapes of birthanddeath processes on young diagrams
 Adv. in Appl. Math
"... Abstract We consider a family of birth processes and birthanddeath processes on Young diagrams of integer partitions of n. This family incorporates three famous models from very different fields: Rost's totally asymmetric particle model (in discrete time), Simon's urban growth model, an ..."
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Abstract We consider a family of birth processes and birthanddeath processes on Young diagrams of integer partitions of n. This family incorporates three famous models from very different fields: Rost's totally asymmetric particle model (in discrete time), Simon's urban growth model, and Moran's infinite alleles model. We study stationary distributions and limit shapes as n tends to infinity, and present a number of results and conjectures.
The Limit Shape of a Stochastic Bulgarian Solitaire
, 2014
"... We consider a stochastic version of Bulgarian solitaire: A number of cards are distributed in piles; in every round a new pile is formed by cards from the old piles, and each card is picked independently with a fixed probability. This game corresponds to a multisquare birthanddeath process on You ..."
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We consider a stochastic version of Bulgarian solitaire: A number of cards are distributed in piles; in every round a new pile is formed by cards from the old piles, and each card is picked independently with a fixed probability. This game corresponds to a multisquare birthanddeath process on Young diagrams of integer partitions. We prove that this process converges in a strong sense to an exponential limit shape as the number of cards tends to infinity. Furthermore, we bound the probability of deviation from the limit shape and relate this to the number of rounds played in the solitaire.