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62
Dyson’s Brownian motions, intertwining and interlacing
 Electron. J. Probab
"... A family of reflected Brownian motions is used to construct Dyson’s process of noncolliding Brownian motions. A number of explicit formulae are given, including one for the distribution of a family of coalescing Brownian motions. 1 ..."
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Cited by 40 (5 self)
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A family of reflected Brownian motions is used to construct Dyson’s process of noncolliding Brownian motions. A number of explicit formulae are given, including one for the distribution of a family of coalescing Brownian motions. 1
A pedestrian’s view on interacting particle systems, KPZ universality, and random matrices
, 2010
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Onedimensional stochastic growth and Gaussian . . .
, 2005
"... In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of ..."
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Cited by 21 (9 self)
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In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multimatrix models, respectively. We also discuss other models which are equivalent to the PNG model: directed polymers, the longest increasing subsequence problem, Young tableaux, a directed percolation model, kinkantikink gas, and Hammersley process.
Ordered Random Walks
, 2006
"... We construct the conditional version of k independent and identically distributed random walks on R given that they stay in strict order at all times. This is a generalisation of socalled noncolliding or nonintersecting random walks, the discrete variant of Dyson’s Brownian motions, which have b ..."
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Cited by 19 (1 self)
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We construct the conditional version of k independent and identically distributed random walks on R given that they stay in strict order at all times. This is a generalisation of socalled noncolliding or nonintersecting random walks, the discrete variant of Dyson’s Brownian motions, which have been considered yet only for nearestneighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob htransform with some positive regular function V that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson’s Brownian motions, the continuous analogue.
Asymptotics of Planchereltype random partitions
 JOURNAL OF ALGEBRA
, 2007
"... We present a solution to a problem suggested by Philippe Biane: We prove that a certain Plancherel–type probability distribution on partitions converges, as partitions get large, to a new determinantal random point process on the set Z+ of nonnegative integers. This can be viewed as an edge limit ..."
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Cited by 16 (6 self)
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We present a solution to a problem suggested by Philippe Biane: We prove that a certain Plancherel–type probability distribution on partitions converges, as partitions get large, to a new determinantal random point process on the set Z+ of nonnegative integers. This can be viewed as an edge limit transition. The limit process is determined by a correlation kernel on Z+ which is expressed through the Hermite polynomials, we call it the discrete Hermite kernel. The proof is based on a simple argument which derives convergence of correlation kernels from convergence of unbounded self–adjoint difference operators. Our approach can also be applied to a number of other probabilistic models. As an example, we discuss a bulk limit for one more Plancherel–type model of random partitions.
MARKOV PROCESSES ON THE PATH SPACE OF THE GELFANDTSETLIN GRAPH AND ON ITS BOUNDARY
, 2010
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