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38
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 62 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
Symmetry of matrixvalued stochastic processes and noncolliding diffusion particle systems
 J. Math. Phys
, 2004
"... As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian randommatrix ensembles, we report a systematic study of hermitian matrixvalued processes and their eigenvalue processes associated with the chiral and nonstandard randommatrix ensembles. In addition to the n ..."
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Cited by 45 (19 self)
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As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian randommatrix ensembles, we report a systematic study of hermitian matrixvalued processes and their eigenvalue processes associated with the chiral and nonstandard randommatrix ensembles. In addition to the noncolliding Brownian motions, we introduce a oneparameter family of temporally homogeneous noncolliding systems of the Bessel processes and a twoparameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the AltlandZirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochasticcalculus proof of a version of the HarishChandra (ItzyksonZuber) formula of integral over unitary group is established. I
A REPRESENTATION FOR NONCOLLIDING RANDOM WALKS
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2002
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NonColliding Random Walks, Tandem Queues, And Discrete Orthogonal Polynomial Ensembles
, 2001
"... We show that the function h(x) = Q i<j (x j x i ) is harmonic for any random walk in R k with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = fx : x 1 < x 2 < < x k g onto a point where h vanishes, we ..."
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Cited by 30 (4 self)
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We show that the function h(x) = Q i<j (x j x i ) is harmonic for any random walk in R k with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = fx : x 1 < x 2 < < x k g onto a point where h vanishes, we define the corresponding Doob htransform. For certain special cases, we show that the marginal distribution of the conditioned process at a fixed time is given by a familiar discrete orthogonal polynomial ensemble. These include the Krawtchouk and Charlier ensembles, where the underlying walks are binomial and Poisson, respectively. We refer to the corresponding conditioned processes in these cases as the Krawtchouk and Charlier processes. In [O'Connell and Yor (2001b)], a representation was obtained for the Charlier process by considering a sequence of M/M/1 queues in tandem. We present the analogue of this representation theorem for the Krawtchouk process, by considering a sequence of discretetime M/M/1 queues in tandem. We also present related results for random walks on the circle, and relate a system of noncolliding walks in this case to the discrete analogue of the circular unitary ensemble (CUE).
Noncolliding Brownian motion and determinantal processes
 J. STAT. PHYS
, 2007
"... A system of onedimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the htransform of absorbing BM in a ..."
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Cited by 29 (14 self)
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A system of onedimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the htransform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The KarlinMcGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the KarlinMcGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrixkernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrixkernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrixkernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.
Random matrices, noncolliding processes and queues
 TO APPEAR IN SÉMINAIRE DE PROBABILITÉS XXXVI
, 2002
"... This is survey of some recent results connecting random matrices, noncolliding processes and queues. ..."
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Cited by 24 (3 self)
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This is survey of some recent results connecting random matrices, noncolliding processes and queues.
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.
Functional central limit theorems for vicious walkers
 STOCH. STOCH. REP
, 2003
"... We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite tim ..."
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Cited by 17 (14 self)
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We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval (0, T] for the first type and in an infinite time interval (0, ∞) for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson’s model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multidimensional generalization of Imhof’s relation, whose original form relates the Brownian meander and the threedimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion