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Discrete Polynuclear Growth and Determinantal processes
 Comm. Math. Phys
, 2003
"... Abstract. We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F1 GOE TracyWidom distribution in terms of the Airy process. ..."
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Cited by 80 (6 self)
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Abstract. We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F1 GOE TracyWidom distribution in terms of the Airy process. We also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem. 1. Introduction and
Integrals over classical groups, random permutations, Toda and Toeplitz lattices
 Comm. Pure Appl. Math
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Differential Equations for Dyson Processes
, 2004
"... We call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n×n Hermitian matrices, and the eigenvalues desc ..."
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Cited by 25 (3 self)
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We call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n×n Hermitian matrices, and the eigenvalues describe n curves. Given sets X1,...,Xm the probability that for each k no curve passes through Xk at time τk is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process (which was introduced by Prähofer and Spohn as the limiting stationary process for a polynuclear growth model) and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process. In earlier work the authors found a system of ordinary differential equations with independent variable ξ whose solution determined the probabilities Pr (A(τ1) <ξ1 + ξ,...,A(τm) <ξm + ξ), where τ → A(τ) denotes the top curve of the Airy process. Our first result is a generalization and strengthening of this. We assume that each Xk is a finite union of intervals and find a system of partial differential equations, with the endpoints of the intervals of the Xk as independent variables, whose solution determines the probability that for each k no curve passes through Xk at time τk. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.
PDEs for the joint distributions of the Dyson, Airy and Sine processes
 Ann. Probab
, 2005
"... In a celebrated paper, Dyson shows that the spectrum of a n × n random Hermitian matrix, diffusing according to an OrnsteinUhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the ..."
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Cited by 21 (4 self)
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In a celebrated paper, Dyson shows that the spectrum of a n × n random Hermitian matrix, diffusing according to an OrnsteinUhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy Process at two different times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process, which itself is based on the joint probability
The Pfaff lattice and skeworthogonal polynomials
 Int. Math. Res. Notices
, 1999
"... Consider a semiinfinite skewsymmetric moment matrix, m ∞ evolving according to the vector fields ∂m/∂tk = Λ k m + mΛ ⊤k, where Λ is the shift matrix. Then The skewBorel decomposition m ∞:= Q −1 JQ ⊤−1 leads to the socalled Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a s ..."
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Cited by 15 (3 self)
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Consider a semiinfinite skewsymmetric moment matrix, m ∞ evolving according to the vector fields ∂m/∂tk = Λ k m + mΛ ⊤k, where Λ is the shift matrix. Then The skewBorel decomposition m ∞:= Q −1 JQ ⊤−1 leads to the socalled Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the affine symplectic algebra. The taufunctions for the system are shown to be pfaffians and the wave vectors skeworthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).
PDEs for the Gaussian ensemble with external source and the Pearcey distribution
 MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 395
, 2007
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Recurrences for elliptic hypergeometric integrals
, 2006
"... In [15], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generali ..."
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Cited by 12 (6 self)
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In [15], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (qhypergeometric) integral identities as limits from the elliptic level.
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 7 (3 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.