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32
Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results
 INT. MATH. RES. NOT
, 2003
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E.: Averages of characteristic polynomials in Random Matrix Theory
 Commun. Pure and Applied Math
, 2006
"... Abstract. We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensemble ..."
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Cited by 24 (3 self)
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Abstract. We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulas by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact pfaffian/determinantal formulas for the discrete averages are proved using standard tools of linear algebra; no application of orthogonal or skeworthogonal polynomials is needed. 1.
Determinantal probability measures
, 2002
"... Abstract. Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent example of which has been the uniform spanning tree measure. Our main results concern relationship ..."
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Cited by 15 (3 self)
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Abstract. Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent example of which has been the uniform spanning tree measure. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology,
Isomonodromy transformations of linear systems of difference equations
"... Abstract. We introduce and study “isomonodromy ” transformations of the matrix linear difference equation Y (z + 1) = A(z)Y (z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Z m(n+1)−1 on the space of coefficients A(z) (here m is the size of matr ..."
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Cited by 13 (2 self)
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Abstract. We introduce and study “isomonodromy ” transformations of the matrix linear difference equation Y (z + 1) = A(z)Y (z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Z m(n+1)−1 on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Schlesinger equations and the Schlesinger transformations known in the isomonodromy theory, can be obtained as limits of our action in two different limit regimes. Similarly to the continuous case, for m = n = 2 the difference Schlesinger equations and their qanalogs yield discrete Painlevé equations; examples include dPII, dPIV, dPV, and qPVI.
Riemann–Hilbert methods in the theory of orthogonal polynomials
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, AMER. MATH. SOC
, 2006
"... In this paper we describe various applications of the RiemannHilbert method to the theory of orthogonal polynomials on the line and on the circle. ..."
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Cited by 11 (0 self)
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In this paper we describe various applications of the RiemannHilbert method to the theory of orthogonal polynomials on the line and on the circle.
Markov processes on partitions
, 2004
"... Abstract. We introduce and study a family of Markov processes on partitions. The processes preserve the socalled zmeasures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have ..."
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Cited by 9 (7 self)
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Abstract. We introduce and study a family of Markov processes on partitions. The processes preserve the socalled zmeasures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels. We also compute the scaling limits of the kernels in two different regimes. The limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random Young diagrams, and the behavior of the Young diagrams near the diagonal. Our results show that recently discovered analogy between random partitions arising in representation theory and spectra of random matrices extends to the associated time–dependent models.
Orbit measures, random matrix theory and interlaced determinantal processes
, 2009
"... A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability ..."
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Cited by 4 (0 self)
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A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal.
Random surface growth with a wall and Plancherel measures for O
, 2009
"... We consider a Markov evolution of lozenge tilings of a quarterplane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local correlations to translationinvariant Gibbs measures in the liquid ..."
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Cited by 4 (2 self)
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We consider a Markov evolution of lozenge tilings of a quarterplane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local correlations to translationinvariant Gibbs measures in the liquid region, and obtain new discrete Jacobi and symmetric Pearcey determinantal point processes near the wall. The model can be viewed as the oneparameter family of Plancherel measures for the infinitedimensional orthogonal group, and we use this interpretation to derive the determinantal formula for the correlation functions