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102
Infinitedimensional diffusions as limits of random walks on partitions
, 2008
"... Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite–dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so–called z–measures, appeared earlier in the prob ..."
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Cited by 18 (6 self)
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Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite–dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so–called z–measures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described.
Direct limits of infinitedimensional Lie groups compared to direct limits in related categories
 J. Funct. Anal
, 2007
"... Let G be a Lie group which is the union of an ascending sequence G1 ⊆ G2 ⊆ · · · of Lie groups (all of which may be infinitedimensional). We study the question when G = lim Gn in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtaine ..."
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Cited by 15 (8 self)
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Let G be a Lie group which is the union of an ascending sequence G1 ⊆ G2 ⊆ · · · of Lie groups (all of which may be infinitedimensional). We study the question when G = lim Gn in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C∞diffeomorphisms of a σcompact smooth manifold M; and for test function groups C ∞ c (M,H) of compactly supported smooth maps with values in a finitedimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie groupvalued analytic maps, or a weak direct product of Lie groups.
Riemann–Hilbert problem and the discrete Bessel kernel
 INTERN. MATH. RESEARCH NOTICES
, 1999
"... We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define a discrete analog of 2 by 2 Riemann–Hilbert problems of special type. We also give an example, explicitly ..."
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Cited by 13 (8 self)
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We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define a discrete analog of 2 by 2 Riemann–Hilbert problems of special type. We also give an example, explicitly solvable in terms of classical special functions, when a discrete Riemann–Hilbert problem converges in a certain scaling limit to a conventional one; the example originates from the representation theory of the infinite symmetric group.
HUATYPE INTEGRALS OVER UNITARY GROUPS AND OVER PROJECTIVE LIMITS OF UNITARY GROUPS
, 2001
"... We discuss some natural maps from a unitary group U(n) to a smaller group U(n−m). (These maps are versions of the Livˇsic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetri ..."
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Cited by 12 (0 self)
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We discuss some natural maps from a unitary group U(n) to a smaller group U(n−m). (These maps are versions of the Livˇsic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces. (Values of the integrals are products of Ɣfunctions.) These integrals generalize Hua integrals. We construct inverse limits of unitary groups equipped with analogues of Haar measure and evaluate some integrals over these inverse limits. Let K be the real numbers R, the complex numbers C, or the algebra of quaternions H. By U(n, K) = O(n), U(n), Sp(n), we denote the unitary group of the space K n = R n, C n, H n. We also use the notation U ◦ (n, K): = SO(n), U(n), Sp(n) for the connected component of the group U(n, K). By σn, we denote the Haar measure on U ◦ (n, K) normalized by the condition σn(U ◦ (n, K)) = 1. Let Q be a matrix over K. By [Q]p, we denote the upper left block of the matrix Q of size p × p. By {Q}p, we denote the lower right block of size p × p. Let us represent a matrix g ∈ U ◦ (n, K) as an (m + (n − m)) × (m + (n − m)) block matrix () P Q. Consider the map
The qGelfandTsetlin graph, Gibbs measures and qToeplitz matrices
 Advances in Math. 229 (2012
"... and q–Toeplitz matrices ..."
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REPRESENTATION THEORY AND RANDOM POINT PROCESSES
, 2004
"... On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite–dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random point pr ..."
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Cited by 11 (8 self)
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On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite–dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random point processes that arise in random matrix theory. The paper is an extended version of the second author’s talk at the Congress.
LAGUERRE AND MEIXNER SYMMETRIC FUNCTIONS, AND INFINITEDIMENSIONAL DIFFUSION PROCESSES
, 2010
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