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101
Infinite wedge and random partitions
 Selecta Mathematica (new series
"... The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example, ..."
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Cited by 94 (6 self)
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The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,
A Fredholm determinant formula for Toeplitz determinants
"... The purpose of this note is to explain how the results of [13] apply to a question raised by A. Its and, independently, P. Deift during the MSRI workshop ..."
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Cited by 75 (10 self)
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The purpose of this note is to explain how the results of [13] apply to a question raised by A. Its and, independently, P. Deift during the MSRI workshop
Distribution on partitions, point processes, and the hypergeometric kernel
 COMM. MATH. PHYS
, 1999
"... We study a 3–parametric family of stochastic point processes on the one–dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. Th ..."
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Cited by 70 (28 self)
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We study a 3–parametric family of stochastic point processes on the one–dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed before, see math.RT/9810015, the correlation functions of these limit processes also have determinantal form with so–called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a ‘discrete integrable operator’. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel for Meixner orthogonal polynomials. This fact
Harmonic analysis on the infinitedimensional unitary group and determinantal point processes
, 2001
"... The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characters of U( ..."
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Cited by 58 (21 self)
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The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite–dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω into the space of point configurations on the real line without 2 points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable’ form and are expressed through the Gauss
Point processes and the infinite symmetric group. Part I: The general formalism and the density function
, 1998
"... We study a 2parametric family of probability measures on an infinite–dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S. Kerov, G. Olshanski and A. Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773778). Our approach i ..."
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Cited by 49 (20 self)
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We study a 2parametric family of probability measures on an infinite–dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S. Kerov, G. Olshanski and A. Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773778). Our approach is to interprete them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes. In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well–known Poisson–Dirichlet distribution. The higher correlation functions are explicitly calculated in the subsequent paper (A. Borodin). In the third part (A. Borodin and G. Olshanski) we discuss some applications and relationships with the random matrix theory. The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.
ZMeasures on Partitions, RobinsonSchenstedKnuth Correspondence, and beta = 2 Random Matrix Ensembles
 RANDOM MATRIX MODELS AND THEIR APPLICATIONS, VOLUME 40 OF MATH. SCI. RES. INST. PUBL
, 1999
"... We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with "fi = 2 ensembles" arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general ..."
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Cited by 37 (11 self)
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We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with "fi = 2 ensembles" arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from socalled generalized regular representations of the infinite symmetric group.
Discrete gap probabilities and discrete Painlevé equations
 DUKE MATH J
, 2003
"... We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm ..."
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Cited by 29 (6 self)
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We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a zmeasure, or as normalized Toeplitz determinants with symbols eη(ζ +ζ −1) and (1 + ξζ)
Harmonic functions on multiplicative graphs and interpolation polynomials
, 1999
"... We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’s S and P ..."
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Cited by 26 (12 self)
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We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’s S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.
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 22 (15 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.