We consider the asymptotics of the Plancherel measures on partitions of n as n goes to infinity. We prove that the local structure of a Plancherel typical partition in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel.On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers [2, 3] and from the combinatorial proof given in [23].Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

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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 is parallel to the degeneration of the Whittaker kernel to the Christoffel-Darboux kernel for Laguerre polynomials.

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 two 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 hypergeometric function.From the analytic point of view, the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by Richard Askey and Peter Lesky. One difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0. We do this by expressing the polynomials in terms of a solution of a discrete Riemann-Hilbert problem and computing the (nonoscillatory) asymptotics of this solution.From the point of view of statistical physics, we study thermodynamic limit of a discrete log-gas system. An interesting feature of this log-gas is that its density function is asymptotically equal to the characteristic function of an interval. Our point processes describe how different the random particle configuration is from the typical 'densely packed' configuration.

Abstract. Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure M n . That is, the weight M n (λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group S n indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (LoganShepp 1977, Vershik-Kerov 1977. In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov's unpublished work notes, 1999.