Abstract not found

Abstract not found

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

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,

Abstract. 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.

Abstract. 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

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 so--called generalized regular representations of the infinite symmetric group.

Abstract. 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.

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 2F1-kernel 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 z-measure, or as normalized Toeplitz determinants with symbols eη(ζ +ζ −1) and (1 + ξζ)

The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point processes satisfy the so-called condition (�λ), which is a general form of Gibbsianness. Under a continuity assumption, the Gibbsian conditional probabilities can be identified explicitly. KEY WORDS: Determinantal point process; fermion point process; Gibbs point process; Papangelou intensity; stochastic domination; percolation.