[2012/02/17]

A Gelfand-Tsetlin scheme of depth $N$ is a triangular array with m integers at level $m$, $m=1,\ldots,N$, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed $N$-th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation).

[2011/11/15]

We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes the Kerov’s operators first appeared in [Okounkov, arXiv:math/0002135], and also stems from the study of duality of graded graphs in [Fomin, 1994].

The talk is based on [8] and describes $\mathfrak{sl}(2,\mathbb{C})$ structures behind Markov jump processes on the Young and related branching graphs

[2011/07/04]

The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.

[2010/10/15]

We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group.

[2010/02/13]

In this note we present new examples of determinantal point processes with infinitely many particles.

The talk describes population genetics perspective behind infinite-dimensional diffusions preserving the two-parameter Poisson–Dirichlet distributions and related models. It is based on [2].

The talk describes algebraic/combinatorial perspective behind infinite-dimensional diffusions preserving the two-parameter Poisson–Dirichlet distributions and related models. It is based on [2], see also [4], [8]

[2009/04/11]

We consider a certain sequence of random walks. The state space of the n-th random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to a continuous-time Markov process.