On Measures on Partitions Arising in Harmonic Analysis for Linear and Projective Characters of the Infinite Symmetric Group
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.
Combinatorics
Course taught in English at the Math in Moscow programme for international undergraduate students at the Independent University of Moscow, Spring 2011.
Pfaffian Stochastic Dynamics of Strict Partitions
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.
Random Strict Partitions and Determinantal Point Processes
In this note we present new examples of determinantal point processes with infinitely many particles.
Infinite-dimensional Diffusions Related to the Two-parameter Poisson-Dirichlet Distributions
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].
Infinite-Dimensional Diffusion Processes Approximated by Finite Markov Chains on Partitions
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]
Random Walks on Strict Partitions
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.