209 citations · 1.2k across the 68 of their papers we have counts for
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Infinite-dimensional -Jacobi Markov processes
Grigori Olshanski
The classical Jacobi polynomials on the interval are eigenfunctions of a second order differential operator. It is well known that this operator generates a diffusion proc…
Macdonald polynomials and extended Gelfand-Tsetlin graph
Grigori Olshanski
Using Okounkov's -integral representation of Macdonald polynomials we construct an infinite sequence of countable sets linked by transition probabilities fro…
Determinantal point processes and fermion quasifree states
Grigori Olshanski
Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the…
A hierarchy of Palm measures for determinantal point processes with gamma kernels
Alexander I. Bufetov, Grigori Olshanski
The gamma kernels are a family of projection kernels on a doubly infinite -dimensional lattice. They are expressed through Euler's gamma function an…
The topological support of the z-measures on the Thoma simplex
Grigori Olshanski
The Thoma simplex is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are a family of probability measures on depending…
Extended Gelfand-Tsetlin graph, its q-boundary, and q-B-splines
Grigori Olshanski
A continuation of the joint work by Vadim Gorin and the author, J. Funct. Anal. 270 (2016), 375-418; arXiv:1504.06832. The extended Gelfand-Tsetlin graph, introduced in that paper,…
Markov dynamics on the dual object to the infinite-dimensional unitary group
Grigori Olshanski
These are notes for a mini-course of 3 lectures given at the St. Petersburg School in Probability and Statistical Physics (June 2012). My aim was to explain, on the example of a pa…
An interacting particle process related to Young tableaux
Alexei Borodin, Grigori Olshanski
We discuss a stochastic particle system consisting of a two-dimensional array of particles living in one space dimension. The stochastic evolution bears a certain similarity to Ham…
Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles
Alexei Borodin, Grigori Olshanski
The Thoma cone is an infinite-dimensional locally compact space, which is closely related to the space of extremal characters of the infinite symmetric group. In another context, t…
Equilibrium Kawasaki dynamics and determinantal point processes
Eugene Lytvynov, Grigori Olshanski
Let "mu" be a point process on a countable discrete space "X". Under assumption that "mu" is quasi-invariant with respect to any finitary permutation of "X", we describe a general…
The two-sided infinite extension of the Mallows model for random permutations
Alexander Gnedin, Grigori Olshanski
We introduce a probability distribution Q on the group of permutations of the set Z of integers. Distribution Q is a natural extension of the Mallows distribution on the finite sym…
Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary
Alexei Borodin, Grigori Olshanski
We construct a four-parameter family of Markov processes on infinite Gelfand-Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces…
The quasi-invariance property for the Gamma kernel determinantal measure
Grigori Olshanski
The Gamma kernel is a projection kernel of the form (A(x)B(y)-B(x)A(y))/(x-y), where A and B are certain functions on the one-dimensional lattice expressed through Euler's Gamma fu…
-exchangeability via quasi-invariance
Alexander Gnedin, Grigori Olshanski
For positive , the -exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for…
A q-analogue of de Finetti's theorem
Alexander Gnedin, Grigori Olshanski
A q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces ove…
Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter
Grigori Olshanski
We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters…
Difference operators and determinantal point processes
Grigori Olshanski
We consider a family {P} of determinantal point processes arising in representation theory and random matrix theory. The processes live on the one-dimensional lattice and their cor…
Infinite-dimensional diffusions as limits of random walks on partitions
Alexei Borodin, Grigori Olshanski
The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measur…
Asymptotics of Plancherel-type random partitions
Alexei Borodin, Grigori Olshanski
We present a solution to a problem suggested by Philippe Biane: We prove that a certain Plancherel-type probability distribution on partitions converges, as partitions get large, t…
Meixner polynomials and random partitions
Alexei Borodin, Grigori Olshanski
The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of ha…
The boundary of the Eulerian number triangle
Alexander Gnedin, Grigori Olshanski
The Eulerian triangle is a classical array of combinatorial numbers defined by a linear recursion. The associated boundary problem asks one to find all extreme nonnegative solution…