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2013 Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles
Alexei Borodin, Grigori Olshanski
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Electron. J. Probab. 18: 1-43 (2013). DOI: 10.1214/EJP.v18-2729

Abstract

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 $S_\infty$. In another context, the Thoma cone appears as the set of parameters for totally positive, upper triangular Toeplitz matrices of infinite size. The purpose of the paper is to construct a family $\{X^{(z,z')}\}$ of continuous time Markov processes on the Thoma cone, depending on two continuous parameters $z$ and $z'$. Our construction largely exploits specific properties of the Thoma cone related to its representation-theoretic origin, although we do not use representations directly. On the other hand, we were inspired by analogies with random matrix theory coming from models of Markov dynamics related to orthogonal polynomial ensembles.

We show that processes $X^{(z,z')}$ possess a number of nice properties, namely: (1) every $X^{(z,z')}$ is a Feller process; (2) the infinitesimal generator of $X^{(z,z')}$, its spectrum, and the eigenfunctions admit an explicit description; (3) in the equilibrium regime, the finite-dimensional distributions of $X^{(z,z')}$ can be interpreted as (the laws of) infinite-particle systems with determinantal correlations; (4) the corresponding time-dependent correlation kernel admits an explicit expression, and its structure is similar to that of time-dependent correlation kernels appearing in random matrix theory.

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Alexei Borodin. Grigori Olshanski. "Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles." Electron. J. Probab. 18 1 - 43, 2013. https://doi.org/10.1214/EJP.v18-2729

Information

Accepted: 13 August 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1293.60074
MathSciNet: MR3091721
Digital Object Identifier: 10.1214/EJP.v18-2729

Subjects:
Primary: 60J25
Secondary: 05E05, 60C05, 60G55, 60J27

JOURNAL ARTICLE
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Vol.18 • 2013
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