## 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_{∞}. 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.

Original language | English |
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Article number | 75 |

Journal | Electronic Journal of Probability |

Volume | 18 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

## Keywords

- Determinantal processes
- Feller processes
- Laguerre polynomials
- Markov intertwiners
- Meixner polynomials
- Thoma cone
- Thoma simplex