Indexing metadata

Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles


 
Dublin Core PKP Metadata Items Metadata for this Document
 
1. Title Title of document Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles
 
2. Creator Author's name, affiliation, country Alexei Borodin; Massachusetts Institute of Technology; United States
 
2. Creator Author's name, affiliation, country Grigori Olshanski; Institute for Information Transmission Problems of the Russian Academy of Sciences; Russian Federation
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) determinantal processes; Feller processes; Thoma simplex; Thoma cone; Markov intertwiners; Meixner polynomials; Laguerre polynomials
 
3. Subject Subject classification 60J25; 60J27; 60G55; 60C05; 05E05
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2013-08-13
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2729
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-2729
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
15. Rights Copyright and permissions The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.

Summary of the Creative Commons Attribution License

You are free
  • to copy, distribute, display, and perform the work
  • to make derivative works
  • to make commercial use of the work
under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.