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Elliptic Determinantal Processes and Elliptic Dyson Models
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13
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Elliptic Determinantal Processes and Elliptic Dyson Models
Інший ID:
2010 Mathematics Subject Classification: 60J65; 60G44; 82C22; 60B20; 33E05; 17B22
DOI:10.3842/SIGMA.2017.079
DOI:10.3842/SIGMA.2017.079
Посилання:
Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.
The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html.
The author would like to thank the anonymous referees whose comments considerably improved
the presentation of the paper. A part of the present work was done during the participation of
the author in the ESI workshop on “Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics” (March 20–24, 2017). The present author expresses his gratitude
for the hospitality of Erwin Schr¨odinger International Institute for Mathematics and Physics
(ESI) of the University of Vienna and for well-organization of the workshop by Christian Krattenthaler, Masatoshi Noumi, Simon Ruijsenaars, Michael J. Schlosser, Vyacheslav P. Spiridonov,
and S. Ole Warnaar. He also thanks Soichi Okada, Masatoshi Noumi, Simon Ruijsenaars, and
Michael J. Schlosser for useful discussion. This work was supported in part by the Grant-in-Aid
for Scientific Research (C) (No. 26400405), (B) (No. 26287019), and (S) (No. 16H06338) of
Japan Society for the Promotion of Science.
Дата:
2017
Переглядів:
545
Завантажень:
232
Анотація:
We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families AN₋₁, BN, CN and DN, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.
