Limit theorems for Bessel and Dunkl processes of large dimensions | Vidéo | Carmin.tv

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Limit theorems for Bessel and Dunkl processes of large dimensions

By Jeannette Woerner

Appears in collection : Modern Analysis Related to Root Systems with Applications / Analyse moderne liée aux systèmes de racines avec applications

We study Bessel and Dunkl processes $\left(X_{t, k}\right)_{t>0}$ on $\mathbb{R}^{N}$ with possibly multivariate coupling constants $k \geq 0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_{N}$ these Bessel processes are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types $\mathrm{A}$ and $\mathrm{B}$ and derive analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N \rightarrow \infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type $\mathrm{B}$ new non-symmetric semicircle-type limit distributions on $\mathbb{R}$ appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko-Pastur limit laws for the empirical measures of the zeroes of Hermite and Laguerre polynomials respectively. (based on joint work with Michael Voit)

Information about the video

Date of recording 19/10/2021
Date of publication 02/12/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19822003
Cite this video Woerner, Jeannette (19/10/2021). Limit theorems for Bessel and Dunkl processes of large dimensions. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19822003
URL https://dx.doi.org/10.24350/CIRM.V.19822003

Domain(s)

Probability

Bibliography

VOIT, Michael et WOERNER, Jeannette HC. Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions. arXiv preprint arXiv:2009.13928, 2020. - https://arxiv.org/abs/2009.13928

MSC codes

Document(s)

https://www.cirm-math.fr/RepOrga/2404/Slides/Woerner.pdf

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