Functional central limit theorems for multivariate Bessel processes in the freezing regime
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Date
2019-01
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Abstract
Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $\beta$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers.Here we provide associated functional central limit theorems which are locally uniform in $t$.The Gaussian limiting processes admit explicit representations in terms of matrix exponentials and the solutions of the associated deterministic dynamical systems.
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interacting particle systems, Calogero-Moser-Sutherland models, functional central limit theorems, zeros of Hermite polynomials, zeros of Laguerre polynomials, Hermite ensembles, Laguerre ensembles, Dyson Brownian motion