Limit theorems for multivariate Bessel processes in the freezing regime

Abstract

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt{k}\cdot x$ for $k\to \infty$ with $x$ in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.

20 pages, 1 figure

References in corpus (3)

• Generalized Bessel function of Type D
• Two-step asymptotics of scaled Dunkl processes
• The Matsumoto and Yor process and infinite dimensional hyperbolic space

Cited by in corpus (7)

• Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices
• Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
• Functional central limit theorems for multivariate Bessel processes in the freezing regime
• Limit theorems for Jacobi ensembles with large parameters
• On the differential equations of frozen Calogero-Moser-Sutherland particle models
• Freezing Limits for Beta-Cauchy Ensembles
• Dunkl jump processes: relaxation and a phase transition