Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices

Abstract

Bessel processes $(X_{t,k}{)}_{t\ge 0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge 0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to \infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigenvectors of the covariance matrices of the Gaussian limits and study applications to CLTs for the intermediate particles for $k\to \infty$ and then $N\to \infty$.

20 pages

References in corpus (10)

• Matrix Models for Beta Ensembles
• The Calogero-Sutherland Model and Generalized Classical Polynomials
• Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics
• Hermite and Laguerre $\beta$-ensembles: asymptotic corrections to the eigenvalue density
• Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime
• Limit theorems for multivariate Bessel processes in the freezing regime
• Two limiting regimes of interacting Bessel processes
• Two-step asymptotics of scaled Dunkl processes

Cited by in corpus (9)

• 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
• Dunkl jump processes: relaxation and a phase transition
• Freezing Limits for Beta-Cauchy Ensembles
• On the differential equations of frozen Calogero-Moser-Sutherland particle models
• Distances of roots of classical orthogonal polynomials
• On the extremal eigenvalues of Jacobi ensembles at zero temperature
• Hua-Pickrell diffusions and differential equations related with pseudo-Jacobi polynomials