Beta Jacobi ensembles and associated Jacobi polynomials

Abstract

Beta ensembles on the real line with three classical weights (Gaussian, Laguerre and Jacobi) are now realized as the eigenvalues of certain tridiagonal random matrices. The paper deals with beta Jacobi ensembles, the type with the Jacobi weight. Making use of the random matrix model, we show that in the regime where $\beta N\to const\in [0,\infty )$, with $N$ the system size, the empirical distribution of the eigenvalues converges weakly to a limiting measure which belongs to a new class of probability measures of associated Jacobi polynomials. This is analogous to the existing results for the other two classical weights. We also study the limiting behavior of the empirical measure process of beta Jacobi processes in the same regime and obtain a dynamic version of the above.

References in corpus (3)

• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices
• Beta Laguerre processes in a high temperature regime

Cited by in corpus (7)

• Full counting statistics for interacting trapped fermions
• Hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schroedinger equation
• On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Matrix addition and the Dunkl transform at high temperature
• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• Large Deviations for Ablowitz-Ladik lattice, and the Schur flow