Beta Laguerre ensembles in global regime

Abstract

Beta Laguerre ensembles which are generalizations of Wishart ensembles and Laguerre ensembles can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart ($\beta =1$) case and the Laguerre ($\beta =2$) case, for fixed $\beta$, it is known that the empirical distribution of the eigenvalues of these ensembles converges weakly to Marchenko--Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter $\beta$ is allowed to vary as a function of the matrix size $N$. We show that the above Marchenko--Pastur law holds as long as $\beta N\to \infty$. When $\beta N\to 2c\in (0,\infty )$, the limit is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.

References in corpus (3)

• Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models
• On central limit theorems in stochastic geometry
• Convergence of the eigenvalue density for beta-Laguerre ensembles on short scales

Cited by in corpus (4)

• 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
• Dyson's disordered linear chain from a random matrix theory viewpoint
• Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform