The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter β is allowed to vary with the matrix size n. In particular, we show that as n→ ∞ with nβ→ ∞, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
- Gaussian beta ensembles
- Martingale difference central limit theorem
- Semicircle law
- Tridiagonal random matrices
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty