The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter (Formula presented.) is allowed to vary with the matrix size n. In particular, we show that as (Formula presented.) with (Formula presented.), 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