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.
60B20 (Primary), 60F05 (Secondary)
|Publication status||Published - 2017 Oct 11|
- Gaussian beta ensembles
- Martingale difference central limit theorem
- Semicircle law
- Tridiagonal random matrices
ASJC Scopus subject areas