Local semicircle law and Gaussian fluctuation for Hermite \(\beta\) ensemble. (Chinese. English summary) Zbl 1488.15066
Summary: Let \(\beta>0\) and considering an \(n\)-point process \(\lambda_1,\lambda_2,\ldots, \lambda_n\) from Hermite \(\beta\) ensemble on the real line \(\mathbb{R}\), some previous authors discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper, we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than \(\sqrt{\log n}\), i.e., local semicircle law holds. And the number of positive states in \((0, \infty)\) is proved to fluctuate normally around its mean \(n/2\) with variance like \(\log n/\pi^2\beta\). The proof of the results relies mainly on the counting states in any interval and the classical martingale method.
MSC:
15B52 | Random matrices (algebraic aspects) |
60F05 | Central limit and other weak theorems |
60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |