The moment convergence rates for largest eigenvalues of \(\beta\) ensembles
The author considers an \(n\times n\) stochastic matrix \(A\) with eigenvalues \(\lambda_1,\dotsc,\lambda_{n}\) whose joint probability density function has the form \[ p(x_1,\dotsc,x_{n})=C_{n,\beta}\prod_{1\leq j<k\leq n}|\lambda_{j}-\lambda_{k}|^{\beta}\exp\biggl\{-n\beta\sum_{k=1}^n\lambda_{k}^2\biggr\}, (x_1,\dotsc,x_{n})\in \mathbb R^n, \] for some constant \(C_{n,\beta}\) and \(\beta>0\). Let \(\lambda_{\max}\) denote the largest eigenvalue of \(A\), and let \(F_{n}\) be its distribution function. \textit{J. Ramírez, B. Rider} and \textit{B. Virág} [J. Am. Math. Soc. 24, 919--944 (2011; Zbl 1239.60005)] proved the existence of a continuous distribution function \(F\) such that \(\lim_{n\to\infty}F_{n}(x/2n^{2/3}+1)=F(x)\), \(x\in \mathbb R\). A main result states that \[ \lim_{\varepsilon\searrow 0}\sqrt\varepsilon\sum_{n\geq 1}\operatorname{E}[\lambda_{\max}-(1+\varepsilon)]_+=(3\sqrt2/4)\int_0^{\infty}\sqrt x(1-F(x))dx. \] The paper contains also three similar theorems.
scientific article; zbMATH DE number 6142990
scientific article