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Universality of general \(\beta\)-ensembles. (English) Zbl 1298.15040
In the theory of random matrices, one considers a random variable valued on the space of matrices \(M_{N\times N}\). The typical probability measure over \(H\in M_{N\times N}\) is of the form \(\exp\left( -N\beta \operatorname{Tr}(V(H)/2) \right)/Z\), where \(V:\mathbb{R}\to\mathbb{R}\) is called the potential and \(\operatorname{Tr}\) is the trace applied to the matrix \(V(H)\). Also, \(\beta>0\) is interpreted as the inverse temperature, and \(Z\) is the normalizing constant.
One fundamental question analyzes the behavior, when \(N\to\infty\), of the associated set of eigenvalues \(\{\lambda_{i}^{(N)}\}_{i=1}^{N} \) (called the log-gas system), which are random for any fixed \(N\). In the case \(V(x):=x^{2}\), called the Gaussian model, it is known that for smooth functions \(G\), the sum \(\sum_{i=1}^{N}G(\lambda_{i}^{(N)})/N\) converges to \(\int_{-2}^{2}G(x)\rho_{sc}(x)\,dx\), where \(\rho_{sc}(x):=\sqrt{4-x^{2}}/2\pi\) is the celebrated semi-circle law.
In this paper, the authors obtain a result of this kind (Theorem 2.1), when \(V\) could be a more general analytic function satisfying the condition \(\inf_{x\in \mathbb{R}}V^{\prime\prime}(x)>0\). This is done after appropriate normalizing of the so-called empirical distribution of the eigenvalues. Their result also represents the universality that one finds in various random matrix models.
To achieve this task, the authors carry out several estimates, and make use of tools such as the so-called Helffe-Sjöstrand functional calculus and the Stieltjes transform, as well as a careful analysis of the location of the eigenvalues. In contrast with other results, they do not make use of the theory of orthogonal polynomials.

MSC:
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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