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On fluctuations of eigenvalues of random Hermitian matrices. (English) Zbl 1039.82504
The author considers quite general random matrix ensembles of \(N \times N\) Hermitian matrices. The density is given by \(c_{N, M} \exp (-M \roman{trace} V(H) )\) where \(V\) is a polynomial of even degree with positive leading coefficient and \(H\) is thought of as a random matrix. Linear statistics problems involve the computation of \((1/N)\sum_{i=1}^{N}f(x_{i})\) where \(x_{i}, i=1, \cdots,N\), are the random eigenvalues of the Hermitian matrices for suitably nice functions \(f\). In particular, one can ask for the distribution of such sums. The main result of the paper is that if \(M/N \rightarrow 1\) then the sums converge in distribution to a normal random variable. The mean, which is zero for suitably normalized \(f\), and variance are directly related to the constants that appear in the asymptotic formula for the strong Szegő limit theorem. The above result is obtained for general even polynomials provided that certain limiting measures have support in a single interval.
The author shows the connection with the Szegő theorem, and also discusses the orthogonal and symplectic ensembles for general \(\beta\) in the statistical mechanics interpretation and finally some results for orthonormal polynomials.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B05 Classical equilibrium statistical mechanics (general)
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