Limit correlation functions at zero for fixed trace random matrix ensembles. (Russian, English. English summary) Zbl 1153.60026

Götze, F.; Gordin, M. I.; Levina, A.

Zap. Nauchn. Semin. POMI 341, 68-80 (2007); translation in J. Math. Sci., New York 147, No. 4, 6884-6890 (2007).

The authors investigate the large-\(N\) limiting behaviour of the eigenvalue correlation functions of \(N\times N\)-Hermitian matrices \({\mathcal H}_N\) chosen at random according to the density \[ g_N(A)= (2\pi)^{-N^2/2}\exp\{-\text{trace}(A^2)/2\}\quad\text{for }A\in{\mathcal H}_N \] on the sphere \(S^{\sqrt{N}}_N= \{A\in{\mathcal H}_N:\text{trace}(A^2)= N\}\) with radius \(\sqrt{N}\), which corresponds to the uniform distribution on this sphere. It is proved that, for any fixed \(n\geq 1\), the suitably normalised \(n\)-point correlation measure of the eigenvalues \(\lambda_1,\dots, \lambda_N\) converges vaguely (as \(N\to\infty\)) to the \(n\)-point correlation measure of a determinantal random point process on the real line whose Lebesgue density coincides with the determinant \(\det (\sin(\pi(t_i- t_j))/\pi(t_i- t_j))_{i,j=1,\dots,n}\).

Reviewer: L. Heinrich (Augsburg)

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05	Central limit and other weak theorems
60G60	Random fields
82B05	Classical equilibrium statistical mechanics (general)

Keywords:

\(n\)-point correlation measure; Wigner’s law; Hermitian matrices; determinantal random point process; weak convergence