Gaussian fluctuations of eigenvalues in the GUE. (English) Zbl 1073.60020

Gustavsson, Jonas

Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 2, 151-178 (2005).

Summary: Under certain conditions on \(k\) we calculate the limit distribution of the \(k\)th largest eigenvalue, \(x_k\), of the Gaussian unitary ensemble (GUE). More specifically, if \(n\) is the dimension of a random matrix from the GUE and \(k\) is such that both \(n-k\) and \(k\) tend to infinity as \(n\to \infty\), then \(x_k\) is normally distributed in the limit. We also consider the joint limit distribution of \(x_{k_1}<\cdots< x_{k_m}\) where we require that \(n-k_i\) and \(k_i\), \(1\leq i\leq m\), tend to infinity with \(n\). The result is an \(m\)-dimensional normal distribution.

MSC:

60F05	Central limit and other weak theorems
15B52	Random matrices (algebraic aspects)

Keywords:

random matrices; limit distribution