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

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
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