Department of Mathematics - Seminar on Probability - How do the eigenvalues of a large random matrix behave?  

We prove that the fluctuations of eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. Due to strong correlations between the eigenvalues, these fluctuations appear on a non-natural scale. We then explore extremal values of this Gaussian field and investigate certain space--time correlations of the eigenvalues.