  Consider an infinite random matrix $H=(h_{ij})_{0<i,j}$ picked from the
Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq
r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $\mu^i_j$. We
show that the configuration of all these eigenvalues $(i,\mu_j^i)$ form a
determinantal point process on $\mathbb{N}\times\mathbb{R}$.
  Furthermore we show that this process can be obtained as the scaling limit in
random tilings of the Aztec diamond close to the boundary. We also discuss the
corresponding limit for random lozenge tilings of a hexagon.
