  We give a new proof of the fact that, near a turning point of the frozen
boundary, the vertical tiles in a uniformly random lozenge tiling of a large
sawtooth domain are distributed like the eigenvalues of a GUE random matrix.
Our argument uses none of the standard tools of integrable probability. In
their place, it uses a combinatorial interpretation of the
Harish-Chandra/Itzykson-Zuber integral as a generating function for
desymmetrized Hurwitz numbers.
