I will report on joint work in progress with Christophe Charlier. We develop a new method to characterize gap probabilities of discrete determinantal point processes in terms of Riemann-Hilbert problems. Simple examples of such discrete point processes arise in domino tilings of Aztec diamonds and lozenge tilings of hexagons. As a first illustration of our approach, we obtain a new explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Padé approximants, by solving the associated Riemann-Hilbert problem. As a second application, we obtain an explicit expression for the number of lozenge tilings of reduced hexagons in terms of Hermite-Padé approximants.
