  We discuss asymptotic properties of a family of discrete probability measures
which may be used to model particle configurations with a wall on a set of
discrete nodes. The correlations are shown to be determinantal and are
expressed in terms of discrete orthogonal polynomials. As an application we
study random tilings of the half-hexagon or, equivalently, configurations of
non-intersecting lattice paths above a wall, so called water melons.
