  Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider
covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is
covered with a predetermined multiplicity. The multinomial tiling model is a
natural probability measure on such configurations (it is the uniform measure
on standard tilings of the corresponding "blow-up" of $G$).
  In the limit of large multiplicities we compute asymptotic growth rate of the
number of multinomial tilings. We show that the individual tile densities tend
to a Gaussian field with respect to an associated discrete Laplacian. We also
find an exact discrete Coulomb gas limit when we vary the multiplicities.
  For tilings of ${\mathbb Z}^d$ with translates of a single tile and a small
density of defects, we study a crystallization phenomena when the defect
density tends to zero, and give examples of naturally occurring quasicrystals
in this framework.
