  Let U be a multiply-connected region in R^2 with smooth boundary. Let
P_epsilon be a polyomino in epsilon Z^2 approximating U as epsilon tends to 0.
We show that, for certain boundary conditions on P_epsilon, the height
distribution on a random domino tiling (dimer covering) of P_epsilon is
conformally invariant in the limit as epsilon tends to
  0, in the sense that the distribution of heights of boundary components only
depends on the conformal type of U. The mean height and all the moments are
explicitly evaluated.
