  We consider tiles (dimers) each of which covers two vertices of a rectangular
lattice. There is a normalized translation invariant weighting on the shape of
the tiles. We study the pressure, p, or entropy, (one over the volume times the
logarithm of the partition function). We let p_0 (easy to compute) be the
pressure in the limit of absolute smoothness (the weighting function is
constant). We prove that as the smoothness of the weighting function, suitably
defined, increases, p converges to p_0, uniformly in the volume. It is the
uniformity statement that makes the result non-trivial. In an earlier paper the
author proved this, but with an additional requirement of a certain fall-off on
the weighting function. Herein fall-off is not demanded, but there is the
technical requirement that each dimer connect a black vertex with a white
vertex, vertices colored as on a checker board. This seems like a very basic
result in the theory of pressure (entropy) of tilings.
