  We study the nearest-neighbour Ising model with a class of random boundary
conditions, chosen from a symmetric i.i.d. distribution. We show for dimensions
4 and higher that almost surely the only limit points for a sequence of
increasing cubes are the plus and the minus state. For d=2 and d=3 we prove a
similar result for sparse sequences of increasing cubes. This question was
raised by Newman and Stein. Our results imply that the Newman-Stein metastate
is concentrated on the plus and the minus state.
