  We consider Ising models in d=2 and d=3 dimensions with nearest neighbor
ferromagnetic and long-range antiferromagnetic interactions, the latter
decaying as (distance)^(-p), p>2d, at large distances. If the strength J of the
ferromagnetic interaction is larger than a critical value J_c, then the ground
state is homogeneous. It has been conjectured that when J is smaller than but
close to J_c the ground state is periodic and striped, with stripes of constant
width h=h(J), and h tends to infinity as J tends to J_c from below. (In d=3
stripes mean slabs, not columns.) Here we rigorously prove that, if we
normalize the energy in such a way that the energy of the homogeneous state is
zero, then the ratio e_0(J)/e_S(J) tends to 1 as J tends to J_c from below,
with e_S(J) being the energy per site of the optimal periodic striped/slabbed
state and e_0(J) the actual ground state energy per site of the system. Our
proof comes with explicit bounds on the difference e_0(J)-e_S(J) at small but
finite J_c-J, and also shows that in this parameter range the ground state is
striped/slabbed in a certain sense: namely, if one looks at a randomly chosen
window, of suitable size l (very large compared to the optimal stripe size
h(J)), one finds a striped/slabbed state with high probability.
