Formation of Stripes and Slabs Near the Ferromagnetic Transition

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 → ∞ as \({J\to J_c^-}\) . (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 \({J\to J_c^-}\) , 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 positive 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 ? (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.