Stripe formation in frustrated Ising models with long-range interactions

I consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling J is larger than a critical value J_c, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of J, it is believed that the ground state consists of a periodic array of stripes (d=2) or slabs (d=3), all of the same size and alternating magnetization. In this talk I present a rigorous proof that the ground state energy per site converges to that of the optimal periodic striped/slabbed state, in the limit that J tends to the ferromagnetic transition point. While our theorem does not prove rigorously that the ground state is precisely striped/slabbed, it does prove that in any suitably large box the ground state is striped/slabbed with high probability. Joint work with E. H. Lieb and R. Seiringer.