Formation of stripes and slabs near the ferromagnetic transition (Q398748)

An Ising model is considered in two and three dimensions on the square lattice with nearest-neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as \((\mathrm{distance})^{-p}\), \(p>-2d\), at large distances. The authors rigorously prove that, if they normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio \(\frac{e_0(J)}{e_s(J)}\) tends to 1 as \(J \rightarrow J^{-}_c\), 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. The proof comes with explicit bounds on the difference \(e_0(J)-e_s(J)\) for small but positive \(J_c-J\), and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: 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.