Periodic striped ground states in Ising models with competing interactions

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let $J$ be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent $p$ of the long range interaction is larger than $d+1$, with $d$ the space dimension, this happens for all values of $J$ smaller than a critical value $J_c(p)$, beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for $p>2d$ and $J$ in a left neighborhood of $J_c(p)$. In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes ($d=2$) or slabs ($d=3$), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.