# Striped states at the ferromagnetic transition

# Striped states at the ferromagnetic transition

We consider a two-dimensional Ising model with nearest neighbor ferromagnetic and long-range, power-law decaying, antiferromagnetic interactions. We assume that the decay exponent of the antiferromagnetic part is larger than 4. The ground state displays a transition as the strength of the ferromagnetic interaction J is increased: if J is smaller than a critical value J_c, then the ground state is non-uniform, while it becomes uniform for all J>J_c. The transition to the uniform state is believed to take place in a ``universal" manner, via a sequence of transitions among periodic striped states. In this talk we report recent rigorous results confirming this picture: namely, the ratio of the ground state energy to the one of the optimal periodic striped state tends to 1 as J-->J_c^-. The proof is based on a combination of localization bounds into mesoscopic boxes and reflection positivity. Joint work with E. Lieb and R. Seiringer.