11 feb 2016 -- 11:30   [open in google calendar]

Aula Magna (Dipartimento di Matematica di Pisa)

Abstract.

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 talk, 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 report a proof 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. We shall explain the key aspects of the proof, which is based on localization bounds combined with reflection positivity. Joint work with Robert Seiringer.

• Miniworkshop Variational and PDE problems in Applied Mathematics