Abstract:
we use the minimizing-movements approach to study an evolution of spin-systems defined on the two-dimensional square lattice of amplitude depending on a vanishing parameter ε. The driven energy is given by a nearest-neighbours anti-ferromagnetic potential.
We analyze both the discrete and the continuous (limit, as ε goes to 0) flow from a geometric point of view identifying spin-systems with the union of lattice squares corresponding to the positive statuses of the system. Through this identification, the anti-ferromagnetic energy corresponds to a negative perimeter. We consider evolutions starting from a single point (nucleation).

We show that, the competition between short-range-repulsion (negative perimeter) and long-range-attraction (dissipation) produces a checkerboard pattern of the minimizers at the discrete level and a "backward" evolution at the continuous one. We prove that the scheme converges to a family of expanding sets with constant velocity. The "shape" of the limit motion depends on the choice of the scale between the time and space parameters and on the norm defining the dissipation term.