A contribution to shape theory for the ising model at low temperature

Summary We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice ^d ,d=1, 2,..., under a negative boundary condition and a small positive external fieldh that decreases linearly with the temperatureT. It is shown that positive and negative spins are separated by a staircase-shaped random boundary. Its explicit distribution is computed in the case that the ratio =h/T exceeds some positive ₀. For < ₀, our results do not rule out infinite negative areas.