A contribution to shape theory for the Ising model at low temperature (Q1182499)

We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice \(\mathbb{Z}^ d\), \(d=1,2,\dots\), under a negative boundary condition and a small positive external field \(h\) that decreases linearly with the temperature \(T\). 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 \(\alpha=h/T\) exceeds some positive \(\alpha_ 0\). For \(\alpha<\alpha_ 0\), our results do not rule out infinite negative areas.