Asymptotic behaviour of ground states for mixtures of ferromagnetic and antiferromagnetic interactions in a dilute regime

We consider randomly distributed mixtures of bonds of ferromagnetic and antiferromagnetic type in a two-dimensional square lattice with probability 1-p and p, respectively, according to an i.i.d. random variable. We study minimizers of the corresponding nearest-neighbour spin energy on large domains in Z2. We prove that there exists a strictly positive critical value such that for p less than this threshold such minimizers are characterized by a majority phase; i.e., they take identically the value 1 or -1 except for small disconnected sets. A deterministic analogue is also proved.