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 \(\mathbb Z^2\). We prove that there exists \(p_0\) such that for \(p\le p_0\) 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.