Asymptotic Behaviour of Ground States for Mixtures of Ferromagnetic and Antiferromagnetic Interactions in a Dilute Regime

Abstract

We consider randomly distributed mixtures of bonds of ferromagnetic and antiferromagnetic type in a two-dimensional square lattice with probability 1−p 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 Z 2 Z2 . We prove that there exists p 0 p0 such that for p≤ p 0 p≤p0 such minimizers are characterized by a majority phase; i.e., they take identically the value 1 or −1 −1 except for small disconnected sets. A deterministic analogue is also proved.