*Published Paper*

**Inserted:** 16 jan 2017

**Last Updated:** 10 sep 2018

**Journal:** J. Stat. Phys.

**Volume:** 171

**Pages:** 1096-1111

**Year:** 2018

**Doi:** 10.1007/s10955-018-2051-8

**Abstract:**

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.

**Keywords:**
ground states, Ising models, random spin systems, asymptotic analysis of periodic media

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