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

## TL;DR

This paper investigates the large-scale behavior of ground states in a two-dimensional lattice with randomly distributed ferromagnetic and antiferromagnetic bonds, showing that below a certain probability threshold, minimizers predominantly exhibit a single phase.

## Contribution

It establishes the asymptotic behavior of minimizers in dilute mixtures of ferromagnetic and antiferromagnetic interactions, including a deterministic analogue.

## Key findings

- Existence of a threshold probability p_0 below which a majority phase dominates.
- Minimizers are mostly uniform with small disconnected exceptions for p ≤ p_0.
- Deterministic analogue of the probabilistic model is also demonstrated.

## 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.