Calculus of Variations and Geometric Measure Theory

A. Braides - A. Causin - A. Piatnitski - M. Solci

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

created by braidesa on 16 Jan 2017
modified on 10 Sep 2018


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


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