Imry-Ma phenomenon for the hard-core model on $\mathbb{Z}^{2}$

TL;DR

This paper demonstrates that even weak disorder prevents crystallization in the 2D hard-core model, extending the Imry-Ma phenomenon known from spin systems to a discrete crystallization model.

Contribution

It adapts the Aizenman-Wehr argument to show disorder destroys phase transitions in the 2D hard-core model, a novel application of the Imry-Ma phenomenon.

Findings

Weak disorder prevents crystal formation in the 2D hard-core model.

The proof adapts spin system symmetry arguments to spatial symmetries.

No phase transition occurs due to disorder in this model.

Abstract

The \emph{Imry-Ma phenomenon} refers to the dramatic effect that disorder can have on first-order phase transitions for two-dimensional spin systems. The most famous example is the absence of a phase transition for the two-dimensional random-field Ising model. This paper establishes that a similar phenomena takes place for the hard-core model, a discrete model of crystallization: arbitrarily weak disorder prevents the formation of a crystal. Our proof of this behaviour is an adaptation of the Aizenman-Wehr argument for the Imry-Ma phenomenon, with the use of internal (spin space) symmetries for spin systems being replaced by the use spatial symmetries.