Periodic striped configurations in the large volume limit

TL;DR

This paper proves the existence of striped pattern formations in large volume limits for a class of antiferromagnetic interaction models, confirming a conjecture about pattern emergence for generic system sizes.

Contribution

It extends previous results by establishing striped pattern formation for arbitrary system sizes, not just specific periodic cases, in a generalized antiferromagnetic model.

Findings

Striped patterns form in large volume limits for certain interaction functionals.

The results confirm the conjecture for all system sizes, not only specific periodic ones.

The study generalizes previous discrete setting results to broader conditions.

Abstract

We show striped pattern formation in the large volume limit for a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension previously considered Goldman-Runa and Daneri-Runa and in Giuliani-Lieb-Lebowitz and Giuliani-Seiringer in the discrete setting. In such a model the relative strength between the short range attractive term favouring pure phases and the long range repulsive term favouring oscillations is modulated by a parameter $\tau$. For $\tau<0$ minimizers are trivial uniform states. It is conjectured that $\forall\,d\geq2$ there exists $0<\bar{\tau}\ll1$ such that for all $0<\tau\leq\bar{\tau}$ and for all $L>0$ minimizers are striped/lamellar patterns. In Daneri-Runa arXiv:1702.07334 the authors prove the above for $L=2kh^*_\tau$, where $k\in\N$ and $h^*_\tau$ is the optimal period of stripes for a given $0<\tau\leq\bar{\tau}$. The purpose of this paper is to show the validity of the conjecture for generic $L$.