Motion of discrete interfaces on the triangular lattice

TL;DR

This paper investigates the evolution of discrete, symmetric convex hexagons on a triangular lattice driven by ferromagnetic interactions, using a coupling of minimizing movements and discrete-to-continuum analysis, and compares it with crystalline curvature motion.

Contribution

It extends the analysis of interface motion from square to triangular lattices, specifically for Wulff-like hexagons, using a novel coupling approach and continuum limit comparison.

Findings

Limit motion matches crystalline curvature evolution with natural mobility.

Established a discrete-to-continuum link for triangular lattice interfaces.

Generalized previous square lattice results to hexagonal symmetry.

Abstract

We study the motion of discrete interfaces driven by ferromagnetic interactions on the two-dimensional triangular lattice by coupling the Almgren, Taylor and Wang minimizing movements approach and a discrete-to-continuum analysis, as introduced by Braides, Gelli and Novaga in the pioneering case of the square lattice. We examine the motion of origin-symmetric convex "Wulff-like" hexagons, i.e. origin-symmetric convex hexagons with sides having the same orientations as those of the hexagonal Wulff shape related to the density of the anisotropic perimeter $\Gamma$-limit of the ferromagnetic energies as the lattice spacing vanishes. We compare the resulting limit motion with the corresponding evolution by crystalline curvature with natural mobility.