Limiting Shapes in Two-Dimensional Ising Ferromagnets
Abstract
We consider an Ising model on a square grid with ferromagnetic spin-spin interactions spanning beyond nearest neighbors. Starting from initial states with a single unbounded interface separating ordered phases, we investigate the evolution of the interface subject to zero-temperature spin-flip dynamics. We consider an interface which is initially (i) the boundary of the quadrant, or (ii) the boundary of a semi-infinite stripe. In the former case the interface recedes from its original location in a self-similar diffusive manner. After a re-scaling by t^{1/2}, the shape of the interface becomes more and more deterministic; we determine this limiting shape analytically and verify our predictions numerically. The semi-infinite stripe acquires a stationary shape resembling a finger, and this finger translates along its axis. We compute the limiting shape and the velocity of the Ising finger.
Cite
@article{arxiv.1303.5128,
title = {Limiting Shapes in Two-Dimensional Ising Ferromagnets},
author = {P. L. Krapivsky and Jason Olejarz},
journal= {arXiv preprint arXiv:1303.5128},
year = {2013}
}
Comments
11 pages, 7 figures; subsection VI.A added; various minor corrections