Limiting Shapes in Two-Dimensional Ising Ferromagnets

TL;DR

This paper analyzes the evolution of interfaces in a two-dimensional Ising ferromagnet with extended interactions, deriving limiting shapes and velocities for different initial configurations under zero-temperature dynamics.

Contribution

It provides an analytical and numerical study of interface limiting shapes and velocities in a 2D Ising model with extended ferromagnetic interactions, including self-similar and stationary shapes.

Findings

The interface in a quadrant recedes in a self-similar diffusive manner.

The scaled interface shape converges to a deterministic limit.

A stationary finger shape moves with a computed velocity.

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.