Limiting shapes for configurations

Limiting shapes in two-dimensional Ising ferromagnets.

We consider an Ising model on a square lattice 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 ...

Classification of Limiting Shapes for Isotropic Curve Flows

with α 6= 0, and initial condition x(p, 0) = x0(p). This produces a family of curves γt = x(S, t). Here κ is the curvature, and n is the outward-pointing unit normal vector. These equations are particularly natural in that they are isotropic (equivariant under rotations in the plane) and homogeneous (equivariant under dilation of space, if time is also scaled accordingly). The main aim of this ...

Limiting shapes of Ising droplets, Ising fingers, and Ising solitons.

We examine the evolution of an Ising ferromagnet endowed with zero-temperature single spin-flip dynamics. A large droplet of one phase in the sea of the opposite phase eventually disappears. An interesting behavior occurs in the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size. In this regime the shape ...

Invariant measures and limiting shapes in sandpile models

Limiting Shapes for a Non-abelian Sandpile Growth Model and Related Cellular Automata

We present limiting shape results for a non-abelian variant of the abelian sandpile growth model (ASGM), some of which have no parallel in the ASGM. One of our limiting shapes is an octagon. In our model, mass spreads from the origin by the toppling rule in Zhang’s sandpile model. Previously, several limiting shape results have been obtained for the ASGM using abelianness and monotonicity as ma...