Fate of 2D Kinetic Ising Ferromagnets and Critical Percolation Crossing Probabilities

TL;DR

This paper reveals a deep connection between the zero-temperature coarsening process in 2D kinetic Ising models and critical percolation crossing probabilities, linking metastable states to percolation theory.

Contribution

It demonstrates that the probabilities of ending in specific metastable stripe states in the 2D kinetic Ising model match exactly with critical percolation crossing probabilities.

Findings

Stripe state probabilities match critical percolation crossing probabilities.

Metastable states are topologically distinct and correspond to percolation paths.

The connection provides exact predictions for the likelihood of different final states.

Abstract

We present evidence for a deep connection between the zero-temperature coarsening of the two-dimensional kinetic Ising model (KIM) and critical continuum percolation. In addition to reaching the ground state, the KIM can also fall into a variety of topologically distinct metastable stripe states. The probability to reach a stripe state that winds a times horizontally and b times vertically on a square lattice with periodic boundary conditions equals the corresponding exactly-solved critical percolation crossing probability P_{a,b} for a spanning path with winding numbers a and b.