Melting of an Ising Quadrant

TL;DR

This paper studies the dynamic evolution of a minority phase in a two-dimensional Ising ferromagnet with zero-temperature spin-flip dynamics, revealing self-similar interface receding and analyzing area fluctuations using advanced probabilistic methods.

Contribution

It introduces a novel mapping of the interface dynamics to the symmetric simple exclusion process and computes cumulants of the invaded area using microscopic and macroscopic approaches.

Findings

Area grows linearly with time

Variance of area is exactly computed

Higher cumulants' asymptotic behavior determined

Abstract

We consider an Ising ferromagnet endowed with zero-temperature spin-flip dynamics and examine the evolution of the Ising quadrant, namely the spin configuration when the minority phase initially occupies a quadrant while the majority phase occupies three remaining quadrants. The two phases are then always separated by a single interface which generically recedes into the minority phase in a self-similar diffusive manner. The area of the invaded region grows (on average) linearly with time and exhibits non-trivial fluctuations. We map the interface separating the two phases onto the one-dimensional symmetric simple exclusion process and utilize this isomorphism to compute basic cumulants of the area. First, we determine the variance via an exact microscopic analysis (the Bethe ansatz). Then we turn to a continuum treatment by recasting the underlying exclusion process into the framework of the macroscopic fluctuation theory. This provides a systematic way of analyzing the statistics of the invaded area and allows us to determine the asymptotic behaviors of the first four cumulants of the area.