Motion by curvature and large deviations for an interface dynamics on $\mathbb{Z}^2$

TL;DR

This paper investigates large deviations for a Markov process modeling interface motion in a 2D lattice, revealing how curvature-driven dynamics and temperature influence typical behavior and deviations, with results matching theoretical predictions.

Contribution

It introduces a tunable interface dynamics model on ^2, proves curvature-driven motion at large , and establishes large deviations bounds with explicit diffusion and mobility parameters.

Findings

Contours follow motion by curvature influenced by 

Large deviations bounds are established for all large enough <

Diffusion coefficient and mobility match theoretical predictions

Abstract

We study large deviations for a Markov process on curves in $\mathbb{Z}^2$ mimicking the motion of an interface. Our dynamics can be tuned with a parameter $\beta$, which plays the role of an inverse temperature, and coincides at $\beta$ = $\infty$ with the zero-temperature Ising model with Glauber dynamics, where curves correspond to the boundaries of droplets of one phase immersed in a sea of the other one. We prove that contours typically follow a motion by curvature with an influence of the parameter $\beta$, and establish large deviations bounds at all large enough $\beta$ < $\infty$. The diffusion coefficient and mobility of the model are identified and correspond to those predicted in the literature.