Motion by mean curvature in interacting particle systems

TL;DR

This paper demonstrates that certain rescaled interacting particle systems with bistable reaction terms and zero-speed traveling waves converge to motion by mean curvature, revealing new phase transition phenomena.

Contribution

It extends the analysis of particle systems to show convergence to motion by mean curvature in cases with zero wave speed, enabling new phase transition insights.

Findings

Convergence to motion by mean curvature in specific particle systems.

Identification of conditions for phase transitions in related models.

Potential to prove discontinuous phase transitions in biological models.

Abstract

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al there were two nontrivial stationary distributions.