Motion by mean curvature in interacting particle systems

• 成果类型: Article
• 署名作者: Huang, Xiangying; Durrett, Rick
• 署名单位: University of British Columbia; Duke University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01082-0
• 发表日期: 2021
• 页码: 489-532
• 关键词: generalized motion; glauber evolution; kac potentials; MODEL

摘要

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, see e.g., Cox et al. (Asterisque 349:1-127, 2013), Durrett (Ann Appl Prob 19:477-496, 2009, Electron J Probab 19:1-64, 2014), Durrett and Neuhauser (Ann Probab 22:289-333, 1994). 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 et al. (Electron J Probab 22:1-40, 2017) 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. (Theoret Pop Biol 55(1999):270-282, 1999) there were two nontrivial stationary distributions.