Rescaled Lotka-Volterra Models Converge to Super Stable Processes

TL;DR

This paper extends the convergence results of rescaled stochastic spatial Lotka-Volterra models to super stable processes, generalizing previous Brownian motion limits and providing new asymptotic insights into the voter model.

Contribution

It demonstrates that under certain conditions, the limit process of rescaled models can be a super stable process, broadening the scope of convergence results beyond Brownian motion.

Findings

Rescaled models converge to super stable processes under stable law attraction.

New asymptotic results for the voter model starting from a single individual.

Improves upon previous results by Bramson and Griffeath (1980).

Abstract

Recently, it has been shown that stochastic spatial Lotka-Volterra models when suitably rescaled can converge to a super Brownian motion. We show that the limit process could be a super stable process if the kernel of the underlying motion is in the domain of attraction of a stable law. The corresponding results in Brownian setting were proved by Cox and Perkins (2005, 2008). As applications of the convergence theorems, some new results on the asymptotics of the voter model started from single 1 at the origin are obtained which improve the results by Bramson and Griffeath (1980).