Perturbations of the Voter Model in One-Dimension

TL;DR

This paper investigates the scaling limits of voter model perturbations in one dimension, including stochastic Potts models, demonstrating convergence to a universal continuum voter model perturbation using duality and reduced graph properties.

Contribution

It introduces a framework for analyzing voter model perturbations and proves convergence from discrete to continuum models, focusing on boundary nucleations and reduced graph structures.

Findings

Discrete and continuum models are connected via duality with killing.

Convergence of discrete nets to continuum nets is established.

Reduced graphs are finite almost surely in the continuum setting.

Abstract

We study the scaling limit of a large class of voter model perturbations in one dimension, including stochastic Potts models, to a universal limiting object, the continuum voter model perturbation. The perturbations can be described in terms of bulk and boundary nucleations of new colors (opinions). The discrete and continuum (space) models are obtained from their respective duals, the discrete net with killing and Brownian net with killing. These determine the color genealogy by means of reduced graphs. We focus our attention on models where the voter and boundary nucleation dynamics depend only on the colors of nearest neighbor sites, for which convergence of the discrete net with killing to its continuum analog was proved in an earlier paper by the authors. We use some detailed properties of the Brownian net with killing to prove voter model perturbations convergence to its continuum counterpart. A crucial property of reduced graphs is that even in the continuum, they are finite almost surely. An important issue is how vertices of the continuum reduced graphs are strongly approximated by their discrete analogues.