Perturbations of voter model in one-dimension

Charles Newman, K. Ravishankar, E. Schertzer

Research output: Contribution to journalArticle

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 converge to their continuum counterparts. 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.

Original languageEnglish (US)
Article number34
JournalElectronic Journal of Probability
Volume22
DOIs
StatePublished - 2017

Fingerprint

Voter Model
One Dimension
Continuum
Perturbation
Nucleation
Graph in graph theory
Analogue
Genealogy
Scaling Limit
Continuum Model
Potts Model
Vote
Stochastic Model
Nearest Neighbor
Limiting
Converge
Model
Color
Graph

Keywords

  • Brownian net
  • Brownian net with killing
  • Brownian web
  • Poisson point process
  • Potts model
  • Scaling limit
  • Voter model

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Perturbations of voter model in one-dimension. / Newman, Charles; Ravishankar, K.; Schertzer, E.

In: Electronic Journal of Probability, Vol. 22, 34, 2017.

Research output: Contribution to journalArticle

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