Open Access
2017 Perturbations of Voter model in one-dimension
C.M. Newman, K. Ravishankar, E. Schertzer
Electron. J. Probab. 22: 1-42 (2017). DOI: 10.1214/17-EJP37


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.


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C.M. Newman. K. Ravishankar. E. Schertzer. "Perturbations of Voter model in one-dimension." Electron. J. Probab. 22 1 - 42, 2017.


Received: 28 July 2016; Accepted: 5 February 2017; Published: 2017
First available in Project Euclid: 18 April 2017

zbMATH: 1362.60088
MathSciNet: MR3646060
Digital Object Identifier: 10.1214/17-EJP37

Primary: 60F17 , 60K35 , 82B20 , 82C22

Keywords: Brownian net , Brownian net with killing , Brownian web , Poisson point process , Potts model , Scaling limit , voter model

Vol.22 • 2017
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