## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 34, 42 pp.

### Perturbations of Voter model in one-dimension

C.M. Newman, K. Ravishankar, and E. Schertzer

#### 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.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 34, 42 pp.

**Dates**

Received: 28 July 2016

Accepted: 5 February 2017

First available in Project Euclid: 18 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1492502428

**Digital Object Identifier**

doi:10.1214/17-EJP37

**Mathematical Reviews number (MathSciNet)**

MR3646060

**Zentralblatt MATH identifier**

1362.60088

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82C22: Interacting particle systems [See also 60K35]

**Keywords**

voter model Potts model scaling limit Poisson point process Brownian web Brownian net Brownian net with killing

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Newman, C.M.; Ravishankar, K.; Schertzer, E. Perturbations of Voter model in one-dimension. Electron. J. Probab. 22 (2017), paper no. 34, 42 pp. doi:10.1214/17-EJP37. https://projecteuclid.org/euclid.ejp/1492502428