Coarsening, nucleation, and the marked Brownian web

L. R G Fontes, M. Isopi, Charles Newman, K. Ravishankar

Research output: Contribution to journalArticle

Abstract

Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing random walks corresponding to color genealogies. Asymptotics for large time and space on the lattice are described via a continuum space-time voter model whose boundary motion is expressed by the Brownian web (BW) of coalescing forward Brownian motions. In this paper, we study how small noise in the voter model, corresponding to the nucleation of randomly colored regions, can be treated in the continuum limit. We present a full construction of the continuum noisy voter model (CNVM) as a random quasicoloring of two-dimensional space time and derive some of its properties. Our construction is based on a Poisson marking of the backward BW within the double (i.e., forward and backward) BW.

Original languageEnglish (US)
Pages (from-to)37-60
Number of pages24
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume42
Issue number1
DOIs
StatePublished - Jan 2006

Fingerprint

Voter Model
Coarsening
Nucleation
Coalescing Random Walk
Continuum
Genealogy
Continuum Limit
Brownian motion
Random walk
Siméon Denis Poisson
Space-time
Motion
World Wide Web
Vote
Color

Keywords

  • Brownian web
  • Coarsening
  • Continuum limit
  • Continuum models
  • Nucleation
  • Poissonian marks
  • Voter model

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

Coarsening, nucleation, and the marked Brownian web. / Fontes, L. R G; Isopi, M.; Newman, Charles; Ravishankar, K.

In: Annales de l'institut Henri Poincare (B) Probability and Statistics, Vol. 42, No. 1, 01.2006, p. 37-60.

Research output: Contribution to journalArticle

@article{c89c57ddcd0944a6ab95b2d646a4107d,
title = "Coarsening, nucleation, and the marked Brownian web",
abstract = "Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing random walks corresponding to color genealogies. Asymptotics for large time and space on the lattice are described via a continuum space-time voter model whose boundary motion is expressed by the Brownian web (BW) of coalescing forward Brownian motions. In this paper, we study how small noise in the voter model, corresponding to the nucleation of randomly colored regions, can be treated in the continuum limit. We present a full construction of the continuum noisy voter model (CNVM) as a random quasicoloring of two-dimensional space time and derive some of its properties. Our construction is based on a Poisson marking of the backward BW within the double (i.e., forward and backward) BW.",
keywords = "Brownian web, Coarsening, Continuum limit, Continuum models, Nucleation, Poissonian marks, Voter model",
author = "Fontes, {L. R G} and M. Isopi and Charles Newman and K. Ravishankar",
year = "2006",
month = "1",
doi = "10.1016/j.anihpb.2005.01.003",
language = "English (US)",
volume = "42",
pages = "37--60",
journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",
issn = "0246-0203",
publisher = "Institute of Mathematical Statistics",
number = "1",

}

TY - JOUR

T1 - Coarsening, nucleation, and the marked Brownian web

AU - Fontes, L. R G

AU - Isopi, M.

AU - Newman, Charles

AU - Ravishankar, K.

PY - 2006/1

Y1 - 2006/1

N2 - Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing random walks corresponding to color genealogies. Asymptotics for large time and space on the lattice are described via a continuum space-time voter model whose boundary motion is expressed by the Brownian web (BW) of coalescing forward Brownian motions. In this paper, we study how small noise in the voter model, corresponding to the nucleation of randomly colored regions, can be treated in the continuum limit. We present a full construction of the continuum noisy voter model (CNVM) as a random quasicoloring of two-dimensional space time and derive some of its properties. Our construction is based on a Poisson marking of the backward BW within the double (i.e., forward and backward) BW.

AB - Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing random walks corresponding to color genealogies. Asymptotics for large time and space on the lattice are described via a continuum space-time voter model whose boundary motion is expressed by the Brownian web (BW) of coalescing forward Brownian motions. In this paper, we study how small noise in the voter model, corresponding to the nucleation of randomly colored regions, can be treated in the continuum limit. We present a full construction of the continuum noisy voter model (CNVM) as a random quasicoloring of two-dimensional space time and derive some of its properties. Our construction is based on a Poisson marking of the backward BW within the double (i.e., forward and backward) BW.

KW - Brownian web

KW - Coarsening

KW - Continuum limit

KW - Continuum models

KW - Nucleation

KW - Poissonian marks

KW - Voter model

UR - http://www.scopus.com/inward/record.url?scp=28644439784&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28644439784&partnerID=8YFLogxK

U2 - 10.1016/j.anihpb.2005.01.003

DO - 10.1016/j.anihpb.2005.01.003

M3 - Article

VL - 42

SP - 37

EP - 60

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 1

ER -