Convergence of coalescing nonsimple random walks to the Brownian Web

Charles Newman, K. Ravishankar, Rongfeng Sun

Research output: Contribution to journalArticle

Abstract

The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time ℝ × ℝ It was first introduced by Arratia, and later analyzed in detail by Tóth and Werner. More recently, Fontes, Isopi, Newman and Ravishankar (FINR) gave a characterization of the BW, and general convergence criteria allowing in principle either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. To date, the general convergence criteria of FINR have not been verified for any case with crossing paths, which appears to be significantly more difficult than the noncrossing paths case. Accordingly, in this paper, we formulate new convergence criteria for the crossing paths case, and verify them for non-simple coalescing random walks satisfying a finite fifth moment condition. This is the first time that convergence to the BW has been proved for models with crossing paths. Several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.

Original languageEnglish (US)
Pages (from-to)21-60
Number of pages40
JournalElectronic Journal of Probability
Volume10
StatePublished - Feb 11 2005

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Coalescing Random Walk
Path
Convergence Criteria
Voter Model
Simple Random Walk
Moment Conditions
Convergence Time
Scaling Limit
Brownian motion
Random walk
World Wide Web
Siméon Denis Poisson
Corollary
Verify

Keywords

  • Brownian Networks
  • Brownian Web
  • Coalescing Random Walks
  • Continuum Limit
  • Invariance Principle

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

Convergence of coalescing nonsimple random walks to the Brownian Web. / Newman, Charles; Ravishankar, K.; Sun, Rongfeng.

In: Electronic Journal of Probability, Vol. 10, 11.02.2005, p. 21-60.

Research output: Contribution to journalArticle

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