### 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 language | English (US) |
---|---|

Pages (from-to) | 21-60 |

Number of pages | 40 |

Journal | Electronic Journal of Probability |

Volume | 10 |

State | Published - Feb 11 2005 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Electronic Journal of Probability*,

*10*, 21-60.

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

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 10, pp. 21-60.

}

TY - JOUR

T1 - Convergence of coalescing nonsimple random walks to the Brownian Web

AU - Newman, Charles

AU - Ravishankar, K.

AU - Sun, Rongfeng

PY - 2005/2/11

Y1 - 2005/2/11

N2 - 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.

AB - 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.

KW - Brownian Networks

KW - Brownian Web

KW - Coalescing Random Walks

KW - Continuum Limit

KW - Invariance Principle

UR - http://www.scopus.com/inward/record.url?scp=15944386675&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15944386675&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:15944386675

VL - 10

SP - 21

EP - 60

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -