Convergence of Coalescing Nonsimple Random Walks to The Brownian Web
Charles M Newman, Courant Institute of Mathematical Sciences, New York University, New York, NY 10
Krishnamurthi Ravishankar, SUNY-New Paltz, New Paltz, NY 12561, USA
Rongfeng Sun, EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlanda
Abstract
Abstract
The Brownian Web (BW) is a family of coalescing Brownian motions
starting from every point in space and time $RtimesR$. It was
first introduced by Arratia, and later analyzed in detail by
T'{o}th 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.
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