文章基本信息

标题：Coalescing directed random walks on the backbone of a 1 + 1-dimensional oriented percolation cluster converge to the Brownian web
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作者：Matthias Birkner ; Nina Gantert ; Sebastian Steiber 等
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2019
卷号：XVI
期号：2
页码：1029-1054
DOI：10.30757/ALEA.v16-37
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We consider the backbone of the infinite cluster generated by supercriticaloriented site percolation in dimension 1 􀀀 1. A directed random walk onthis backbone can be seen as an “ancestral lineage” of an individual sampled inthe stationary discrete-time contact process. Such ancestral lineages were investigatedin Birkner et al. (2013) where a central limit theorem for a single walker wasproved. Here, we consider infinitely many coalescing walkers on the same backbonestarting at each space-time point. We show that, after diffusive rescaling, the collectionof paths converges in distribution (under the averaged law) to the Brownianweb. Hence, we prove convergence to the Brownian web for a particular system ofcoalescing random walks in a dynamical random environment. An important toolin the proof is a tail bound on the meeting time of two walkers on the backbone,started at the same time. Our result can be interpreted as an averaging statementabout the percolation cluster: apart from a change of variance, it behaves as thefull lattice, i.e. the effect of the “holes” in the cluster vanishes on a large scale.
关键词：Oriented percolation; coalescing random walks; Brownian web