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Brownian web in the scaling limit of supercritical oriented percolation in dimension \(1 + 1\). (English) Zbl 1290.60107

Let \(\mathbb{Z}^2_{\text{even}}=\{(x,i)\in \mathbb{Z}^2: x+i \text{ is even}\}\) be a space-time lattice, with oriented edges leading from \((x,i)\) to \((x\pm 1, i+1)\) for all \((x,i)\in \mathbb{Z}^2_{\text{even}}\). In the paper the authors proved that after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on \(\mathbb{Z}^2_{\text{even}}\) converges in distribution to the Brownian web. This proves a conjecture of X.-Y. Wu and Y. Zhang [Ann. Probab. 36, No. 3, 862–875 (2008; Zbl 1139.60345)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation

Citations:

Zbl 1139.60345
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