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arXiv · 2412.20781

The Planar Lattice Two-Neighbor Graph Percolates

David Coupier·Benoît Henry·Benedikt Jahnel·Jonas Köppl
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Abstract

The k-neighbor graph is a directed percolation model on the hypercubic lattice Z d in which each vertex independently picks exactly k of its 2d nearest neighbors at random, and we open directed edges towards those. We prove that the 2-neighbor graph percolates on Z 2 , i.e., that the origin is connected to infinity with positive probability. The proof rests on duality, an exploration algorithm, a comparison to i.i.d. bond percolation under constraints as well as enhancement arguments. As a byproduct, we show that i.i.d. bond percolation with forbidden local patterns has a strictly larger percolation threshold than 1/2. Additionally, our main result provides further evidence that, in low dimensions, less variability is beneficial for percolation.

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David Coupier, Benoît Henry, Benedikt Jahnel, Jonas Köppl. 2024-12-30. The Planar Lattice Two-Neighbor Graph Percolates. https://arxiv.org/abs/2412.20781

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