2024/12/30 by David Coupier, Coupier, David, Benoît Henry +5 · 2 citations
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2412.20781
openalex publication_date 2024/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.