Percolation in lattice $k$-neighbor graphs

Abstract

We define a random graph obtained via connecting each point of $\mathbb{Z}^d$ independently to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed $k$-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional $k$-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed $k$-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for $k=1$ even the undirected $k$-neighbor graph never percolates, but the directed one percolates whenever $k \geq d+1$, $k \geq 3$ and $d \geq 5$, or $k \geq 4$ and $d=4$. We also show that the undirected $2$-neighbor graph percolates for $d=2$, the undirected $3$-neighbor graph percolates for $d=3$, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the $k$-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.