Percolation in lattice $k$-neighbor graphs

TL;DR

This paper investigates percolation phenomena in directed and undirected lattice $k$-neighbor graphs, establishing conditions under which infinite paths exist, thus advancing understanding of discrete analogues to continuum percolation models.

Contribution

It provides new percolation thresholds and behaviors for directed, undirected, and bidirectional $k$-neighbor graphs on integer lattices, connecting discrete models to continuum percolation.

Findings

Undirected $1$-neighbor graph never percolates for $d eq 2$.

Directed $k$-neighbor graphs percolate for $k eq 1$ under certain conditions.

Percolation occurs in undirected $2$-neighbor graph at $d=2$ and in $3$-neighbor graph at $d=3$.

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.