Random nearest neighbor graphs: the translation invariant case

TL;DR

This paper investigates the structure of translation invariant nearest neighbor graphs with distinct weights on $\\mathbb{Z}^d$, showing they lack doubly-infinite paths and can have a controlled number of infinite components, contrasting with the i.i.d. case.

Contribution

It characterizes all translation invariant nearest neighbor graphs with distinct weights, proving the absence of doubly-infinite paths and describing possible numbers of infinite components.

Findings

No doubly-infinite directed paths exist in these graphs.

In dimension two, graphs have either one or two infinite components.

In higher dimensions, graphs can have any finite or countably infinite number of infinite components.

Abstract

If $(\omega(e))$ is a family of random variables (weights) assigned to the edges of $\mathbb{Z}^d$, the nearest neighbor graph is the directed graph induced by all edges $\langle x,y \rangle$ such that $\omega(\{x,y\})$ is minimal among all neighbors $y$ of $x$. That is, each vertex points to its closest neighbor, if the weights are viewed as edge-lengths. Nanda-Newman introduced nearest neighbor graphs when the weights are i.i.d. and continuously distributed and proved that a.s., all components of the undirected version of the graph are finite. We study the case of translation invariant, distinct weights, and prove that nearest neighbor graphs do not contain doubly-infinite directed paths. In contrast to the i.i.d. case, we show that in this stationary case, the graphs can contain either one or two infinite components (but not more) in dimension two, and $k$ infinite components for any $k \in [1,\infty]$ in dimension $\geq 3$. The latter constructions use a general procedure to exhibit a certain class of directed graphs as nearest neighbor graphs with distinct weights, and thereby characterize all translation invariant nearest neighbor graphs. We also discuss relations to geodesic graphs from first-passage percolation and implications for the coalescing walk model of Chaika-Krishnan.