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If $(\omega(e))$ is a family of random variables (weights) assigned to theedges of $\mathbb{Z}^d$, the nearest neighbor graph is the directed graphinduced by all edges $\langle x,y \rangle$ such that $\omega(\{x,y\})$ isminimal among all neighbors $y$ of $x$. That is, each vertex points to itsclosest neighbor, if the weights are viewed as edge-lengths. Nanda-Newmanintroduced nearest neighbor graphs when the weights are i.i.d. and continuouslydistributed and proved that a.s., all components of the undirected version ofthe graph are finite. We study the case of translation invariant, distinctweights, and prove that nearest neighbor graphs do not contain doubly-infinitedirected paths. In contrast to the i.i.d. case, we show that in this stationarycase, the graphs can contain either one or two infinite components (but notmore) in dimension two, and $k$ infinite components for any $k \in [1,\infty]$in dimension $\geq 3$. The latter constructions use a general procedure toexhibit a certain class of directed graphs as nearest neighbor graphs withdistinct weights, and thereby characterize all translation invariant nearestneighbor graphs. We also discuss relations to geodesic graphs fromfirst-passage percolation and implications for the coalescing walk model ofChaika-Krishnan.

Random nearest neighbor graphs: the translation invariant case