New results for the random nearest neighbor tree

Abstract

In this paper, for d ∈ N, we study the online nearest neighbor random tree in dimension d (d-NN tree for short) defined as follows. Fix the flat torus T d n of dimension d and area n, equipped with the metric inherited from the Euclidean metric of R d . Then, embed consecutively n vertices in T d n uniformly at random and let each vertex but the first one connect to its (already embedded) nearest neighbor. We show that a.a.s. the number of vertices of degree at least k ∈ N in the d-NN tree decreases exponentially with k, and we obtain en passant that the maximum degree of G n is of order Θ d (log n) a.a.s. Moreover, the number of copies of any finite tree in the d-NN tree is shown to be a.a.s. Θ d (n), and all aforementioned counts are, by a general concentration lemma that we show, tightly concentrated around their expectations. We also give explicit bounds for the number of leaves that are independent of the dimension. Next, we show that a.a.s. the height of a uniformly chosen vertex in)) log n independently of the dimension, and the diameter of T d n is a.a.s. (2e + o(1)) log n, again, independently of the dimension. Finally, we define a natural infinite version of a d-NN tree G ∞ via a Poisson Point Process in R d , where the vertices are equipped with uniform arrival times in [0, 1],and show that it corresponds to the local limit of the sequence of finite graphs (G n ) n≥1 . Moreover, we prove that a.s. G ∞ is locally finite, that the simple random walk on G ∞ is a.s. recurrent, and that a.s. G ∞ is connected.To the best of our knowledge, ours are the first rigorous results both in the finite and infinite setup of this model.