Abstract
In this paper, we study the online nearest neighbor random tree in dimension \(d\in {\mathbb {N}}\) (called d-NN tree for short) defined as follows. We fix the torus \({\mathbb {T}}^d_n\) of dimension d and area n and equip it with the metric inherited from the Euclidean metric in \({\mathbb {R}}^d\). Then, embed consecutively n vertices in \({\mathbb {T}}^d_n\) uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph \(G_n\). We show multiple results concerning the degree sequence of \(G_n\). First, we prove that typically the number of vertices of degree at least \(k\in {\mathbb {N}}\) in the d-NN tree decreases exponentially with k and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of \(G_n\) is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in \(G_n\) is \((1+o(1))\log n\) and the diameter of \({\mathbb {T}}^d_n\) is \((2e+o(1))\log n\), independently of the dimension. Finally, we define a natural infinite analog \(G_{\infty }\) of \(G_n\) and show that it corresponds to the local limit of the sequence of finite graphs \((G_n)_{n \ge 1}\). Furthermore, we prove almost surely that \(G_{\infty }\) is locally finite, that the simple random walk on \(G_{\infty }\) is recurrent, and that \(G_{\infty }\) is connected.
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Notes
The choice of the torus as an ambient space might not be the most natural. However, it avoids the need for boundary considerations and, in most part, does not modify the results. Indeed, the main proof ideas can be applied for other geometric spaces but at the cost of an increased level of technicality.
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Acknowledgements
The authors would like to thank Gábor Lugosi and Vasiliki Velona for bringing the topic to our attention and for discussions in an early stage of this paper. The authors would also like to thank Bas Lodewijks for a careful proofreading, and to David Aldous and Andrew Wade for bringing several additional references to our attention. The first author would like to thank Ivailo Hartarsky for a discussion around the connectivity of \(G_{\infty }\). We are also grateful for many useful comments and remarks by the two anonymous referees.
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Dieter Mitsche has been partially supported by grant Fondecyt Grant 1220174 and by Grant GrHyDy ANR-20-CE40-0002.
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