2024/01/15 by Caraceni, Alessandra, Curien, Nicolas, Stephenson, Robin
#05C05 #05C80 #60G18 #60G57 #FOS: Mathematics #G.3 #Probability (math.PR)
paper · doi:10.48550/arxiv.2401.07891
We study a model of random binary trees grown "by the leaves" in the style of Luczak and Winkler. If τn is a uniform plane binary tree of size n, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure ντn such that the tree obtained by adding a cherry on a leaf sampled according to ντn is still uniformly distributed on the set of all plane binary trees with size n+1. It turns out that the measure ντn, which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree τn. In fact, we prove that, as n → ∞, with high probability it is almost entirely supported by a subset of only n3 ( 2 - √(3))+o(1) ≈ n0.8038... leaves. In the continuous setting, we construct the scaling limit of this measure, which is a probability measure on the Brownian Continuum Random Tree supported by a fractal set of dimension 6 (2 - √(3)). We also compute the full (discrete) multifractal spectrum. This work is a first step towards understanding the diffusion limit of the discrete leaf-growth procedure.