Where do trees grow leaves?
Consider the problem of growing a binary tree uniformly at random by the leaves, that is, of coupling a uniform random binary tree with n internal vertices with a uniform binary tree with one extra internal vertex, in such a way that the latter is obtained from the former by growing two new children of a random leaf. The fact that a probability measure on the leaves for which this uniform growth procedure is possible is due to Luczak and Winkler [1], and some more information about what this measure must look like was obtained in joint work with Alexandre Stauffer [2]. In work with Nicolas Curien and Robin Stephenson [3], we extend this uniform growth procedure to the continuous setting of the Brownian Tree, and explore the interesting multifractal properties of the resulting leaf growth measure. [1] M. Luczak, P. Winkler, Building uniformly random subtrees, Random Structures and Algorithms, 24(2004), 420 443. [2] A. Caraceni, A. Stauffer, Polynomial mixing time of edge flips on quadrangulations, Probab. Theory Related Fields, 176(2020), 35 76. [3] A. Caraceni, N. Curien, R. Stephenson, Where do (random) trees grow leaves, ArXiv preprint.
Area: IS1 - A promenade through integrable system (Alessandra Occelli)
Keywords: Random trees
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