The critical beta-splitting random tree: exchangeable partitions and Mellin transform analysis

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The family of beta-splitting random trees was introduced by David Aldous in 1993. The tree is a binary tree with a given number of leaves, and is constructed by recursively splitting the set of leaves into two random subsets with sizes having a distribution given by a certain formula including a parameter beta. The "critical" case beta = -1 turns out to be particularly interesting, and it has recently attracted renewed interest by Aldous and others, including myself. I will talk about a few of these results, in particular a representation using exchangeable random partitions of N, and, using this, an analysis of leaf height using Mellin transforms.

(Joint work with David Aldous, see arXiv:2412.09655 and arXiv:2412.12319.)

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