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arXiv · 2607.15370

Limit laws of random simplex tree-child networks

Víctor J. Maciá·Benedikt Stufler
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Abstract

We prove that the longer and shorter Sackin indices of a uniformly random simplex tree-child network with $n$ taxa admit joint distributional limits after rescaling by $n^{-7/4}$. The limiting distributions are described by functionals of a Brownian excursion. We also identify the limiting law of the height after rescaling by $n^{-3/4}$, thereby answering a question of Zhang~(2022). Moreover, we establish sharp tail bounds for the height, which imply convergence of all moments in the above distributional limits. We further obtain a scaling limit for the entire height profile of the leaves. Finally, we determine the local limits of large simplex networks around the fixed root, a uniformly random vertex, and a uniformly random leaf.

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Víctor J. Maciá, Benedikt Stufler. 2026-07-16. Limit laws of random simplex tree-child networks. https://arxiv.org/abs/2607.15370

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