Limit laws of random simplex tree-child networks

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.