Scaling limits of discrete snakes with stable branching

Cyril Marzouk

doi:10.1214/19-aihp970

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arXiv · 1802.08137

Scaling limits of discrete snakes with stable branching

Cyril Marzouk

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Abstract

We consider so-called discrete snakes obtained from size-conditioned critical Bienaym\'e-Galton-Watson trees by assigning to each node a random spatial position in such a way that the increments along each edge are i.i.d. When the offspring distribution belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$, we give a necessary and sufficient condition on the tail distribution of the spatial increments for this spatial tree to converge, in a functional sense, towards the Brownian snake driven by the $\alpha$-stable L\'evy tree. We also study the case of heavier tails, and apply our result to study the number of inversions of a uniformly random permutation indexed by the tree.

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Cyril Marzouk. 2018-02-22. Scaling limits of discrete snakes with stable branching. https://doi.org/10.1214/19-aihp970

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