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arXiv · 2202.08666

Scaling limits of random looptrees and bipartite plane maps with prescribed large faces

Cyril Marzouk
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Abstract

We first rephrase and unify known bijections between bipartite plane maps and labelled trees with the formalism of looptrees, which we argue to be both more relevant and technically simpler since the geometry of a looptree is explicitly encoded by the depth-first walk (or Łukasiewicz path) of the tree, as opposed to the height or contour process for the tree. We then construct continuum analogues associated with any càdlàg path with no negative jump and derive several invariance principles. We especially focus on uniformly random looptrees and maps with prescribed face degrees and study their scaling limits in the presence of macroscopic faces, which complements a previous work in the case of no large faces. The limits (along subsequences for maps) form new families of random metric measured spaces related to processes with exchangeable increments with no negative jumps and our results generalise previous works which concerned the Brownian and stable Lévy bridges.

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Cyril Marzouk. 2022-02-17. Scaling limits of random looptrees and bipartite plane maps with prescribed large faces. https://doi.org/10.1214/23-aihp1387

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