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Looptrees

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Random Geometry

A looptree of a plane tree is the graph obtained by replacing each vertex of the tree by a discrete loop of length equal to its degree, and by gluing these loops according to the tree structure. We will be interested in the scaling limits, for the Gromov-Hausdorff topology, of looptrees associated with different classes of random trees: – random trees built by preferential attachment: in this case, the scaling limit is the Brownian rabbit (joint work with N. Curien, T. Duquesne and I. Manolescu); – critical Galton-Watson trees with finite variance: in this case the scaling limit is a multiple of the CRT (joint work with N. Curien and B. Haas) – critical Galton-Watson trees with infinite variance and heavy tail offspring distribution: in this case, the scaling limits are the so-called stable looptrees, which are informally the dual graphs of stable Lvy trees. We will see that the scaling limit of the boundary of large critical site percolation clusters on the UIPT is the random stable looptree of index 3/2 (joint works with N. Curien).

This talk is part of the Isaac Newton Institute Seminar Series series.

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