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Brownian excursions, conformal loop ensembles and critical Liouville quantum gravity

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In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be described by the mating of two continuum random trees. In this talk I will discuss the counterpart of their result for critical LQG and SLE . More precisely, I will explain how, as we approach criticality from the subcritical regime, the space-filling SLE degenerates to the uniform CLE _4 exploration introduced by Werner and Wu, together with a collection of independent coin tosses indexed by the branch points of the exploration. Furthermore, although the pair of continuum random trees collapse to a single continuum random tree in the limit we can apply an appropriate affine transform to the encoding Brownian motions before taking the limit, and get convergence to a Brownian half-plane excursion. I will try to explain how observables of interest in the critical CLE decorated LQG picture are encoded by a growth fragmentation naturally embedded in the Brownian excursion. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun.

This talk is part of the Probability series.

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