Critical Liouville Quantum Gravity and Brownian Half-Plane Excursions

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.

Please report video problems to itsupport@slmath.org.

See more of our Streaming videos on our main VMath Videos page.