Schramm -- Loewner Evolution and Liouville Quantum Multifractality

Schramm -- Loewner Evolution and Liouville Quantum Multifractality

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Bertrand Duplantier, Institut de Physique Theorique, France
Jadwin Hall 343

We describe some recent advances in the study of the fundamental coupling of a canonical model of random paths, the Schramm--Loewner Evolution (SLE), to a canonical model of random surfaces, Liouville Quantum Gravity (LQG). The latter is expected to be the conformally invariant continuum limit of various models of random planar maps. Via the KPZ relation the multifractal spectra of planar SLE morph into natural quantum analogues in LQG. We make this explicit for extreme nesting in the Conformal Loop Ensemble (CLE) in the plane and on a random planar map, and for the SLE joint harmonic measure and winding spectrum. Based on joint work with G. Borot (MPI Bonn), J. Miller (Cambridge) and S. Sheffield (MIT).