Morris Ang: Moduli of Annuli in Random Conformal Geometry

I will discuss results on two fundamental objects in 2D random geometry. The first is Werner's measure on self-avoiding loops, which is the conjectural scaling limit of self-avoiding walks in the plane. We obtain an exact formula for its annulus partition function, confirming a prediction of Cardy (2006) via conformal field theory methods. Second, the Brownian annulus is the random surface arising as the scaling limit of planar maps with annular topology. We derive the law of its conformal modulus ("thickness"), as anticipated from the ghost partition function in bosonic string theory. Our arguments depend on the deep interplay between Schramm-Loewner evolution curves and Liouville quantum gravity surfaces.