Probability Seminar

Speaker #1 (Lawler): I will present a somewhat novel approach to known relationships (in works of Sheffield, Miller, and others) between SLE and GFF, the exponential of the GFF (quantum length/area), and Minkowski content of paths. The Neumann GFF is defined as the real part of a stochastic integral with respect to a complex Brownian motion. This viewpoint helps illuminate the relationship between boundary length and the stationary object invariant under “zipping up”.

Speaker #2 (Ang): Sheffield showed that conformally welding a $\gamma$-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter $\kappa = \gamma^2$ as the interface, and Duplantier-Miller-Sheffield proved similar stories for $\kappa = 16/\gamma^2$ for $\gamma$-LQG surfaces with boundaries decorated by looptrees of disks or by continuum random trees. We study these dynamics for LQG surfaces coming from Liouville conformal field theory (LCFT). At stopping times depending only on the curve, we give an explicit description of the surface and curve in terms of LCFT and SLE. This has applications to both LCFT and SLE. We prove the boundary BPZ equation for LCFT, which is crucial to solving boundary LCFT. With Yu we prove the reversibility of whole-plane SLE for $\kappa ≥ 8$ via a novel radial mating-of-trees.