SLE/GFF Coupling, Zipping Up, and Quantum Length

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”.