Analysis and Probability seminar

Abstract: The Gaussian free field (GFF) can be viewed as the two-dimensional time version of a Brownian motion and arises as the scaling limit of many discrete two-dimensional random models. While we typically think of the GFF as a random surface, it is actually not a random function and does not make sense pointwise, but rather takes values in a space of distributions. Nevertheless, one can still make sense of 'level sets' of the GFF. Using the GFF and its level sets, we construct a conformally invariant random metric on a planar domain, in which, heuristically, the length of a curve is given by the maximal value of the GFF on said curve. This turns out to be very natural object from the GFF perspective: the metric determines the GFF and it provides a (somewhat) canonical definition of height of the GFF. Furthermore, we expect that this metric should be the renormalised limit of the Liouville quantum gravity metric when the parameter $\xi$ is sent to infinity.