Probability Seminar: Linan Chen

We generalize part of the recent 2D results of Duplantier and Sheffield on Liouville quantum gravity to four dimensions. To be specific, we construct a random measure on R^4  which can be "formally" written as "exp(h(x))dx" where "dx" is the Lebesgue measure and "h" is an instance of the Gaussian free field defined on R^4. In this setting, we also prove the KPZ (Knizhnik-Polyakov-Zamolodchikov) relation, which is the quadratic relation between the Euclidean Hausdorff dimension of a bounded set in R^4 and its counterpart under the random measure. This is joint work with Dmitry Jakobson.