Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$: Neyman Seminar

Seminar | November 13 | 4-5 p.m. | 1011 Evans Hall

 Jason Miller, University of Cambridge

 Department of Statistics

Liouville quantum gravity (LQG) is in some sense the canonical model of a two-dimensional Riemannian manifold and is defined using the (formal) metric tensor
\[ e^{\gamma h(z)} (dx^2 + dy^2)\]
where $h$ is an instance of some form of the Gaussian free field and $\gamma \in (0,2)$ is a parameter. This expression does not make literal sense since $h$ is a distribution and not a function, so cannot be exponentiated. Previously, the associated metric (distance function) was constructed only in the special case $\gamma=\sqrt{8/3}$ in joint work with Sheffield. In this talk, we will show how to associate with LQG a canonical conformally covariant metric for all $\gamma \in (0,2)$. It is obtained as a limit of certain approximations which were recently shown to be tight by Ding, Dub\’edat, Dunlap and Falconet.

Based on joint work with Ewain Gwynne.

 Berkeley, CA 94720, 5106422781