The critical and supercritical Liouville quantum gravity metrics

Mon April 4th 2022, 4:00pm
Sequoia 200
Ewain Gwynne, University of Chicago

Liouville first passage percolation (LFPP) with parameter $\xi>0$ is the family of random distance functions on the plane obtained by integrating $e^{\xi h_\epsilon}$ along paths, where $h_\epsilon$ for $\epsilon>0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller showed that there is a critical value $\xi_{\text{crit}}>0$ such that for $\xi<\xi_{\text{crit}}$, LFPP converges under appropriate rescaling to a random metric on the plane which induces the same topology as the Euclidean metric: the so-called $\gamma$-Liouville quantum gravity metric for $\gamma=\gamma(\xi)\in(0,2)$.

Recently, Jian Ding and I showed that LFPP also converges when $\xi\geq\xi_{\text{crit}}$. For $\xi>\xi_{\text{crit}}$, the subsequential limiting metrics do not induce the Euclidean topology. Rather, there is an uncountable, dense, Lebesgue measure-zero set of points $z\in\mathbb{C}$ such that $D_h(z,w)=\infty$ for every $w\in\mathbb{C}\setminus\{z\}$. These metrics are related to a supercritical phase of Liouville quantum gravity, corresponding to matter central charge in (1,25).

I will discuss the properties of the limiting metrics, their connection to Liouville quantum gravity, and some open problems. The talk will assume no prior knowledge of Liouville quantum gravity.