Ewain Gwynne

Tightness of supercritical Liouville first passage percolation

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-Dub\'edat-Dunlap-Falconet and Gwynne-Miller showed that there is a critical value $\xi_{\mathrm{crit}} > 0$ such that for $\xi < \xi_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling 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 admits non-trivial subsequential scaling limits for all $\xi > 0$.
For $\xi >\xi_{\mathrm{crit}}$, the subsequential limiting metrics do \emph{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 $w\in\mathbb C\setminus \{z\}$.
We expect that the subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.