Tightness of supercritical Liouville first passage percolation

Abstract

Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions $\{D_h^{ϵ}{\}}_{ϵ>0}$ on the plane obtained by integrating $e^{\xi h_{ϵ}}$ along paths, where $h_{ϵ}$ for $ϵ>0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller has shown 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 $\gamma =\gamma (\xi )\in (0,2)$).

We show that for all $\xi >0$, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For $\xi >{\xi }_{\mathrm{crit}}$, every possible subsequential limit $D_h$ is a metric on the plane which does 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 these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.