Tightness of supercritical Liouville first passage percolation

Abstract

Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions $\{D_h^\epsilon\}_{\epsilon >0}$ 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 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 $\gamma$-\emph{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 \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 these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.