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–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 $(1,25)$.