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 $\gamma$-\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 $(1,25)$.

72 pages, 9 figures; to appear in JEMS