# Tightness of supercritical Liouville first passage percolation

### Jian Ding

University of Pennsylvania, Philadelphia, USA### Ewain Gwynne

University of Chicago, USA

## Abstract

*Liouville first passage percolation (LFPP)* with parameter $ξ>0$ is the family of random distance functions ${D_{h}}_{ϵ>0}$ on the plane obtained by integrating $e_{ξh_{ϵ}}$ 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 $ξ_{crit}>0$ such that for $ξ<ξ_{crit}$, 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 $γ$-*Liouville quantum gravity metric* for $γ=γ(ξ)∈(0,2)$).

We show that for all $ξ>0$, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For $ξ>ξ_{crit}$, 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 $z∈C$ such that $D_{h}(z,w)=∞$ for every $w∈C∖{z}$. We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.

## Cite this article

Jian Ding, Ewain Gwynne, Tightness of supercritical Liouville first passage percolation. J. Eur. Math. Soc. 25 (2023), no. 10, pp. 3833–3911

DOI 10.4171/JEMS/1273