Tightness of supercritical Liouville first passage percolation

Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions ${D\_h^\epsilon}{\epsilon >0}$ on plane obtained by integrating $e^{\xi h\epsilon}$ along paths, where $h\_\epsilon$ for $\epsilon a smooth mollification planar Gaussian free field. Previous work Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller has shown that there critical value $\xi\_{\mathrm{crit}} > 0$ such < \xi\_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to metric which induces same topology as Euclidean (the so-called $\gamma$-Liouville quantum gravity $\gamma = \gamma(\xi)\in (0,2)$). We show all 0$, metrics are tight respect lower semicontinuous functions. For every possible subsequential limit $D\_h$ does not induce topology: rather, an uncountable, dense, Lebesgue measure-zero set points $z\in\mathbb C$ $D\_h(z,w) \infty$ $w\in\mathbb C\setminus {z}$. expect these limiting related matter central charge in $(1,25)$.