Tightness of supercritical Liouville first passage percolation
Jian Ding, Ewain Gwynne
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) .