Weak LQG metrics and Liouville first passage percolation

Abstract

For ${\hat{I}}^3\in (0,2)$, we define a weak ${\hat{I}}^3$-Liouville quantum gravity (LQG) metric to be a function $h\mapsto D_h$ which takes in an instance of the planar Gaussian free field (GFF) and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding-Dubédat-Dunlap-Falconet (2019). It is also known that these axioms are satisfied for the $\sqrt{8/3}$-LQG metric constructed by Miller and Sheffield (2013-2016). For any weak ${\hat{I}}^3$-LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak ${\hat{I}}^3$-LQG metric is unique for each ${\hat{I}}^3\in (0,2)$, which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when ${\hat{I}}^3=\sqrt{8/3}$.

57 pages, 5 figures; to appear in PTRF