Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to nonfederal websites. Their policies may differ from this site.

null (Ed.)Abstract For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma $$ γ  Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar Gaussian free field 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 et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. ArXiv eprints, arXiv:1904.08021 ). It is also known that these axioms are satisfied for the $$\sqrt{8/3}$$ 8 / 3 LQG metric constructed by Miller and Sheffield (2013–2016). For any weak $$\gamma $$ γ LQG metric, we obtain moment bounds for diameters of sets as well as pointtopoint, settoset, and pointtoset distances. We also show that any such metric is locally biHölder 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 $$\gamma $$ γ LQG metric is unique for each $$\gamma \in (0,2)$$ γ ∈ ( 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 $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 .more » « less