Liouville first passage percolation (LFPP) with parameter ξ > 0 is the family of random distance functions {D h } >0 on the plane obtained by integrating e ξh along paths, where h for > 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).1 As per the discussion in Section 1.3 below, ξ = 1/ √ 6 corresponds to Liouville quantum gravity with parameter γ = 8/3 (equivalently, matter central charge cM = 0) and the fact that Q(1/ √ 6) = 5/ √ 6 is a consequence of the fact that 8/3-LQG has Hausdorff dimension 4.

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 e-prints, 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 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 $$\gamma $$ 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 .

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We study the Liouville metric associated to an approximation of a logcorrelated Gaussian field with short range correlation. We show that below a parameter γ c > 0, the left-right length of rectangles for the Riemannian metric e γφ0,n ds 2 with various aspect ratio is concentrated with quasi-lognormal tails, that the renormalized metric is tight when γ < min(γ c , 0.4) and that subsequential limits are consistent with the Weyl scaling.2010 Mathematics Subject Classification. Primary: 60K35. Secondary: 60G60.

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