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 .

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge c M ∈ (−∞, 1]. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating γ times a variant of the planar Gaussian free field, where γ ∈ (0, 2] satisfies c M = 25 − 6(2/γ + γ/2) 2 . Physics considerations suggest that LQG should also make sense in the regime when c M > 1. However, the behavior in this regime is rather mysterious in part because the corresponding value of γ is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting.We introduce and study a discretization of LQG which makes sense for all values of c M ∈ (−∞, 25). Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same "LQG size" with respect to the Gaussian free field. We prove that several formulas for dimension-related quantities are still valid for c M ∈ (1, 25), with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for c M ∈ (1, 25), which gives a finite quantum dimension if and only if the Euclidean dimension is at most (25 − c M )/12. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius r grows faster than any power of r (which suggests that the Hausdorff dimension of LQG in the case when c M ∈ (1, 25) is infinite).We include a substantial list of open problems.

We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation. arXiv:1409.4312v2 [math.PR]

For γ ∈ (0, 2), U ⊂ C, and an instance h of the Gaussian free field (GFF) on U , the γ-Liouville quantum gravity (LQG) surface associated with (U, h) is formally described by the Riemannian metric tensor e γh (dx 2 + dy 2 ) on U . Previous work by the authors showed that one can define a canonical metric D h on U associated with a γ-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for γ-LQG surfaces. That is, if U, U are domains, φ : U → U is a conformal transformation, Q = 2/γ + γ/2,This proves that D h is intrinsic to the quantum surface structure of (U, h), i.e., it does not depend on the particular choice of parameterization.

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