Introduction to the Liouville quantum gravity metric

One can define LQG surfaces with the topology of any orientable surface (disks, spheres, torii, etc.), and all have the same local geometry. We will be primarily interested in the local geometry, so for simplicity we will focus on LQG surfaces with the topology of the whole plane.¹¹1See [DMS21, DKRV16, GRV19, DRV16, Rem18] for constructions of canonical LQG surfaces with various topologies.

To define LQG, we first need to define the Gaussian free field. The whole-plane Gaussian free field (GFF) is the centered Gaussian process $h$ with covariances²²2Our choice of covariance function corresponds to normalizing $h$ so that its average over the unit circle is zero; see, e.g., [Var17, Section 2.1.1].

Since $\lim_{w\rightarrow z}G(z,w)=\infty$, the GFF is not a function. However, it still makes sense as a generalized function (i.e., a distribution). That is, if $\phi:\mathbbm{C}\rightarrow\mathbbm{R}$ is smooth and compactly supported, then one can define the $L^{2}$ inner product $(h,\phi)=\int_{\mathbbm{C}}h(z)\phi(z)\,d^{2}z$ as a random variable. These random variables have covariances

The reader can consult [She07, WP21, BP] for more background on the GFF. We have included a simulation of the GFF in Figure 1, left.

More generally, we say that a random generalized function $h$ on $\mathbbm{C}$ is a GFF plus a nice function if $h=\widetilde{h}+f$, where $\widetilde{h}$ is the whole-plane GFF and $f:\mathbbm{C}\rightarrow\mathbbm{R}$ is a (possibly random and $\widetilde{h}$-dependent) function which is continuous except at finitely many points.

Let $\gamma\in(0,2]$, which will be the parameter for our LQG surfaces. A $\gamma$-LQG surface parametrized by $\mathbbm{C}$ is the random two-dimensional Riemannian manifold with Riemannian metric tensor

where $dx^{2}+dy^{2}$ denotes the Euclidean metric tensor and $h$ is the whole-plane GFF, or more generally a whole-plane GFF plus a nice function.

The Riemannian metric tensor (1.1) is not well-defined since $h$ is not defined pointwise, so $e^{\gamma h}$ does not make literal sense. However, it is possible to make sense of various objects associated with (1.1) rigorously using regularization procedures. The idea is to consider a collection of continuous functions $\{h_{\varepsilon}\}_{\varepsilon>0}$ which converge to $h$ in some sense as $\varepsilon\rightarrow 0$, define objects associated with the Riemannian metric tensor (1.1) with $h_{\varepsilon}$ in place of $h$, then take a limit as $\varepsilon\rightarrow 0$. In this paper, we will discuss two objects which can be constructed in this way: the LQG area measure (to be discussed just below) and the LQG metric (which is the main focus of the paper). Other examples include the LQG length measure on Schramm-Loewner evolution-type curves [She16, Ben18], Liouville Brownian motion [GRV16, Ber15], and the correlation functions for the random “fields” $e^{\alpha h}$ for $\alpha\in\mathbbm{R}$ [KRV20].

For simplicity, let us restrict attention to the case when $h$ is a whole-plane GFF. A convenient choice of $\{h_{\varepsilon}\}$ is the convolution of $h$ with the heat kernel. For $t>0$ and $z\in\mathbbm{C}$, we define the heat kernel $p_{t}(z):=\frac{1}{2\pi t}e^{-|z|^{2}/2t}$ and we define

where the integral is interpreted in the sense of distributional pairing.

The easiest non-trivial object associated with (1.1) to construct rigorously is the LQG area measure, or volume form. This is a random measure $\mu_{h}$ on $\mathbbm{C}$ which is defined as the a.s. limit, with respect to the vague topology,³³3In the case when $\gamma=2$, there is a log correction in the scaling factor, see [DRSV14a, DRSV14b, Pow18].

where $d^{2}z$ denotes Lebesgue measure on $\mathbbm{C}$. The reason for the normalizing factor $\varepsilon^{\gamma^{2}/2}$ is that $\mathbbm{E}[e^{\gamma h_{\varepsilon}^{*}(z)}]\approx\varepsilon^{-\gamma^{2}/2}$. The existence of the limit in (1.3) is a special case of the theory of Gaussian multiplicative chaos (GMC) [Kah85, RV14]. There are a variety of different ways of approximating $\mu_{h}$ which are all known to converge to the same limit; see [DS11, Sha16] for some results in this direction.

The measure $\mu_{h}$ is mutually singular with respect to Lebesgue measure. In fact, it is supported on a dense subset of $\mathbbm{C}$ of Hausdorff dimension $2-\gamma^{2}/2$; see, e.g., [DS11, Section 3.3]. However, it has no atoms and assigns positive mass to every open subset of $\mathbbm{C}$.

The LQG area measure also satisfies a conformal covariance property. Let $U,\widetilde{U}\subset\mathbbm{C}$ be open and let $f:\widetilde{U}\rightarrow U$ be a conformal (bijective, holomorphic) map. Let

Then $\widetilde{h}$ is a random generalized function on $\widetilde{U}$ whose law is locally absolutely continuous with respect to the law of $h$, so $\mu_{\widetilde{h}}$ can be defined. It is shown in [DS11, Proposition 2.1] that a.s.

We can think of the pairs $(U,h|_{U})$ and $(\widetilde{U},\widetilde{h})$ as representing two different parametrizations of the same LQG surface. The relation (1.5) implies that the LQG area measure is an intrinsic function of the surface, i.e., it does not depend on the choice of parametrization.

The main focus of this article is the LQG metric, i.e., the Riemannian distance function associated with the Riemannian metric tensor (1.1). This metric can be constructed via a similar regularization procedure as the measure, but the proof of convergence is much more involved. See Section 2 for details.

LQG was first studied by Polyakov [Pol81] in the 1980s in the context of string theory (we discuss Polyakov’s motivation in Remark 2.11). LQG is also of interest in conformal field theory since it is closely connected to Liouville conformal field theory, one of the simplest non-trivial conformal field theories. See [Var17] for an overview of recent mathematical work on Liouville conformal field theory.

One of the most important applications of LQG theory is the so-called Knizhnik-Polyakov-Zamolodchikov (KPZ) formula [KPZ88], which gives a relationship between critical exponents for statistical mechanics models in random geometries and deterministic geometries.⁴⁴4The KPZ formula discussed here has no relation with Kardar-Parisi-Zhang equation from [KPZ86], except that the initials of the authors for the two papers are the same. For example, this formula was used by Duplantier to give non-rigorous predictions for the Brownian intersection exponents [Dup98] (the exponents were predicted earlier by Duplantier and Kwon [DK88]). These predictions were later verified rigorously by Lawler, Schramm, and Werner in [LSW01a, LSW01b, LSW02] using SLE techniques. We discuss the KPZ formula in the context of the LQG metric in Section 3.5.

Another reason to study LQG is that, at least conjecturally, it describes the large-scale behavior of discrete random geometries, such as random planar maps. A planar map is a graph embedded in the plane so that no two edges cross, viewed modulo orientation-preserving homeomorphisms of the plane. See Figure 1, middle, for an illustration. There are various interesting types of random planar maps, such as the following.

• •

  Uniform planar maps: consider the (finite) set of planar maps with a specified number $n\in\mathbbm{N}$ of edges and choose an element of this set uniformly at random.

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  Uniform planar maps with local constraints, such as triangulations (resp. quadrangulations), where each face has exactly 3 (resp. 4) edges.

• •

  Decorated planar maps. Suppose, for example, that we want to sample a uniform pair $(M,T)$ consisting of a planar map $M$ with $n$ edges and a spanning tree $T$ on $M$ (i.e., a subgraph of $M$ which includes every vertex of $M$ and has no cycles). Under this probability measure, the marginal law of $M$ is not uniform; rather, the probability of seeing any particular planar map with $n$ edges is proportional to the number of spanning trees it admits. One can similarly consider planar maps decorated by statistical physics models (such as the Ising model or the FK model) or by various types of orientations on their edges.

It is believed that a large class of different types of planar maps converge to LQG in some sense. The parameter $\gamma$ depends on the type of planar map under consideration. Uniform planar maps, including maps with local constraints, correspond to $\gamma=\sqrt{8/3}$. This case is sometimes called “pure gravity” in the physics literature. Other values of $\gamma$ correspond to planar maps decorated by statistical physics models. This case is sometimes called “gravity coupled to matter”. For example, the spanning tree-decorated maps discussed above are expected to converge to LQG with $\gamma=\sqrt{2}$.

For this article, the most relevant conjectured mode of convergence of random planar maps toward LQG is the following. View a planar map as a compact metric space, equipped with the graph distance. If we re-scale distances in this metric space appropriately, then as the number of edges tends to $\infty$ it should converge in the Gromov-Hausdorff sense to an LQG surface equipped with its LQG metric. So far, this type of convergence has only been proven for $\gamma=\sqrt{8/3}$, see Section 2.4. However, weaker connections between random planar maps and $\gamma$-LQG have been established rigorously for all $\gamma\in(0,2)$ using so-called mating of trees theory. See [GHS19] for a survey of this theory.

In analogy with the approximation scheme for the LQG measure in (1.3), for a parameter $\xi>0$, we define

where the infimum is over all piecewise continuously differentiable paths $P:[0,1]\rightarrow\mathbbm{C}$ from $z$ to $w$. The metrics $D_{h}^{\varepsilon}$ are sometimes referred to as $\varepsilon$-Liouville first passage percolation (LFPP).

We want to choose the parameter $\xi$ in a manner depending on $\gamma$ so that the LFPP metrics (2.1) converge to the distance function associated with the metric tensor (1.1). To determine what $\xi$ should be, we use a heuristic scaling argument. From (1.3), we see that scaling areas by $C>0$ corresponds to replacing $h$ by $h+\frac{1}{\gamma}\log C$. On the other hand, from (2.1) we see that replacing $h$ by $h+\frac{1}{\gamma}\log C$ scales distances by a factor of $C^{\xi/\gamma}$. Hence $\xi/\gamma$ is the scaling exponent relating areas and distances. In other words, we want $\gamma/\xi$ to be the “dimension” of an LQG surface.

It was shown in [DZZ19, DG18] that there is an exponent $d_{\gamma}>2$ which arises in various discrete approximations of LQG and which can be interpreted as the dimension of LQG. For example, $d_{\gamma}$ is the ball volume exponent for certain random planar maps [DG18, Theorem 1.6]. Once the LQG metric has been constructed, one can show that $d_{\gamma}$ is its Hausdorff dimension [GP22] (see Theorem 3.1). The value of $d_{\gamma}$ is not known explicitly except that $d_{\sqrt{8/3}}=4$. Computing $d_{\gamma}$ for general $\gamma\in(0,2]$ is one of the most important open problems in LQG theory.

The above discussion suggests that one should take

It is shown in [DG18, Proposition 1.7] that $\xi$ is an increasing function of $\gamma$, so for $\gamma\in(0,2]$, $\xi$ takes values in $(0,2/d_{2}]$. Estimates for $d_{\gamma}$ [DG18, GP19] show that $2/d_{2}\approx 0.41$.

The definition of LFPP in (2.1) also makes sense for $\xi>2/d_{2}$. In this regime, LFPP metrics do not correspond to $\gamma$-LQG with $\gamma\in(0,2]$. Rather, as we will explain in Section 2.3.2, LFPP for $\xi>2/d_{2}$ converges to a metric which is related to LQG with matter central charge ${\mathbf{c}_{\mathrm{M}}}\in(1,25)$, or equivalently $\gamma\in\mathbbm{C}$ with $|\gamma|=2$.

In the special case when $\gamma=\sqrt{8/3}$, there is an earlier construction of the $\sqrt{8/3}$-LQG metric due to Miller and Sheffield [MS20, MS21b, MS21c]. We will comment briefly on the main idea of this construction. See Miller’s ICM paper [Mil18] for a more detailed overview.

The idea of the Miller-Sheffield construction is to first construct a candidate for LQG metric balls, then show that these balls are in fact the metric balls for a unique metric on $\mathbbm{C}$. The candidates for LQG metric balls are generated using a random growth process called quantum Loewner evolution (QLE), which is produced by “re-shuffling” an SLE₆ curve in a random manner depending on $h$. The construction of this growth process and the proof that one can generate a metric from it both rely crucially on special symmetries for $\sqrt{8/3}$-LQG which are established in [DMS21, MS19], so the construction does not work for any other value of $\gamma$.

The Miller-Sheffield metric satisfies the conditions of Definition 2.5, so Theorem 2.6 implies that it agrees with the $\sqrt{8/3}$-LQG metric constructed using LFPP. On the other hand, the construction using QLE gives a number of properties of the $\sqrt{8/3}$-LQG metric which are not apparent from the LFPP construction, for example various Markov properties for LQG metric balls and the fact that $d_{\sqrt{8/3}}=4$. These properties can be proven directly using QLE, or can alternatively be deduced from analogous properties of the Brownian map together with the equivalence between the Brownian map and $\sqrt{8/3}$-LQG discussed just below.

The papers [MS20, MS21b] also establish a link between the $\sqrt{8/3}$-LQG metric and uniform random planar maps. This link comes by combining two big results:

• •

  Le Gall [Le 13] and Miermont [Mie13] showed independently that certain types of uniform random planar maps (namely, uniform $k$-angulations for $k=3$ or $k$ even), equipped with their graph distance, converge in the Gromov-Hausdorff sense to a random metric space called the Brownian map. See [LG14, Le 19] for a survey of this work.

• •

  Miller and Sheffield showed that there is a certain special variant of the GFF on $\mathbbm{C}$ (corresponding to the so-called quantum sphere) such that the sphere $\mathbbm{C}\cup\{\infty\}$, equipped with the $\sqrt{8/3}$-LQG metric, is isometric to the Brownian map. This is done using the axiomatic characterization of the Brownian map from [MS21a].

For $\Delta>0$, the $\Delta$-Hausdorff content of a compact metric space $(X,d)$ is

and the Hausdorff dimension of $(X,d)$ is the infimum of the values of $\Delta$ for which the $\Delta$-Hausdorff content is zero.

The following theorem follows from the combination of [GP22, Corollary 1.7] and [Pfe21, Proposition 1.14].

As noted above, the value of $d_{\gamma}$ is not known except that $d_{\sqrt{8/3}}=4$, but upper and lower bounds for $d_{\gamma}$ have been proven in [DG18, GP19, Ang19] (see Figure 5). It is shown in [DG18, Theorem 1.2] that $\gamma\mapsto d_{\gamma}$ is increasing and $\lim_{\gamma\rightarrow 0}d_{\gamma}=2$. Hence, Theorem 3.1 implies that the LQG metric gets “rougher” as $\gamma$ increases. We expect that the dimension of $\mathbbm{C}$ with respect to the critical ($\gamma=2$) LQG metric is $d_{2}=\lim_{\gamma\rightarrow 2}d_{\gamma}\approx 4.8$, but this has not been proven.

It was shown in [AFS20] that for $\gamma\in(0,2)$, the Minkowski dimension of $(\mathbbm{C},D_{h})$ is also equal to $d_{\gamma}$. We expect that in this case, the $d_{\gamma}$-Minkowski content measure for $D_{h}$ exists and is equal to the $\gamma$-LQG area measure $\mu_{h}$ from (1.3). Similarly, the Hausdorff measure associated with $D_{h}$, for an appropriate gauge function, should exist and be equal to $\mu_{h}$. This has been proven for the Brownian map (which is equivalent to $\sqrt{8/3}$-LQG, recall Section 2.4) in [Le 21].

The optimal Hölder exponents relating $D_{h}$ and the Euclidean metric can be computed in terms of $\xi$ and $Q$. For the subcritical (resp. supercritical) case, see [DFG⁺20, Theorem 1.7] (resp. [Pfe21, Proposition 1.10]).

In the critical case when $\xi=2/d_{2}$, equivalently $Q=2$, the metric $D_{h}$ is continuous with respect to the Euclidean metric but not Hölder continuous. Rather, the optimal upper bound for $D_{h}(z,w)$ is a power of $1/\log(|z-w|^{-1})$ [DG21b].

We also have moment bounds for point-to-point distances, set-to-set distances, and diameters. The following is a compilation of several results from [DFG⁺20, Pfe21].

The moment bound for diameters is related to the fact that the LQG area measure has finite moments up to order $4/\gamma^{2}$ (see, e.g., [RV14, Theorem 2.11]).

Using basic metric space theory, one can show that a.s. for any two points $z,w\in\mathbbm{C}$ with $D_{h}(z,w)<\infty$, there is a $D_{h}$-geodesic from $z$ to $w$, i.e., a path of minimal $D_{h}$-length (see, e.g., [BBI01, Corollary 2.5.20] for the subcritical case and [Pfe21, Proposition 1.12] for the supercritical case). If $z$ and $w$ are fixed, then a.s. this geodesic is unique [MQ20b, Theorem 1.2]. We give a short proof of this fact in Lemma 4.2 below.

It can be shown that the $D_{h}$-geodesics started from a specified point have a tree-like structure: two geodesics with the same starting point and different target points stay together for a non-trivial initial time interval. The property is called confluence of geodesics, and can be seen in the simulations from Figure 2.

We emphasize that confluence of geodesics is not true for a smooth Riemannian metric (such as the Euclidean metric). Rather, two geodesics for a smooth Riemannian metric with the same starting points and different target points typically intersect only at their starting point.

Confluence of geodesics for the LQG metric was established in the subcritical case ($\xi<2/d_{2}$) in [GM20a] and for general $\xi>0$ in [DG21a]. Let us now state a precise version of this result, which is illustrated in Figure 4 For $s>0$ and $z\in\mathbbm{C}$, let $\mathcal{B}_{s}(z;D_{h})$ be the $D_{h}$-metric ball of radius $s$ centered at $z$.

Theorem 3.4 only holds a.s. for a fixed center point $z\in\mathbbm{C}$. Almost surely, there is a Lebesgue measure zero set of points in $\mathbbm{C}$ where Theorem 3.4 fails. For example, if $P:[0,T]\rightarrow\mathbbm{C}$ is a $D_{h}$-geodesic, then the conclusion of Theorem 3.4 fails for each $z\in P((0,T))$.

Confluence of geodesics is used in the proof of the uniqueness of the $\gamma$-LQG metric $\gamma\in(0,2)$ in [GM21b]. Roughly speaking, confluence is used to establish near-independence for events which depend on small neighborhoods of far-away points on a $D_{h}$-geodesic, despite the fact that $D_{h}$-geodesics are non-Markovian and do not depend locally on $h$. See [GM21b] for details. The proof of the uniqueness of the LQG metric for general $\xi>0$ in [DG21c] does not use confluence of geodesics.

Various extensions of the confluence property for $\gamma\in(0,2)$ are proven in [GPS22, Gwy21] and for $\gamma=\sqrt{8/3}$ in [AKM17, MQ20a].

Little is known about the geometry of a single LQG geodesic. For example, we do not know the Hausdorff dimension of such a geodesic w.r.t. the Euclidean metric (the dimension w.r.t. the LQG metric is trivially equal to 1), and we do not have any exact description of its law. The strongest current results in this direction are an upper bound for the Euclidean dimension of an LQG geodesic [GP22, Corollary 1.10], which is not expected to be optimal; and the fact LQG geodesics do not locally look like SLE_κ curves for any value of $\kappa$ [MQ20b]. We do not have a non-trivial lower bound for the Euclidean Hausdorff dimension of an LQG geodesic, but we expect that it is strictly greater than 1 (see [DZ19] for a closely related result for the geodesics for a version of LFPP). Finally, we mention the very recent work [BBG21], which constructs a local limit of the GFF near a typical point of an LQG geodesic.

From the simulations in Figure 2, one can see that LQG metric balls have a fractal-like geometry. Almost surely, the complement of each LQG metric ball has infinitely many connected components, in the subcritical, critical, and supercritical cases [GPS22, Pfe21]. In fact, a.s. “most” points on the boundary of the ball do not lie on any complementary connected component, but rather are accumulation points of arbitrarily small complementary connected components [GPS22, Theorem 1.14], [DG21a, Theorem 1.4].

In the subcritical and critical cases, i.e., when $\xi=\gamma/d_{\gamma}$ for $\gamma\in(0,2]$, the LQG metric induces the same topology as the Euclidean metric so a.s. each closed LQG metric ball is equal to the closure Euclidean interior. In contrast, in the supercritical case a.s. each LQG metric ball has empty Euclidean interior but positive Lebesgue measure. This is a consequence of the fact that the set of singular points from (2.13) is Euclidean-dense but has Lebesgue measure zero.

In the subcritical case, it is shown in [Gwy20, GPS22] that a.s. the Hausdorff dimension of the boundary of a $\gamma$-LQG metric ball for $\gamma\in(0,2)$ w.r.t. the Euclidean (resp. LQG) metric is $2-\xi Q+\xi^{2}/2$ (resp. $d_{\gamma}-1$). We expect that these formulas are also valid for $\gamma=2$ (equivalently, $\xi=2/d_{2}$).

In the supercritical case $\xi>2/d_{2}$, the LQG metric $D_{h}$ does not induce the Euclidean topology, so one has to make a distinction between the boundary with respect to the Euclidean topology or with respect to $D_{h}$. The boundary of a closed $D_{h}$-metric ball with respect to the Euclidean topology is equal to the ball itself (since the ball is Euclidean closed and has empty Euclidean interior), whereas the boundary with respect to $D_{h}$ is a proper subset of the ball [DG21a, Section 1.2]. It is shown in [Pfe21, Proposition 1.14] that for $\xi>2/d_{2}$, a.s. the Euclidean boundary of a $D_{h}$-metric ball (i.e., the whole $D_{h}$-metric ball) is not compact with respect to $D_{h}$ and has infinite Hausdorff dimension w.r.t. $D_{h}$. We expect that the same is true for the $D_{h}$-boundary of a $D_{h}$-metric ball. The Hausdorff dimension of the Euclidean boundary of a $D_{h}$-metric ball with respect to the Euclidean metric is 2 since the metric ball has positive Lebesgue measure. The Hausdorff dimension of the $D_{h}$-boundary of a $D_{h}$-metric ball with respect to the Euclidean metric has not been computed rigorously.

It is also of interest to consider the boundary of a single complementary connected component of an LQG metric ball. The Hausdorff dimension of such a boundary component w.r.t. the Euclidean or LQG metric is not known. However, it is known that, even in the supercritical case, each boundary component is a Jordan curve and is compact and finite-dimensional w.r.t. $D_{h}$ [DG21a, Theorem 1.4].

The (geometric) Knizhnik-Polyakov-Zamolodchikov (KPZ) formula [KPZ88] is a formula which relates the “Euclidean dimension” and the “LQG dimension” of a deterministic set $X\subset\mathbbm{C}$, or a random set independent from the GFF $h$. The first rigorous versions of the KPZ formula appeared in [DS11, RV11]. These papers defined the “LQG dimension” in terms of the LQG area measure. There are several different versions of the KPZ formula in the literature which use different notions of dimension (see, e.g., [Aru15, GHM20, BGRV16, BJRV13, BS09]). Here, we state what is perhaps the most natural version of the KPZ formula, where we compare the Hausdorff dimensions of a set w.r.t. the LQG metric and the Euclidean metric. We start with the subcritical case, which is [GP22, Theorem 1.4].

Theorem 3.6 does not apply if $X$ is not independent from $h$. For example, the KPZ formula does not hold for the Hausdorff dimensions of LQG metric ball boundaries w.r.t. the Euclidean and LQG metrics, as discussed in Section 3.4. However, one has inequalities relating the Hausdorff dimensions of an arbitrary set with respect to the Euclidean and LQG metrics, see [GP22, Theorem 1.8].

It is shown in [Pfe21, Theorem 1.15] that the KPZ formula of Theorem 3.6 extends to the case when $\xi\geq 2/d_{2}$ (modulo some technicalities about the particular notion of “fractal dimension” involved), with the following important caveat. When $\xi>2/d_{2}$, we have $Q\in(0,2)$ and the right side of the formula (3.1) is non-real when $\Delta_{0}>Q^{2}/2$. The extension of the KPZ formula to the supercritical case coincides with (3.1) when $\Delta_{0}<Q^{2}/2$, and gives $\Delta_{h}=\infty$ when $\Delta_{0}>Q^{2}/2$ (the case when $\Delta_{0}=Q^{2}/2$ is not treated).

Suppose that $E$ is an event depending on the LQG metric $D_{h}$. For example, maybe we have two points $z,w\in\mathbbm{C}$ and $E$ is the event that $D_{h}(z,w)>100$, or that the $D_{h}$-geodesic from $z$ to $w$ stays in some specified open set. For many choices of $E$, it is straightforward to show that $\mathbbm{P}[E]>0$ via the following method. Let $\phi$ be a deterministic smooth, compactly supported function. It is easy to see from basic properties of the GFF that the laws of $h$ and $h+\phi$ are mutually absolutely continuous. See, e.g., [MS16, Proposition 3.4] for a proof. Using Weyl scaling (Axiom 2.8), we can choose $\phi$ so that with high probability, the event $E$ occurs with $h+\phi$ in place of $h$. The absolute continuity of the laws of $h+\phi$ and $h$ then implies that $\mathbbm{P}[E]>0$. Let us illustrate this idea by showing that an LQG geodesic stays in a specified open set with positive probability.

In a similar vein, it is sometimes useful to add a random bump function to $h$ in order to show that $D_{h}$ has certain “typical” behavior with probability 1. To be more precise, again let $\phi$ be a smooth compactly supported bump function and let $X$ be a random variable which is uniform on $[0,1]$, sampled independently from $h$. Then the laws of $h$ and $h+X\phi$ are mutually absolutely continuous. So, if $E$ is an event depending on $D_{h}$, then to show that $\mathbbm{P}[E]=0$ it suffices to show that the probability that $E$ occurs with $h+X\phi$ in place of $h$ is zero. To show this latter statement, it suffices to show that a.s. the Lebesgue measure of the set of $x\in[0,1]$ such that $E$ occurs with $h+x\phi$ in place of $h$ is zero. Usually, it is possible to show that this set consists of at most a single point. Let us illustrate this technique by proving the uniqueness of $D_{h}$-geodesics between typical points.