Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces which can be thought of as the canonical models of random two-dimensional Riemannian manifolds, in the same sense that Brownian motion is the canonical model of a random path. LQG surfaces are the continuum limits of discrete random surfaces called random planar maps. In this expository article, we discuss the definition of random planar maps and LQG, the sense in which random planar maps converge to LQG, and the motivations for studying these objects. We also mention several open problems. We do not assume any background knowledge beyond that of a second-year mathematics graduate student.
We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path. In the special case that $γ=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable Lévy process (with only upward jumps) to produce a pair of t…
This paper is concerned with computing the spectral dimension of 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as announced by physicists (see Ambj\orn and al. in \cite{amb}) fifteen years ago…
We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.
We prove that for any metric which one can associate with a Liouville quantum gravity (LQG) surface for $γ\in (0,2)$ satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point $z$, a.s.\ any two $γ$-LQG geodesics started from distinct points other than $z$ must merge into each other and subsequently coincide until they reach $z$. This is analogous to the confluence of geodesics property for the Brownian map proven by Le Gall (2010). Our results apply for the subsequential limits of Liouville first passage percolation and are an important inp…
We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLE$_κ$. These hypotheses are satisfied by the LQG metric for $γ=\sqrt{8/3}$ constructed by the first author and Sheffield, and subsequent work by Gwynne and the first author has shown that there is a unique metric which satisfies these hypotheses for each $γ\in (0,2)$. As a consequence of our analysis, we also establish certain regularity properties of LQG geodesics which imply, among ot…
Consider two critical Liouville quantum gravity surfaces (i.e., $γ$-LQG for $γ=2$), each with the topology of $\mathbb{H}$ and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent SLE$_4$. Combined with the proof of uniqueness for such a welding, recently established by McEnteggart, Miller, and Qian (2018), this shows that the welding operation is well-defined. Our result is a critical analogue of Sheffiel…
The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, these notes contain a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. These notes are rather short and require no prior knowledge on the topic.
We show that every possible metric associated with critical ($γ=2$) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric. More precisely, we show that the optimal modulus of continuity of the critical LQG metric with respect to the Euclidean metric is a power of $1/\log(1/|\cdot|)$. Our result applies to every possible subsequential limit of critical Liouville first passage percolation, a natural approximation scheme for the LQG metric which was recently shown to be tight.
In this paper, we rigorously construct $2d$ Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov "Quantum Geometry of bosonic strings". We also establish some of its fundamental properties like conformal covariance under PSL$_2(\mathbb{C})$-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quantum Gravity.
In this work we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists initiated by Polyakov in 1981. It is also a very important example of a conformal field theory (CFT). Results have already been obtained on the Riemann sphere and on the unit disk so this paper will follow the same approach. The case of the annulus contains two difficulties: it is a surface with two boundaries and it has a non-trivial moduli space. We recover the Weyl anomaly - a formula verified …
We prove that for each $γ\in (0,2)$, there is an exponent $d_γ> 2$, the "fractal dimension of $γ$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $γ$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $γ$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $d_γ$ is a continuous, strictly increasing function of $γ$ and prove upper and lower bounds for $d_γ$ which in some cases greatly improve on previ…
We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ …
In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov. Our approach follows the construction carried out by the authors together with A. Kupiainen in the case of the Riemann sphere. The difference is here that the moduli space for complex tori is non trivial. Modular properties of LQFT are thus investigated. This allows us to sum up the LQFT on complex tori over the moduli space, to compute the law of the random Liouville modulus, therefore recovering (and extending) formulae obtained by physicists, and mak…
For $γ\in (0,2)$, the quantum disk and $γ$-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling SLE$_{16/γ^2}$ curve on a quantum disk or on a $γ$-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation $-\cos(πγ^2/4)$. This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case…
The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In the current article we confirm the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is stirctly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter $ξ$. We deduce this from a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQ…
In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.
Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk.
Liouville quantum gravity (LQG) is a one-parameter family of models of random fractal surfaces which first appeared in the physics literature in the 1980s. Recent works have constructed a metric (distance function) on an LQG surface. We give an overview of the construction of this metric and discuss some of its most important properties, such as the behavior of geodesics and the KPZ formula. We also discuss some of the main techniques for proving statements about the LQG metric, give examples of their use, and discuss some open problems.
Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the family of random distance functions $\{D_h^ε\}_{ε>0}$ on the plane obtained by integrating $e^{ξh_ε}$ along paths, where $\{h_ε\}_{ε>0}$ is a smooth mollification of the planar Gaussian free field. Recent works have shown that for all $ξ> 0$ the LFPP metrics, appropriately re-scaled, admit non-trivial subsequential limiting metrics. In the case when $ξ< ξ_{\mathrm{crit}} \approx 0.41$, it has been shown that the subsequential limit is unique and defines a metric on $γ$-Liouville quantum gravity $γ= γ(ξ) \in (0,2)$. We prov…
Previous works in this series have shown that an instance of a $\sqrt{8/3}$-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $\sqrt{8/3…
We numerically compute the heat kernel on a square lattice torus equipped with the measure corresponding to Liouville quantum gravity (LQG). From the on-diagonal heat kernel we verify that the spectral dimension of LQG is 2. Furthermore, when diffusion is started from a high point of the underlying Gaussian free field, our numerics indicates superdiffusive space-time scaling with respect to the Euclidean metric in the small space-to-time regime. The implications of this result require further investigation, but seem to coincide with the notion that the Euclidean metric is not the right geodesi…
For $γ\in (0,2)$, $U\subset \mathbb C$, 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 (distance function) $D_h$ 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 $U,\widetilde{U}$ are domains, $ϕ\colon U \to \widetild…
Recent work has shown that for $γ\in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In particular, we completely classify the possible networks of geodesics from a typical point on the surface to an arbitrary point on the surface, as well as the types of networks of geodesics joining two points which occur for a dense set of pairs of points on the surface. This latter result is the $γ$-LQG analog of the classification of geodesic networks in the Brownian map due to Angel, Kolesnik, and Mie…
Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2π)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures ε^{γ^2/2} e^{γh_ε(z)}dz, where dz is Lebesgue measure on D and h_ε(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these t…
Let $Q$ be a free Boltzmann quadrangulation with simple boundary decorated by a critical ($p=3/4$) face percolation configuration. We prove that the chordal percolation exploration path on $Q$ between two marked boundary edges converges in the scaling limit to chordal SLE$_6$ on an independent $\sqrt{8/3}$-Liouville quantum gravity disk (equivalently, a Brownian disk). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. We also obtain analogous scaling limit results for fa…
Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $γ$, and it has long been believed that when $γ= \sqrt{8/3}$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other's structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with th…
We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain $D$ of the complex plane for which the Riemannian metric tensor at a point $z \in D$ is given by $\exp (γh(z) - \frac12 γ^2 \E (h(z)^2))$. Here $h$ is an instance of the Gaussian Free Field on $D$ and $γ\in (0,2)$ is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian Free Fie…
We prove that SLE$_κ$ for $κ\in (4,8)$ on an independent $γ=4/\sqrtκ$-Liouville quantum gravity (LQG) surface is uniquely characterized by the form of its LQG boundary length process and the form of the conditional law of the unexplored quantum surface given the explored curve-decorated quantum surface up to each time $t$. We prove variants of this characterization for both whole-plane space-filling SLE$_κ$ on an infinite-volume LQG surface and for chordal SLE$_κ$ on a finite-volume LQG surface with boundary. Using the equivalence of Brownian and $\sqrt{8/3}$-LQG surfaces, we deduce that SLE$_…
We prove that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorate…
In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone, and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane, and disk). We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane…
We consider the metric growth in Liouville quantum gravity (LQG) for $γ\in (0,2)$. We show that a process associated with the trace of the free field on the boundary of a filled LQG ball is stationary, for every $γ\in (0,2)$. The infinitesimal version of this stationarity combined with an explicit expression of the generator of the evolution of the trace field $(h_t)$ provides a formal invariance equation that a measure on trace fields must satisfy. When considering a modified process corresponding to an evolution of LQG surfaces, we prove that the invariance equation is satisfied by an explic…
Liouville Quantum Field Theory can be seen as a probabilistic theory of 2d Riemannian metrics $e^{ϕ(z)}dz^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $ϕ$ in LQFT depends on weights $α\in \mathbb{R}$ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in \cite{DKRV} in the case when the weights are below the so called Seiberg bound: $α<Q$ where $Q$ parametrizes the random surface model in question. These correspond to conical singularities in th…
We study the volume of metric balls in Liouville quantum gravity (LQG). For $γ\in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \in (-\infty, 4/γ^2)$. Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $γ= \sqrt{8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d…
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $γ\in (0,2)$. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a form…
We show that for each $γ\in (0,2)$, there is a unique metric (i.e., distance function) associated with $γ$-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) $h$, there is a unique random metric $D_h$ associated with the Riemannian metric tensor "$e^{γh} (dx^2 + dy^2)$" on $\mathbb C$ which is characterized by a certain list of axioms: it is locally determined by $h$ and it transforms appropriately when either adding a continuous function to $h$ or applying a conformal automorphism of $\mathbb C$ (i.e., a complex affine transformation). …
We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between …
We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin--Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield's hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including…
We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $γ=\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with …
We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a h…
Let $γ\in (0,2)$, let $h$ be the planar Gaussian free field, and consider the $γ$-Liouville quantum gravity (LQG) metric associated with $h$. We show that the essential supremum of the Hausdorff dimension of the boundary of a $γ$-LQG metric ball with respect to the Euclidean (resp. $γ$-LQG) metric is $2 - \fracγ{d_γ}\left(\frac{2}γ + \fracγ{2} \right) + \frac{γ^2}{2d_γ^2}$ (resp. $d_γ-1$), where $d_γ$ is the Hausdorff dimension of the whole plane with respect to the $γ$-LQG metric. For $γ= \sqrt{8/3}$, in which case $d_{\sqrt{8/3}}=4$, we get that the essential supremum of Euclidean (resp. $\s…
We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{γX(z)}\,dz^2$, $γ<γ_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_t$ depending on the local behavior of the Liouville measure "$M_γ(dz)=e^{γX(…
Let $γ\in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $γ$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \subset \mathbb{C}$ which is independent from $h$, the Hausdorff dimensions of $X$ with respect to the Euclidean metric and with respect to the $γ$-LQG metric $D_h$ are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum $γ$-LQG metric is equal to the exponent $d_γ> 2$ studied by Ding and Gwynne (2018), which describes distances in discrete approximation…
