RRANDOM SURFACES AND LIOUVILLE QUANTUM GRAVITY
EWAIN GWYNNE
Abstract.
Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces whichcan be thought of as the canonical models of random two-dimensional Riemannian manifolds, in thesame sense that Brownian motion is the canonical model of a random path. LQG surfaces are thecontinuum 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 mapsconverge to LQG, and the motivations for studying these objects. We also mention several openproblems. We do not assume any background knowledge beyond that of a second-year mathematicsgraduate student.
What is the most natural way of choosing a random surface (two-dimensional Riemannian man-ifold)? If we are given a finite set X , the easiest way to choose a random element of X is uniformly,i.e., by assigning equal probability to each element of X . More generally, if we are given a set X ⊂ R n with finite, positive Lebesgue measure, the simplest way of choosing a random element of X is by sampling from Lebesgue measure normalized to have total mass one. However, the space ofall surfaces is infinite dimensional for any reasonable notion of dimension, so it is not immediatelyobvious whether there is a canonical way of choosing a random surface.Nevertheless, there is a class of canonical models of “random surfaces” called Liouville quantumgravity (LQG) surfaces . The reason for the quotations is that LQG surfaces are not Riemannianmanifolds in the literal sense since they are too singular to admit a smooth structure. Instead,LQG surfaces are defined as random topological surfaces equipped with a measure, a metric, anda conformal structure. These surfaces are fractal in the sense that the Hausdorff dimension of anLQG surface, viewed as a metric space, is strictly bigger than 2.LQG surfaces have a rich geometric structure which is still not fully understood (see Section 4).Furthermore, such surfaces are important in statistical mechanics, string theory, and conformalfield theory and have deep connections to other mathematical objects such as Schramm-Loewnerevolution [Sch00] (see, e.g., [She16a]), random matrix theory (see, e.g., [Web15]), and randomplanar maps. 1.
Discrete random surfaces
Before we discuss random surfaces, let us first, by way of analogy, consider the simpler problem offinding a canonical way to choose a random curve in the plane. As in the case of surfaces, the spaceof all planar curves is infinite-dimensional and does not admit a canonical probability measure inan obvious way. To get around this, we discretize the problem. Let us consider for each n ∈ N the set of all nearest-neighbor paths in the integer lattice Z with n steps. This is a finite set (ofcardinality 4 n ), so we can choose a uniformly random element S n from it. This discrete randompath S n is called the simple random walk .By linearly interpolating, we may view S n as a curve from [0 , n ] to R . There is a classical theoremin probability due to Donsker which states that the re-scaled random walk paths t (cid:55)→ n − / S n ( nt )converge in distribution (with respect to the uniform topology) to a limiting random continuouscurve called Brownian motion , which can be thought of as the canonical random planar curve.Brownian motion has a much richer structure than an ordinary smooth curve. A Brownian motioncurve is nowhere differentiable and crosses itself in every interval of time. It has zero Lebesguemeasure, but Hausdorff dimension 2. a r X i v : . [ m a t h . P R ] S e p EWAIN GWYNNE
Spanningtree
Figure 1.
Left:
A planar map.
Right:
A planar map decorated by a spanning tree.One can define a canonical random surface via a similar approach. Let us first define thediscrete random surfaces which we will consider. A planar map is a graph (multiple edges andself-loops allowed) embedded into the plane C in such a way that no two edges cross, viewedmodulo orientation-preserving homeomorphisms C → C . Planar maps have vertices, edges, andfaces, but these objects do not correspond to particular points / sets in C since we do not specifya particular embedding. See Figure 1 for an illustration of a planar map.We can think of a planar map as a discrete random surface, where each face is equipped with theflat Riemannian metric coming from a polygon in C with the appropriate number of edges and unitside length. Equivalently, a planar map is obtained by starting with a collection of polygons withunit side length and identifying pairs of their sides (in such a way that the identification betweenany two edges is a Euclidean isometry) to produce a surface, subject to the constraint that thereare no holes or handles. The surface obtained in this way always has the topology of the sphere,but we can similarly obtain random surfaces with other topologies (we are primarily interested inthe local geometry of random surfaces rather than their topology).For each n ∈ N , there are only finitely many planar maps with n edges. Hence it makes senseto choose such a map uniformly at random. One can also consider uniform planar maps with localconstraints, such as triangulations (resp. quadrangulations), which are required to have three (resp.four) edges on the boundary of each face.It is also natural to consider planar maps weighted by some sort of additional structure on themap. Indeed, suppose, for example, that we are interested in planar maps M decorated by a spanning tree , i.e., a connected subgraph of M which contains every vertex of M and has no cycles.Then we might want sample a uniform random pair ( M, T ) consisting of a planar map with n edgesand a spanning tree on it. In this case, the marginal distribution of M is not uniform: rather,the probability that M is equal to any fixed planar map M with n edges is proportional to thenumber of possible spanning trees of M . In a similar vein, one might want to look at planar mapssampled with probability proportional to the number of certain types of orientations on the edgesof M , or the partition function of a statistical mechanics model on M (like the Ising model or theFortuin-Kasteleyn model).Since planar maps can be thought of as discrete surfaces, it is natural to expect that randomplanar maps converge, in some sense, to limiting random surfaces as the total number of edges tendsto ∞ . In other words, if we sample a large random planar map and “zoom out” so that we see onlyits large-scale structure, we should get something which looks like some sort of (continuum) randomsurface. For a certain class of random planar maps, including uniform planar maps and planar mapssampled with probability proportional to the partition function of a statistical mechanics model There is some ambiguity when counting pairs (
M, T ) since there can be automorphisms M → M which do notfix T . In practice, this ambiguity is removed by specifying a distinguished edge e of M : there are no non-trivialautomorphisms of M which fix e . ANDOM SURFACES AND LIOUVILLE QUANTUM GRAVITY 3 with parameters set to the critical values, the particular types of surfaces which arise in this wayare LQG surfaces, as discussed above and defined precisely in Section 2. We will discuss the precisesense in which this convergence occurs in Section 3.2.
Liouville quantum gravity
Isothermal coordinates.
To define LQG surfaces, we first recall some facts from the theoryof deterministic surfaces. Suppose S is a continuously differentiable surface, i.e., in local coordinates S can be represented by E dx + F dx dy + G dy for some continuously differentiable functions E, F, G of the parameters ( x, y ). We will be primarily interested in the local geometry of S , soby possibly replacing S by an open subset of S , we can assume that S is homeomorphic to theopen unit disk D ⊂ R . A standard theorem in Riemannian geometry (see, e.g., [Che55]) assertsthat, at least locally, S can be parametrized by isothermal coordinates . This means that we canparametrize S by coordinates ( x, y ) in D in such a way that the Riemannian metric tensor takesthe form e h ( z ) ( dx + dy ) for some continuously differentiable function h : D → R . Here, z = x + iy and dx + dy is the Euclidean Riemannian metric tensor.It is easy to describe areas and distances with respect to isothermal coordinates. For a Lebesguemeasurable set A ⊂ D , the area of the corresponding subset of the surface S is given by(2.1) (cid:90) A e h ( z ) d z, where d z = dx dy is two-dimensional Lebesgue measure. The S -distance between any two pointsin z, w ∈ D is given by(2.2) inf P : z → w (cid:90) ba e h ( P ( t )) / | P (cid:48) ( t ) | dt, where the inf is over all piecewise continuously differentiable paths P : [ a, b ] → D from z to w .2.2. The Gaussian free field.
We want to define an LQG surface as a random surface parametrizedby isothermal coordinates by making a random choice of h . Since LQG surfaces should describe thelarge-scale behavior of random planar maps, by analogy with the central limit theorem a naturalfirst guess is that h should be a “standard Gaussian random variable” taking values in the spaceof differentiable functions on D . To explain what this means, suppose that we are given a finite-dimensional Hilbert space H and let { x , . . . , x n } be an orthonormal basis for H . We can define a standard Gaussian random variable on H by(2.3) n (cid:88) j =1 α j x j where the α j ’s are i.i.d. standard Gaussian random variables on R , i.e., they are sampled from theprobability measure with density √ π e − x / .For an open domain U ⊂ C , consider the infinite-dimensional Hilbert space H ( U ) which is theHilbert space completion of the space of smooth, compactly supported functions on D with respectto the Dirichlet inner product (2.4) ( f, g ) ∇ = 12 π (cid:90) U ∇ f ( z ) · ∇ g ( z ) d z, where ∇ denotes the gradient and · denotes the dot product. This is sometimes called the firstorder Sobolev space on U , with zero boundary conditions. EWAIN GWYNNE
The
Gaussian free field (GFF) is the standard Gaussian random variable on H ( U ). That is, let { f j } j ∈ N be an orthonormal basis for H ( U ). By analogy with (2.3), we define the GFF h by(2.5) h := n (cid:88) j =1 α j f j where the α j ’s are i.i.d. standard Gaussian random variables. The sum (2.5) a.s. does not convergepointwise, so the GFF does not have well-defined pointwise values.However, it is not hard to show that the GFF makes sense as a random distribution (generalizedfunction) on H ( U ) [She07]. This means that for any f ∈ H ( U ) the Dirichlet inner product “( h, f ) ∇ ”and the L inner product ( h, φ ) = “ (cid:82) U h ( z ) f ( z ) d z ” are well-defined. The random variables ( h, f ) ∇ and ( h, g ) ∇ for f, g ∈ H ( U ) are jointly centered Gaussian with covariances Cov(( h, f ) ∇ , ( h, g ) ∇ ) =( f, g ) ∇ .Although we will primarily be interested in the two-dimensional case, we remark that the aboveconstruction of the GFF also makes sense in other dimensions. In dimension 1, one gets a Brownianbridge (a one-dimensional Brownian motion defined on an interval and conditioned to be zero atthe endpoints). Hence the GFF can be seen as a generalization of Brownian motion with two timedimensions. In dimension at least three, the GFF is in some sense rougher, i.e., further from beinga function, than in dimension 2 and hence regularization procedures such as the ones describedbelow do not give interesting objects. Rather, it is more natural to consider so-called log-correlated Gaussian fields [DRSV14a] (the GFF is log-correlated only in dimension 2).2.3.
Liouville quantum gravity surfaces.
Let h be the GFF on a domain U ⊂ C and let γ ∈ (0 , γ -Liouvillequantum gravity (LQG) surface associated with ( U, h ) to be the random surface parametrized by U with Riemannian metric tensor “ e γh ( dx + dy )”. This definition does not make literal sensesince h is not a function, so it cannot be exponentiated. However, one can define LQG surfacesrigorously via regularization procedures which we will describe shortly.The parameter γ controls the “roughness” of the surface: γ = 0 corresponds to a smooth surface,and larger values of γ correspond to surfaces which are more fractal and less Euclidean-like. Theparameter γ is also related to the type of random planar map model under consideration. The casewhen γ = (cid:112) / γ (“gravity coupled to matter”) correspond to random planar mapssampled with probability proportional to some sort of additional structure on the map, such as thenumber of spanning trees ( γ = √
2) or the Ising model partition function ( γ = √ Certain special LQG surfaces are the topic of study in Liouvilleconformal field theory, the simplest non-rational conformal field theory. See, e.g., [DKRV16,KRV17]and the references therein for rigorous works on LQG from the conformal field theory perspective.We now explain how to make rigorous sense of LQG surfaces as random metric measure spaces.The basic idea is to consider a family of continuous functions { h ε } ε> which approximate the GFFas ε →
0, define approximate notions of area and distance by replacing h with a multiple of h ε in (2.1) and (2.2), then send ε → Roughly speaking, for c ∈ N an evolving string in R c − traces out a two-dimensional surface embedded in space-time R c − × R , called a world sheet . Polyakov wanted to develop a theory of integrals over all possible surfacesembedded in R c as a string-theoretic generalization of the Feynman path integral (which is an integral over allpossible paths). To do this one needs to define a probability measure on surfaces. The most natural way of doingthis turns out to only work when the “dimension of the space into which the surface is embedded” (a.k.a. the centralcharge) c lies in ( −∞ , γ -LQG with c = 25 − /γ + γ/ . ANDOM SURFACES AND LIOUVILLE QUANTUM GRAVITY 5 associated with γ -LQG. One possible choice of h ε is the convolution with the heat kernel,(2.6) h ε ( z ) = (cid:90) U h ( z ) p ε ( z, w ) d w for p ε ( z, w ) := 1 πε e −| z − w | /ε where the integral is interpreted as the distributional pairing of h with p ε ( z, · ). Note that p ε ( z, w ) d w approximates a point mass at z as ε →
0, so h ε is close to h (e.g., in the distributional sense) when ε is small. Other possible choices for h ε include convolutions with other mollifiers, averages overcircles, truncated versions of the orthonormal basis expansion (2.5), etc. γ = 0 . γ = 1 . γ = 1 . Figure 2.
Simulations of the γ -LQG measure on the unit square produced by J.Miller. The square is divided into dyadic sub-squares which all have approximatelythe same γ -LQG mass. Squares are colored according to their Euclidean size. Notethat as γ increases, the Euclidean sizes of these squares become more variable.2.4. The γ -LQG area measure. For several possible choices of { h ε } ε> , one can define the γ -LQG area measure as the a.s. limit(2.7) µ h = lim ε → ε γ / e γh ε ( z ) d z, with respect to the weak topology on measures on U , where d z denotes Lebesgue measure asabove. Indeed, this construction is a special case of a more general theory of regularized randommeasures called Gaussian multiplicative chaos , which was initiated in the work of Kahane [Kah85];see [RV14, Ber17, Aru17] for surveys of this theory. Several important properties of the measure(including the convergence of the circle average approximation and the so-called
KPZ formula )were estalbished by Duplantier and Sheffield [DS11]. The γ -LQG measure has no point massesand assigns positive mass to every open subset of U , but it is mutually singular with respect tothe Lebesgue measure. In fact, it assigns full mass to a set of Hausdorff dimension 2 − γ / γ -LQG measure.In the case when γ ≥
2, the limit (2.7) is identically zero, which is why we restrict to γ ∈ (0 , γ = 2 one can construct a measure with similar properties butan additional logarithmic correction is needed in the scaling factor; see [DRSV14b, DRSV14c]. For γ >
2, one can make sense of the γ -LQG measure as a purely atomic measure (i.e., a countable sumof point masses) which is closely related to γ (cid:48) -LQG for γ (cid:48) = 4 /γ ∈ (0 ,
2) [Dup10, BJRV13, RV14,DMS14].2.5.
The γ -LQG metric. The γ -LQG metric D h can be constructed in an analogous way to (2.7),but the proof that the approximations converge is much more involved than in the case of themeasure. Intuitively, the reason for this is that if we replace h by a multiple of h ε in (2.2), thenthe near-minimal paths could in principle be very different for different values of ε (although onegets a posteriori that this is not the case). EWAIN GWYNNE
Before describing the construction of the metric we need to introduce an exponent d γ > γ -universality class, a graph distance ball of radius r ∈ N in the maptypically has of order r d γ vertices. Once the γ -LQG metric D h is constructed, it is possible to showthat d γ is the Hausdorff dimension of the metric space ( U, D h ) [GP19c].It can be shown using special symmetries for uniform planar maps or (cid:112) / d √ / = 4.However, d γ is not known (even at a heuristic level) for γ ∈ (0 , \ { (cid:112) / } ; determining its valueis one of the most important open problems in the theory of LQG. See Section 4 for more on d γ .To approximate LQG distances, we let D εh ( z, w ) for z, w ∈ U and ε > γ/d γ ) h ε in place of h , i.e.,(2.8) D εh ( z, w ) = inf P : z → w (cid:90) ba e γdγ h ε ( P ( t )) | P (cid:48) ( t ) | dt. The reason why we have γ/d γ instead of γ/ d γ is the dimension of the γ -LQG surface, scaling γ -LQG areas by C > γ -LQG distances by C /d γ . By (2.7), scaling areas by C corresponds to adding the constant γ log C to h . By (2.8), thisscales D εh by C /d γ , as desired.It was shown by Ding-Dub´edat-Dunlap-Falconet [DDDF19] that there are constants { a ε } ε> such that the re-scaled metrics a − ε D εh are tight with respect to the local uniform topology on U ,and every subsequential limit is bi-H¨older continuous with respect to the Euclidean metric on U .Building on this and [GM19d, DFG +
19, GM19a], Gwynne and Miller [GM19c] showed that in fact a − ε D εh converges in probability (not just subsequentially) to a limiting metric D h which is definedto be the γ -LQG metric. They also proved an axiomatic characterization of D h which implies thatit is the only possible metric associated with γ -LQG.The metric D h induces the same topology on U as the Euclidean metric, but the Hausdorffdimension of the metric space ( U, D h ) is d γ >
2. Moreover, many of its geometric properties (e.g.,scaling properties and the behavior of geodesics) are quite different from those of the Euclideanmetric or indeed any smooth Riemannian metric on U . See Figure 3 for simulations of LQG metricballs.There is also an earlier construction of the LQG metric in the special case when γ = (cid:112) / (cid:112) / γ (cid:54) = (cid:112) /
3, but it gives additional information about (cid:112) / Conformal coordinate change.
Just like for deterministic surfaces, it is possible to pa-rametrize LQG surfaces in different ways. Suppose φ : (cid:101) U → U is a conformal (i.e., bijective andholomorphic) map. Let h be the GFF on U and let(2.9) (cid:101) h := h + Q log | φ (cid:48) | where Q := 2 γ + γ . Then (cid:101) h is a random distribution on (cid:101) U . It is shown in [DS11,GM19b] that the γ -LQG area measuresand metrics associated with h and (cid:101) h are a.s. related by µ h ( φ ( A )) = µ (cid:101) h ( A ), for each Borel measurableset A ⊂ (cid:101) U and D h ( φ ( z ) , φ ( w )) = D (cid:101) h ( z, w ) for each z, w ∈ (cid:101) U . We think of ( U, h ) and ( (cid:101)
U , (cid:101) h ) asrepresenting different parametrizations of the same LQG surface. The coordinate change relation ANDOM SURFACES AND LIOUVILLE QUANTUM GRAVITY 7 γ = 0 . γ = 1 . γ = 1 . Figure 3.
Simulations of γ -LQG metric balls w.r.t. the same GFF instance,produced by J. Miller. The colors indicate distances to the center of the ball.Geodesics from points in a grid back to the center point are shown in black. Note thatthese geodesics have a tree-like structure: unlike geodesics for a smooth Riemannianmetric, LQG geodesics with different starting points and targeted at 0 merge intoone another before reaching 0. This was proven to be the case in [GM19a].for µ h and D h shows that these objects depend only on the LQG surface, not on the choice ofparametrization. 3. LQG as the limit of random planar maps
We now discuss the senses in which random planar maps should converge to LQG surfaces, andthe extent to which each type of convergence has been proven.
Figure 4.
Left:
Simulation of a large uniform quadrangulation embedded into R in such a way that the embedding is in some sense as close as possible to being anisometry, made by J. Bettinelli. The Gromov-Hausdorff limit of these triangulationsis a (cid:112) / Middle:
Simulation of acircle packing of a uniform triangulation made by J. Miller.
Right:
Simulation ofthe Tutte embedding of an instance of the √ √ EWAIN GWYNNE
Gromov-Hausdorff convergence.
Let K be the set of all compact metric spaces. The Gromov-Hausdorff (GH) distance on K is the metric on K defined by(3.1) d GH (( X , d ) , ( X , d )) := inf ( Y,D ) ,ι ,ι d H ( ι ( X ) , ι ( X ))where d H denotes the Hausdorff distance on compact subsets of Y and the infimum is over allcompact metric spaces ( Y, D ) and isometric embeddings ι : ( X , d ) → ( Y, D ) and ι : ( X , d ) → ( Y, D ). A planar map can be viewed as a compact metric space equipped with its graph distance(see Figure 4, left). One can then ask whether large random planar maps, with their graph distancere-scaled appropriately, converge in distribution to γ -LQG surfaces w.r.t. the GH topology. We canadditionally equip the map with its re-scaled counting measure and ask for convergence in the Gromov-Hausdorff-Prokhorov (GHP) topology , the analog of the GH topology for metric measurespaces.So far, GH or GHP convergence has only been established for uniform planar maps (includinguniform maps with local constraints), which we recall correspond to γ = (cid:112) /
3. The first suchconvergence results were obtained independently by Le Gall [Le 13] and Miermont [Mie13]. Theyshowed that certain types of uniform planar maps (with graph distances re-scaled by n − / andthe counting measure on vertices scaled by 1 /n ) converge in the GHP sense to the Brownianmap , a random metric measure space which can be constructed from a continuum random treevia an explicit metric quotient procedure. Miller and Sheffield [MS15, MS16a, MS16b] showedthat a certain special (cid:112) / quantum sphere , has the same distributionas the Brownian map viewed as a metric measure space modulo measure-preserving isometries.Hence uniform planar maps converge to (cid:112) / (cid:112) / d √ / = 4). So, the convergence proofs do not extendto non-uniform maps and γ (cid:54) = (cid:112) / Embedding convergence.
Although a planar map is defined only modulo orientation-preservinghomeomorphisms of C , there are various ways of embedding the map into C , i.e., associating eachvertex (resp. edge) with a point (resp. curve) in C in such a way that no two edges cross. Examplesinclude circle packing , where the planar map is realized as the tangency graph of a collection ofcircles in the plane (Figure 4, middle); and Tutte embedding (a.k.a. harmonic embedding or baycen-tric embedding), which is defined by the condition that the position of each vertex is the average(barycenter) of the positions of its neighbors (Figure 4, right). Once we have chosen an embedding,we can ask, e.g., whether the counting measure on the vertices of the planar map, re-scaled by thetotal number of vertices, converges in distribution (w.r.t. the weak topology on C ) to a variant ofthe γ -LQG area measure.So far, there are three results establishing this type of convergence for random planar maps.The first result [GMS17] establishes embedding convergence for a one-parameter family of randomplanar maps called mated-CRT maps , one for each γ ∈ (0 , (cid:112) / Cardy embedding , which is defined using crossing probabilities for per-colation on the map [HS19]; and for the Poisson-Voronoi approximation of the Brownian map underthe Tutte embedding [GMS18]. In the case of uniform triangulations, it is in fact shown in [HS19](see also [GHS19b]) that one has convergence in the GHP sense (as discussed in Section 3.1) andthe mating-of-trees sense (as discussed in Section 3.3 below) simultaneously with the convergence
ANDOM SURFACES AND LIOUVILLE QUANTUM GRAVITY 9 of the embedded map. It is a major open problem to establish these types of convergence resultsfor other random planar map models and/or other embeddings (see Problem 2).3.3.
Mating-of-trees convergence.
There are several combinatorial bijections which encode arandom planar map decorated by some additional structure by means of a random walk on Z . Thesimplest example of such a bijection is the Mullin bijection [Mul67,Ber07b], which encodes a planarmap decorated by a spanning tree by a nearest-neighbor random walk on Z . There are other suchbijections, with different walks, that encode planar maps decorated by e.g., percolation [Ber07a,BHS18], bipolar orientations [KMSW19], or the Fortuin-Kasteleyn model [She16b]. These bijectionsare called mating-of-trees bijections since the planar map is constructed from the walk by gluingtogether, or mating, the discrete random trees associated with the two coordinates of the walk. Inthe case of the Mullin bijection, the two trees are the spanning tree on the map and its correspondingdual spanning tree.For random planar maps which are related to γ -LQG, it can be shown that the encoding walkconverges in distribution to a two-dimensional Brownian motion Z = ( L, R ) with the correlationbetween the coordinates L t and R t given by − cos( πγ /
4) for each time t . A fundamental theoremof Duplantier, Miller, and Sheffield [DMS14] shows that one can construct a γ -LQG surface fromthis two-dimensional Brownian motion Z via a continuum analog of a mating-of-trees bijection.Hence the convergence of the encoding walks in the discrete mating-of-trees bijections toward Z can be viewed as a convergence statement for random planar maps toward LQG surfaces in acertain topology: the one where two surfaces are close if their encoding paths are close. This typeof convergence is referred to as mating-of-trees or peanosphere convergence.Mating-of-trees convergence is not strictly weaker than Gromov-Hausdorff convergence or embed-ding convergence, but it is arguably less natural than these other modes of convergence. However,mating-of-trees convergence can still be used to extract a substantial amount of useful informationabout the random planar map. This includes scaling limits for various functionals of the map andthe computation of exponents related to partition functions, graph distances, random curves onthe map, etc. Moreover, for many types of random planar maps mating-of-trees convergence isthe only scaling limit result available, and it can sometimes be used as an intermediate step inproving one of the other types of convergence (as is done in [HS19]). See [GHS19a] for a survey ofmating-of-trees theory and its applications.4. Open problems
Here, we discuss some of the most important open problems in the theory of Liouville quantumgravity. Additional open problems can be found, e.g., in [GM19c, Section 7]. Our first problemwas alluded to in Section 2.3.
Problem 1.
What is the Hausdorff dimension d γ of a γ -LQG surface, viewed as a metric space,for γ ∈ (0 , \ { (cid:112) / } ?Recall that the Hausdorff dimension of (cid:112) / d γ for general γ ∈ (0 , d γ , dueto Watabiki [Wat93], was proven to be incorrect, at least for small values of γ , by Ding andGoswami [DG16]. However, it is known that γ (cid:55)→ d γ is strictly increasing [DG18] and there arereasonably sharp upper and lower bounds for d γ [DG18, GP19a, Ang19]. For example, one has3 . ≤ d √ ≤ . d γ . Manyquantities associated with random planar maps and LQG can be expressed in terms of d γ (see,e.g., [DG18, GHS17, DFG +
19, GP19c, GM17, GH18, GP19b]), so computing d γ would yield manyadditional results. Problem 2.
Show that weighted random planar map models of the type discussed in Section 1converge in distribution to γ -LQG surfaces with γ ∈ (0 , \ { (cid:112) / } in both the Gromov-Hausdorffsense and under suitable embeddings into C .As noted in Section 3, both types of convergence have already been established for certain typesof uniform planar maps toward (cid:112) / n -dimensional manifolds for any n ∈ N ; see [RV14,Ber17]. However, the associated metrichas only been constructed in dimension 2. Problem 3.
Is it possible to construct random metrics associated with log-correlated Gaussianfields on R n , or on n -dimensional manifolds, for n ≥ Additional expository references
We mention a few more expository references in addition to the papers cited above. See [Ber]for introductory lecture notes on the GFF and LQG which go into substantially more detail thanthis article. See [Le 14] for a survey on the geometry of random planar maps and the Brownianmap and [Cur16] for lecture notes on random planar maps, emphasizing the applications of thespatial Markov property. See [Var17] for lecture notes on the conformal field theory (path integral)approach to LQG. Although not emphasized in this article, Liouville quantum gravity is closelyrelated to
Schramm-Loewner evolution (SLE), a family of random fractal curves introduced bySchramm [Sch00] (see, e.g., [She16a, DMS14] for relationships between SLE and LQG). For anintroduction to SLE see the lecture notes [Wer04, BN] and the textbook [Law05].
Acknowledgements.
We thank two anonymous referees and also Nina Holden, Jason Miller, andScott Sheffield for helpful comments. We thank J´er´emie Bettinelli and Jason Miller for allowingus to use their beautiful simulations in this article. The author was supported by a Clay ResearchFellowship and a Junior Research Fellowship at Trinity College, Cambridge.
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