Harmonic functions on mated-CRT maps

A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma$-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos(\pi\gamma^2/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps --- including mated-CRT maps and the UIPT --- the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_n(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.


B Index of notation 52
1 Introduction

Overview
There has been substantial interest in random planar maps in recent years. One reason for this is that random planar maps are the discrete analogs of γ-Liouville quantum gravity (LQG) surfaces for γ ∈ (0, 2). Such surfaces have been studied in the physics literature since the 1980's [Pol81a,Pol81b], and can be rigorously defined as metric measure spaces with a conformal structure [DS11,MS15,MS16b,MS16c,GM19]. The parameter γ depends on the particular type of random planar map model under consideration. For example, γ = 8/3 for uniform random planar maps, γ = √ 2 for spanning-tree weighted maps, and γ = 4/3 for bipolar-oriented maps.
Central problems in the study of random planar maps include describing the large-scale behavior of graph distances; analyzing statistical mechanics models on the map; and understanding the conformal structure of the map, which involves studying the simple random walk on the map and various ways of embedding the map into C. Here we will focus on this last type of question for a particular family of random planar maps called mated-CRT maps, which (as we will discuss more just below) are directly connected to many other random planar map models and to LQG.
To define mated-CRT maps, fix γ ∈ (0, 2) (which corresponds to the LQG parameter) and let (L, R) : R → R 2 be a pair of correlated, two-sided standard linear Brownian motions normalized so that L 0 = R 0 = 0 with correlation − cos(πγ 2 /4), i.e., corr(L t , R t ) = − cos(πγ 2 /4) for each t ∈ R \ {0} and (L t , R t ) can be obtained from a standard planar Brownian motion by applying an appropriate linear transformation. We note that the correlation ranges from −1 to 1 as γ ranges from 0 to 2. The mated CRT map is the random planar map obtained by mating, i.e., gluing together, discretized versions of the continuum random trees (CRT's) constructed from L and R [Ald91a, Ald91b,Ald93]. More precisely, the ε-mated-CRT map 1 associated with (L, R) is the random graph with vertex set εZ, with two vertices x 1 , x 2 ∈ εZ with x 1 < x 2 connected by an edge if and only if either (1.1) or the same holds with R in place of L. If |x 2 − x 1 | > ε and (1.1) holds for both L and R, then x 1 and x 2 are connected by two edges. We note that the law of the planar map G ε does not depend on ε due to Brownian scaling, but for reasons which will become apparent just below it is convenient to think of the whole collection of maps {G ε } ε>0 coupled together with the same Brownian motion (L, R). See Figure 1 for an illustration of the definition of G ε and an explanation of how to endow it with a canonical planar map structure under which it is a triangulation.
εZ L C − R εZ ε 2ε 3ε 4ε 5ε 6ε 7ε 8ε 9ε 10ε 11ε 12ε Figure 1: Left: To construct the mated-CRT map G ε geometrically, one can draw the graph of L (red) and the graph of C − R (blue) for some large constant C > 0 chosen so that the parts of the graphs over some time interval of interest do not intersect. Here, this time interval is [0, 12ε]. One then divides the region between the graphs into vertical strips (boundaries shown in orange). Each vertical strip corresponds to the vertex x ∈ εZ which is the horizontal coordinate of its rightmost points. Vertices x 1 , x 2 ∈ εZ are connected by an edge if and only if the corresponding vertical strips are connected by a horizontal line segment which lies under the graph of L or above the graph of C − R. For each pair of vertices for which the condition holds for L (resp. C − R), we have drawn the lowest (resp. highest) segment for which joins the corresponding vertical strips in green. Equivalently, for each x ∈ εZ, we let t x be the time in [x − ε, x] at which L attains its minimum value and we draw in green the longest horizontal segment under the graph of L which contains (t x , L tx ); and we perform a similar procedure for R. Note that consecutive vertices are always joined by an edge. Right: One can draw the graph G ε in the plane by connecting two vertices x 1 , x 2 ∈ εZ by an arc above (resp. below) the real line if the corresponding vertical strips are connected by a horizontal segment above (resp. below) the graph of L (resp. C − R); and connecting each pair of consecutive vertices of εZ by an edge. This gives G ε a planar map structure. With this planar map structure, each face of G ε corresponds to a horizontal strip below the graph of L or above the graph of C − R which is bounded by two horizontal green segments and two segments of either the graph of L or the graph of C − R. Almost surely, neither L nor R attains a local minimum at any point in εZ and neither L nor R has two local minima where it attains the same value. From this, it follows that a.s. the boundary of each horizontal strip intersects the boundaries of exactly three vertical strips (two of these intersections each consist of a segment of the graph of L or C − R, and one is a single point). This means that a.s. each face of G ε has exactly three vertices on its boundary, so G ε is a triangulation.
Mated-CRT maps are an especially natural family of random planar maps to study. One reason for this is that these maps provide a bridge between many other interesting random planar map models and their continuum analogs: LQG surfaces. Let us now explain the precise sense in which this is the case, starting with the link between mated-CRT maps and other random planar map models; see Figure 2 for an illustration.
A number of random planar maps can be bijectively encoded by pairs of discrete random trees (equivalently, two-dimensional random walks) by discrete versions of the above definition of the mated-CRT map. Consequently, the mated-CRT map (with γ depending on the particular model) can be viewed as a coarse-grained approximation of any of these random planar maps. For example, Mullin's bijection [Mul67] (see [Ber07b,She16b,Che17] for more explicit expositions) shows that if we replace (L, R) by a two-sided simple random walk on Z 2 and construct a graph with adjacency defined by a direct discrete analog of (1.1), then we obtain the infinite-volume local limit of random planar maps sampled with probability proportional to the number of spanning trees they admit. The left-right ordering of the vertices corresponds to the depth-first ordering of the spanning tree. There are similar bijective constructions, with different laws for the random walk, which produce the uniform infinite planar triangulation (UIPT) [Ber07a,BHS18] as well as a number of natural random planar maps decorated by statistical mechanics models [She16b,GKMW18,KMSW15,LSW17].
At least in the case when the encoding walk has i.i.d. increments, one can use a strong coupling result for random walk and Brownian motion [KMT76,Zai98], which says that the random walk Z and the Brownian motion Z can be coupled together so that max −n≤j≤n |Z j − Z j | = O(log n) with high probability, to couple one of these other random planar maps with the mated-CRT map. This allows us to compare the maps directly. This approach is used in [GHS17] to couple the maps in such a way that graph distances differ by at most a polylogarithmic factor, which allows one to transfer the estimates for graph distances in the mated-CRT map from [GHS19] to a larger class of random planar map models. A similar approach is used in [GM17,GH18] to prove estimates for random walk on these same random planar map models.
On the other hand, the mated-CRT map possesses an a priori relationship with SLE-decorated Liouville quantum gravity. We will describe this relationship in more detail in Section 1.2, but let us briefly mention it here. Suppose h is the random distribution on C which describes a γ-quantum cone, a particular type of γ-LQG surface. Let η be a whole-plane space-filling SLE κ from ∞ to ∞ with 2 κ = 16/γ 2 > 4, sampled independently of h and then parameterized by γ-LQG mass with respect to h (we recall the definition and basic properties of space-filling SLE in Section 2.1.3). It follows from [DMS14,Theorem 1.9] that if we let G ε for ε > 0 be the graph whose vertex set is εZ, with two vertices x 1 , x 2 ∈ εZ connected by an edge if and only if the corresponding cells η([x 1 − ε, x 1 ]) and η([x 2 − ε, x 2 ]) share a non-trivial boundary arc, then {G ε } ε>0 has the same law as the family of mated-CRT maps defined above.
The above construction gives us an embedding of the mated-CRT map into C by mapping each vertex to the corresponding space-filling SLE cell. It is shown in [GMS17] that the simple random walk on G ε under this embedding converges in law to Brownian motion modulo time parameterization (which implies that the above SLE/LQG embedding is close when ε is small to the so-called Tutte embedding). The main theorem of [GMS17] is proven using a general scaling limit result for random walk in certain random environments [GMS18], which in turn is proven using ergodic theory. The theorem gives us control on the large-scale behavior of random walk and harmonic functions on G ε under the SLE/LQG embedding, but provides very little information about their behavior at smaller scales and no quantitative bounds for rates of convergence.
The goal of this paper is to prove quantitative estimates for discrete harmonic functions on G ε , which can be applied at mesoscopic scales and which include polynomial bounds for the rate of convergence of the probabilities that the estimates hold. In particular, we obtain estimates for the Dirichlet energy and the modulus of continuity of a large class of such discrete harmonic functions. See Section 1.5 for precise statements. We will not use the main theorem of [GMS17] in our proofs.
Instead, we will rely on a quantitative law-of-large-numbers type bound for integrals of functions defined on C against certain quantities associated with the cells η([x − ε, x]).
Our results provide a general toolbox for the study of random walk on mated-CRT maps, and thereby random walk on other random planar maps thanks to the coupling results discussed above. For example, our results give an independent proof that the random walk on the mated-CRT map is recurrent (Theorem 1.4; this can also be deduced from the general criterion of Gurel-Gurevich and Nachmias [GGN13], see Section 2.2). We also obtain a polynomial (in ε) upper bound for the maximum length of the edges of the mated-CRT map under the so-called Tutte embedding with identity boundary data (Corollary 1.6). We note that [GMS17] shows only that the maximum length of these embedded edges tends to zero as ε → 0, but does not give any quantitative bound for the rate of convergence.
The results of this paper will also play a crucial role in the subsequent work [GM17], which proves that the spectral dimension of a large class of random planar maps -including mated-CRT maps, spanning-tree weighted maps, and the UIPT -is two (i.e., the return probability after n steps is n −1+on(1) ) and also proves a lower bound for the graph distance displacement of the random walk on these maps which is correct up to polylogarithmic errors (the complementary upper bound is proven in [GH18]). We expect that our results may also have eventual applications to the study of discrete conformal embedddings of random planar maps, e.g., to the problem of showing that the maximal size of the faces of certain random planar maps -like uniform triangulations and spanning tree-weighted maps -under the Tutte embedding tends to 0. See the discussion just after Corollary 1.6.
One way to think about the approach used in this paper is as follows. A powerful technique for studying random walk and harmonic functions on random planar maps is to embed the map into C in some way, then consider how the embedded map interacts with paths and functions in C. A number of recent works have used this technique with the embedding given by the circle packing of the map [Ste03]; see, e.g., [BS01, GGN13, ABGGN16, GR13, AHNR16, Lee17,Lee18]. Here, we study random walk and harmonic functions on the mated-CRT map using the embedding of this map coming from SLE/LQG instead of the circle packing. For many quantities of interest, one can get stronger estimates using this embedding than using circle packing since we have good estimates for the behavior of space-filling SLE and the γ-LQG measure.

Mated-CRT maps and SLE-decorated Liouville quantum gravity
We now describe the connection between mated-CRT maps and SLE-decorated LQG, as alluded to at the end of Section 1.1. This connection gives an embedding of the mated-CRT map into C, which will be our main tool for analyzing mated-CRT maps. Moreover, most of our main results will be stated in terms of this embedding. See Section 2 for additional background on the objects involved.
Heuristically speaking, for γ ∈ (0, 2) a γ-LQG surface parameterized by a domain D ⊂ C is the random two-dimensional Riemannian manifold with metric tensor e γh dx ⊗ dy, where h is some variant of the Gaussian free field (GFF) on D [She07, SS13, MS16d, MS17] and dx ⊗ dy is the Euclidean metric tensor. This does not make literal sense since h is a random distribution, not a pointwise-defined function. Nevertheless, one can make literal sense of γ-LQG in various ways. Duplantier and Sheffield [DS11] constructed the volume form associated with a γ-LQG surface, a measure µ h which is the limit of regularized versions of e γh(z) dz, where dz denotes Lebesgue measure. One can similarly define a γ-LQG boundary length measure ν h on certain curves in D, including ∂D and SLE κ -type curves for κ = γ 2 [She16a]. These measures are a special case of a more general theory called Gaussian multiplicative chaos; see [Kah85,RV14,Ber17].
Mated-CRT maps are related to SLE-decorated LQG via the peanosphere (or mating-of-trees)  Left: Various random planar maps (e.g., the UIPT or spanning-tree weighted maps) can be encoded by means of a two-dimensional random walk via a discrete version of the construction of the mated-CRT map (we will not use these bijections in this paper). Right: The mated-CRT map is defined using a pair of Brownian motions and has an embedding into C as the adjacency graph on the "cells" η([x − ε, x]) for x ∈ εZ of a space-filling SLE parameterized by γ-LQG mass. This paper proves estimates for the mated-CRT map under this embedding. One can transfer these estimates to other random planar maps (up to a polylogarithmic error) using a strong coupling of the encoding walk for the other planar map and the Brownian motion used to define the mated-CRT map; see [GHS17,GM17].
construction of [DMS14, Theorem 1.9], which we now describe. Suppose h is the random distribution on C corresponding to the particular type of γ-LQG surface called a γ-quantum cone. Then h is a slight modification of a whole-plane GFF plus −γ log | · | (see Section 2.1.2 for more on this field). Also let κ = 16/γ 2 > 4 and let η be a whole-plane space-filling SLE κ curve from ∞ to ∞ sampled independently from h and then parameterized in such a way that η(0) = 0 and the γ-LQG mass satisfies µ h (η([t 1 , t 2 ])) = t 2 − t 1 whenever t 1 , t 2 ∈ R with t 1 < t 2 (see Section 2.1.3 and the references therein more on space-filling SLE).
Let ν h be the γ-LQG length measure associated with h and define a process L : Figure 3: Illustration of the definition of the left/right boundary length process (L, R) for space-filling SLE on a γ-quantum cone. The left figure corresponds to the case when κ ≥ 8, so that η([t 1 , t 2 ]) is simply connected. The right figure corresponds to the case when κ ∈ (4, 8), in which case the topology is more complicated since the left and right boundaries of the curve can intersect each other, but the definition of the left/right boundary length process is the same. In both cases, the intersection of the left (resp. right) outer boundaries of η((−∞, t 1 ]) and η([t 2 , ∞)) is shown in red (resp. blue). The black dots on the boundary correspond to the endpoints η(t 1 ) and η(t 2 ) and the points where η((−∞, t 1 ]) ∩ η([t 1 , t 2 ]) and η([t 2 , ∞)) ∩ η([t 1 , t 2 ]) meet. These latter two points are hit by η at the times when L and R, respectively, attain their minima on [t 1 , t 2 ].
way that L 0 = 0 and for t 1 , t 2 ∈ R with t 1 < t 2 , Define R t similarly but with "right" in place of "left" and set Z t = (L t , R t ). See Figure 3 for an illustration. It is shown in [DMS14, Theorem 1.9] that Z evolves as a correlated two-dimensional Brownian motion with correlation − cos(πγ 2 /4), i.e., Z has the same law as the Brownian motion used to construct the mated-CRT map with parameter γ (up to multiplication by a deterministic constant, which does not affect the definition of the mated-CRT map). Moreover, by [DMS14, Theorem 1.11], Z a.s. determines (h, η) modulo rotation and scaling. We can re-phrase the adjacency condition (1.1) in terms of (h, η). In particular, for x 1 , x 2 ∈ εZ with x 1 < x 2 , (1.1) is satisfied if and only if the cells η([x 1 − ε, x 1 ]) and η([x 2 − ε, x 2 ]) intersect along a non-trivial connected arc of their left outer boundaries; and similarly with "R" in place of "L" and "left" in place of "right". Indeed, this follows from the explicit description of the curve-decorated topological space (C, η) in terms of (L, R) given in [DMS14,Section 8.2].
Consequently, the mated-CRT map G ε is precisely the graph with vertex set εZ, with two vertices connected by an edge if and only if the corresponding cells η([x 1 − ε, x 1 ]) and η([x 2 − ε, x 2 ]) share a non-trivial connected boundary arc. The graph on cells is sometimes called the ε-structure graph of the curve η since it encodes the topological structure of the cells. The identification of G ε with the ε-structure graph of η gives us an embedding of G ε into C by sending each vertex x ∈ εZ to the point η(x). See Figure 4 for an illustration.

Basic notation
We write N for the set of positive integers and N 0 = N ∪ {0}. x ∈ εZ. This figure looks like what we would expect to see for a space-filling SLE κ parameterized by γ-quantum mass when κ ≥ 8, since this is the range when the curve does not make and fill in bubbles (see Section 2.1.3). Top right: Same as top-left but with an orange path showing the order in which cells are hit by η. Bottom left: A point in each cell is shown in red, and is connected to each adjacent cell by a red edge. As explained in Figure 1, G ε can be viewed as a planar triangulation. In the present picture, the faces correspond to the points where three of the black curves meet. Note that we cannot have more than three black curves meeting at a single point or we would have a face of degree greater than three (this can also be seen directly from the geometry of space-filling SLE; see [DMS14, Section 8.2]). Bottom right: If we forget the original cells η([x − ε, x]) but keep the red edges we get an embedding of G ε into C. For K ⊂ C, we write area(K) for the Lebesgue measure of K and diam(K) for its Euclidean diameter. For r > 0 and z ∈ C we write B r (z) be the open disk of radius r centered at z. For K ⊂ C, we also write B r (K) for the (open) set of points z ∈ C which lie at Euclidean distance less 8 than r from K.
If a and b are two "quantities" (i.e., functions from any sort of "configuration space" to the real numbers) we write a b (resp. a b) if there is a constant C > 0 (independent of the values of a or b and certain other parameters of interest) such that a ≤ Cb (resp. a ≥ Cb). We write a b if a b and a b. We typically describe dependence of implicit constants in lemma/proposition statements and require constants in the proof to satisfy the same dependencies.
If a and b are two quantities depending on a variable x, we write a = O x (b) (resp. a = o x (b)) if a/b remains bounded (resp. tends to 0) as x → 0 or as x → ∞ (the regime we are considering will be clear from the context). We write a = o ∞ For a graph G, we write V(G) and E(G), respectively, for the set of vertices and edges of G, respectively. We sometimes omit the parentheses and write VG = V(G) and EG = E(G). For v ∈ V(G), we write deg(v; G) for the degree of v (i.e., the number of edges with v as an endpoint).

Setup
In this subsection, we describe the setup we consider throughout most of the paper and introduce some relevant notation. Let h be a random distribution on C whose γ-quantum measure µ h is well-defined and has infinite total mass. We will most frequently consider the case when h is the distribution corresponding to a γ-quantum cone, since this is the case for which the corresponding structure graph coincides with a mated-CRT map. However, we will also have occasion to consider a choice of h which does not have a γ-log singularity at the origin-in particular, we will sometimes take h to be either a whole-plane GFF or the distribution corresponding to a 0-quantum cone.
Let η be a whole-plane space-filling SLE κ sampled independently from h and then parameterized by γ-quantum mass with respect to h. For ε > 0, we let G ε be the graph with vertex set εZ, with two vertices x 1 , x 2 ∈ εZ connected by an edge if and only if the cells η([x 1 − ε, x 1 ]) and η([x 2 − ε, x 2 ]) share a non-trivial boundary arc.
We abbreviate cells by and for z ∈ C, we define the vertex z is the (a.s. unique) structure graph cell containing z.
For a set D ⊂ C, we write G ε (D) for the sub-graph of G ε with vertex set with two vertices connected by an edge if and only if they are connected by an edge in G ε . See Figure 5 for an illustration of the above definitions.

Main results
Suppose we are in the setting of Section 1.4 with h equal to the circle-average embedding of a γ-quantum cone (i.e., h is the random distribution from Definition 2.3 with α = γ), so that the graphs {G ε } ε>0 are the same in law as the ε-mated CRT maps defined in Section 1.1.
We will study discrete harmonic functions on sub-maps of G ε corresponding to domains in C. normalization of the field, 3 so we fix ρ ∈ (0, 1) and restrict attention to B ρ (0). Let D ⊂ B ρ (0) be an open set and let f : D → C be a continuous function.
Recall the sub-graph G ε (D) ⊂ G ε from (1.5) and let f ε : VG ε (D) → R be the function such that and f ε is discrete harmonic on VG ε (D) \ VG ε (∂D). The first main result of this paper shows that the discrete Dirichlet energy of f ε can be bounded above by a constant times the Dirichlet energy of f .  There are constants α > 0 (depending only on γ) and A > 0 (depending only on D, γ, and the Lipschitz constants for f and ∇f ) such that with probability at least 1 − O ε (ε α ), the discrete and continuum Dirichlet energies of f ε and f are related by (1.7) We will actually prove a more quantitative version of Theorem 1.3 below (see Theorem 3.2), which makes the dependence of A more explicit.
One reason why bounds for Dirichlet energy are important is that one can express many quantities related to random walk on the graph -such as the Green's function, effective resistances, and return probabilities -in terms of the discrete Dirichlet energy of certain functions (see, e.g., [LP16, Section 2]). These relationships together with Theorem 1.3 lead to a lower bound for the Green's function of random walk on G ε on the diagonal, or equivalently for the effective resistance to the boundary of a Euclidean ball (Theorem 1.4 just below). Further applications of our Dirichlet energy estimates will be explored in [GM17].
For n ∈ N and ε > 0, let Gr ε n (·, ·) be the Green's function of G ε at time n, i.e., Gr ε n (x, y) for vertices x, y ∈ VG ε gives the (conditional given G ε ) expected number of times that simple random walk on G ε started from x hits y before time n.
Theorem 1.4 (Green's function on the diagonal). Fix ρ ∈ (0, 1) and let τ ε for ε > 0 be the exit time of simple random walk on G ε from VG ε (B ρ (0)). There exists α > 0 (depending only on γ) and A > 0 (depending only on ρ and γ) such that Furthermore, the simple random walk on G ε is a.s. recurrent.
The recurrence of random walk on G ε can also be deduced from the general recurrence criterion for random planar maps due to Gurel-Gurevich and Nachmias [GGN13] (see Section 2.2), but our results give an independent proof. Note, however, that our results do not give an independent proof of the recurrence of random walk on other planar maps, such as the UIPT.
Our next main result gives a Hölder continuity bound for the functions f ε for ε > 0 in terms of the Euclidean metric.
As in the case of Theorem 1.3, we will prove a more quantitative version of Theorem 1.5; see Theorem 3.9. The proof of this theorem proceeds by way of a "uniform ellipticity" type estimate for simple random walk on G ε , which says that the walk has uniformly positive probability to stay close to a fixed path in C (Proposition 3.6).
Theorem 1.5 gives a polynomial bound for the rate at which the maximal length of an edge of the graph G ε (D) under the so-called Tutte embedding with identity boundary data converges to 0 as ε → 0 (since G ε is a triangulation, this is equivalent to the analogous statement with faces in place of edges). Note that [GMS17] shows that the Tutte embedding with identity boundary data converges to the identity, but gives no quantitative bound on the maximal length of the embedded edges.
To state this more precisely, let Φ ε 1 be the function f ε from above with f (z) = Re z and let Φ ε 2 be defined analogously with f (z) = Im z. Then Φ ε := (Φ ε 1 , Φ ε 2 ) : VG ε (D) → R 2 is discrete harmonic on the interior of G ε (D) and approximates the map x → η(x) on VG ε (∂D). The function Φ ε is called the Tutte embedding of G ε (D) with identity boundary data.
It is easy to see that the maximal size of the cells H ε x for x ∈ VG ε (D) is at most some positive power of ε with probability tending to 1 as ε → 0 (Lemma 2.7). Applying Theorem 1.5 to each coordinate of Φ ε and considering vertices x and y which are connected by an edge in G ε yields the following.
Corollary 1.6 (Maximal length of embedded edges). Define the Tutte embedding Φ ε with identity boundary data as above. If D ⊂ B ρ (0) is simply connected, then there exists ξ = ξ (ρ, γ) > 0 such that with probability tending to 1 as ε → 0, It is a major open problem to prove that the maximal length of the embedded edges of other types of random planar maps-e.g., uniform random planar maps or planar maps sampled with probability proportional to the number of spanning trees-under the Tutte embedding (or under other embeddings, like the circle packing [Ste03]) tends to 0 as the total number of vertices tends to 0. Indeed, this is believed to be a key obstacle to proving that such embedded maps converge to γ-LQG in various senses, as conjectured, e.g., in [DS11,She16a,DKRV16,Cur15]. Corollary 1.6 suggests a possible approach to proving that the maximal edge length for various additional types of embedded random planar maps, besides just the mated-CRT map, also tends to zero. The reason for this is that in many cases it is possible to transfer estimates from the mated-CRT map to estimates for other random planar maps modulo polylogarithmic multiplicative errors. So far, this has been done for graph distances [GHS17], random walk speed [GM17,GH18], and random walk return probabilities [GM17]. However, we have not yet found a way to transfer modulus of continuity bounds for harmonic functions, which is what is needed to deduce an analog of Corollary 1.6 for other planar map models. Figure 6 shows a diagram of the logical connections between the main results related to this paper. In Section 2, we will review some facts from the theory of SLE and LQG, prove that the law of the degree of a vertex of the mated-CRT map has an exponential tail (Lemma 2.5), and prove that the maximum diameter of the cells of G ε which intersect a fixed Euclidean ball decays polynomially in ε (Lemma 2.7). We then state an estimate (Proposition 2.10) which says that if D ⊂ C and f : D → R is a sufficiently regular function, then except on an event of probability decaying polynomially in ε,

Outline
where here we recall that H ε x ε z is the cell of G ε containing z. The proof of this estimate is deferred to Section 4. Intuitively, (1.11) says that the measure which assigns mass each z ∈ C is not too much different from Lebesgue measure, which in turn is a consequence of the fact that is of constant order for most z ∈ D. In Section 3, we assume the aforementioned estimate (1.11) and deduce our main results. We first prove in Section 3.1 an estimate to the effect that if f : D → R is as in (1.11), then with high which follows from (1.11) by breaking up the integral in (1.11) into integrals over individual cells.
The bound (1.12) is used in Section 3.2 to prove an upper bound for the discrete Dirichlet energy of x → f (η(x)), which in turn implies (a more precise version of) Theorem 1.3 since discrete harmonic functions minimize Dirichlet energy. In Section 3.3, we deduce Theorem 1.4 from this more general bound. In Section 3.4, we use our Dirichlet energy bound to show that the simple random walk on G ε has uniformly positive probability to stay close to a fixed Euclidean path, even if we condition on G ε . The basic idea is to first prove a lower bound for the probability of hitting the inner boundary of an annulus before the outer boundary (using Dirichlet energy estimates and the Cauchy-Schwarz inequality) then cover a path by such annuli. In Section 3.5, we use the result of Section 3.4 to prove a Hölder continuity estimate for harmonic functions on G ε (D) which includes Theorem 1.5 as a special case.
In Section 4, we prove (1.11), taking the moment bounds for the squared diameter over area and degree of the cells of G ε from [GMS17, Theorem 4.1] as a starting point. Heuristically, these moment bounds say that cells are not too likely to be "long and skinny" and are not too likely to have large degree. The proof is outlined in Section 4.1, and is based on using long-range independence properties for the GFF to bound the variance of the integral appearing in (1.11).
Appendix A contains some basic estimates for the GFF which are needed in our proofs. Appendix B contains an index of notation. [GMS17]: Moment bound for diameter 2 /area times degree. Proposition 2.10: Estimate for integrals against diameter 2 /area times degree.  Theorems 1.5 and 3.9: Modulus of continuity bound.

3.1:
Estimate for sum of diameter 2 times degree of cells.

[GM17]:
Spectral dimension and displacement for random walk on random planar maps. Theorem 1.4: Green's function bound and recurrence. 2 Preliminaries 2.1 Background on GFF, LQG, and SLE Throughout this paper, we always fix an LQG parameter γ ∈ (0, 2) and a corresponding SLE parameter κ = 16/γ 2 > 4. Here we provide some background on the main continuum objects involved in this paper, namely the Gaussian free field, Liouville quantum gravity, and space-filling SLE κ . A reader who is already familiar with these objects can safely skip this subsection.

The Gaussian free field
Here we give a brief review of the definition of the zero-boundary and whole-plane Gaussian free fields. We refer the reader to [She07]  For an open domain D ⊂ C with harmonically non-trivial boundary (i.e., Brownian motion started from a point in D a.s. hits ∂D), we define H(D) be the Hilbert space completion of the set of smooth, compactly supported functions on D with respect to the Dirichlet inner product, In the case when D = C, constant functions c satisfy (c, c) ∇ = 0, so to get a positive definite norm in this case we instead take H(C) to be the Hilbert space completion of the set of smooth, compactly supported functions φ on C with C φ(z) dz = 0, with respect to the same inner product (2.1).
The (zero-boundary) Gaussian free field on D is defined by the formal sum where the X j 's are i.i.d. standard Gaussian random variables and the φ j 's are an orthonormal basis for H(D). The sum (2.2) does not converge pointwise, but for each fixed φ ∈ H(D), the formal inner product (h, φ) ∇ := ∞ j=1 X j (φ j , φ) ∇ is a centered Gaussian random variable and these In the case when D = C and D has harmonically non-trivial boundary, one can use integration by parts to define the ordinary L 2 inner products (h, φ) . This allows one to define the GFF as a distribution (generalized function). See [She07, Section 2] for some discussion about precisely which spaces of distributions the GFF takes values in. For z ∈ D and r > 0 such that B r (z) ⊂ D, we write h r (z) for the circle average of h over ∂B r (z), as in [DS11, Section 3.1]. Following [DS11, Section 3.1], to define this circle average precisely, one can let ξ z r (w) := − log max{r, |w − z|}, so that −∆ξ z r (defined in the distributional sense) is 2π times the uniform measure on ∂B r (z). One then defines h r (z) to be the Dirichlet inner product (h, ξ z r ) ∇ .
In the case when D = C, one can similarly define (h, φ) , so the whole-plane GFF is only defined as a distribution modulo a global additive constant; that is, h can be viewed as an equivalence class of distributions under the equivalence relation whereby two distributions are equivalent if their difference is a constant. We will typically fix the additive constant for the GFF (i.e., choose a particular equivalence class representative) by requiring that the circle average h 1 (0) over ∂D is zero. That is, we consider the field h − h 1 (0), which is well-defined not just modulo additive constant. The law of the whole-plane GFF is scale and translation invariant modulo additive constant, which means that for z ∈ C and we can define the restriction of h to an open set U ⊂ D as the restriction of the distributional pairing φ → (h, φ) to test functions φ which are supported on V . It does not make literal sense to restrict the GFF to a closed set K ⊂ D, but the σ-algebra generated by h| K can be defined as ε>0 σ(h| Bε(K) ), where B ε (K) is the Euclidean ε-neighborhood of K. Hence it makes sense to speak of, e.g., "conditioning on h| K ".
The zero-boundary GFF on D possesses the following Markov property (see, e.g., [She07, Section 2.6]). Let U ⊂ D be a sub-domain with harmonically non-trivial boundary. Then we can write h = h +h, where h is a random distribution on D which is harmonic on U and is determined by h| D\U ; andh is a zero-boundary GFF on U which is independent from h| D\U . The restrictions of these distributions to U are called the harmonic part and zero-boundary part of h| U , respectively.
In the whole-plane case, one has a slightly more complicated Markov property due to the need to fix the additive constant. We state two versions of this Markov property, one with the field viewed modulo additive constant and one with the additive constant fixed. The first version is a re-statement of [MS17, Proposition 2.8].
Lemma 2.1. Let h be a whole-plane GFF viewed modulo additive constant. For each open set U ⊂ C with harmonically non-trivial boundary, we have the decomposition where h is a random distribution viewed modulo additive constant which is harmonic on U and is determined by h | C\U , viewed modulo additive constant; andh is a zero-boundary GFF on C \ U which is determined by the equivalence class of h | C\U modulo additive constant.
We refer to the distributions h | U andh | U as the harmonic part and zero-boundary part of h | U , respectively. Now suppose we want to fix the additive constant for the field so that h 1 (0) = 0, i.e., we want to consider h − h 1 (0). In the setting of Lemma 2.1, the distributions h − h 1 (0) andh are not independent if ∂D ∩ U = ∅ since h 1 (0) depends onh . Nevertheless, it turns out that a slight modification of these distributions are independent.
where h is a random distribution which is harmonic on U and is determined by h| C\U andh is independent from h and has the law of a zero-boundary GFF on U minus its average over ∂D ∩ U .
If U is disjoint from ∂D, thenh is a zero-boundary GFF and is independent from h| C\U .
Proof. Let h be a whole-plane GFF viewed modulo additive constant, so that h = h − h 1 (0). Write h = h +h as in Lemma 2.1. Leth 1 (0) be the average ofh over ∂D (equivalently, over ∂D ∩ U ). Also let h 1 (0) = h 1 (0) −h 1 (0) be the average of h over ∂D. We define Then h is a harmonic function in U and is well-defined (not just modulo additive constant) andh is a zero-boundary GFF in U minus its average over ∂D∩U . By definition, we have h+h = h −h 1 (0) = h. Furthermore, h (resp.h) is determined by h (resp.h ), so h andh are independent. Since h is determined by h | C\U , viewed modulo additive constant, it follows that h is determined by h| C\U . Hence h is determined by h| C\U . If ∂D is disjoint from U , thenh 1 (0) = 0 so h| C\U = h| C\U . This implies thath is a zero-boundary GFF andh is independent from h| C\U .

Liouville quantum gravity
(2.6) One can similarly define a γ-LQG surface with k ∈ N marked points. This is an equivalence class of k + 2-tuples (D, h, x 1 , . . . , x k ) with the equivalence relation defined as in (2.6) except that the map f is required to map the marked points of one surface to the corresponding marked points of the other. We call different choices of the distribution h corresponding to the same LQG surface different embeddings of the surface.
If h is a random distribution on D which can be coupled with a GFF on D in such a way that their difference is a.s. a continuous function, then one can define the γ-LQG area measure µ h on D, which is defined to the a.s. limit µ h = lim ε→0 ε γ 2 /2 e γhε(z) dz in the Prokhorov distance (or local Prokhorov distance, if D is unbounded) as ε → 0 along powers of 2 [DS11]. Here h ε (z) is the circle-average of h over ∂B ε (z), as defined in [DS11, Section 3.1] and discussed in Section 2. One can similarly define a boundary length measure ν h on certain curves in D, including ∂D [DS11] and SLE κ type curves for κ = γ 2 which are independent from h [She16a].
If h and h are related by a conformal map as in (2.6), then f * µ h = µ h and f * ν h = ν h . Hence µ h and ν h can be viewed as measures on the LQG surface (D, h). We note that there is a more general theory of regularized measures of this type, called Gaussian multiplicative chaos which originates in work of Kahane [Kah85]. See [RV14,Ber17] for surveys of this theory.
In this paper, we will be interested in two different types of γ-LQG surface. The first and most basic type of LQG surface we consider is the one where h is a whole-plane GFF, as in Section 2.1.1. We will typically fix the additive constant for the whole-plane GFF by requiring that the circle average over ∂D is 0.
The other type of γ-LQG surface with the topology of the plane which we will be interested in is the α-quantum cone for α ∈ (−∞, Q), which is a doubly marked LQG surface (C, h, 0, ∞) introduced in [DMS14, Definition 4.10]. Roughly speaking, the α-quantum cone is obtained by starting with a whole-plane GFF plus α log(1/| · |) then "zooming in" near the origin and rescaling [DMS14, Proposition 4.13(ii) and Lemma A.10].
We will not need the precise definition of the α-quantum cone in this paper, but we recall it here for completeness. Recall the Hilbert space H(C) used in the definition of the whole-plane GFF. Let H 0 (C) (resp. H † (C)) be the subspace of H(C) consisting of functions which are constant (resp. have mean zero) on each circle ∂B r (0) for r > 0. By [DMS14,Lemma 4.9], H(C) is the orthogonal direct sum of H 0 (C) and H † (C).
Definition 2.3 (Quantum cone). For α < Q, the α-quantum cone is the LQG surface (C, h, 0, ∞) with the distribution h defined as follows. Let B be a standard linear Brownian motion and let B be a standard linear Brownian motion conditioned so that Then the projection of h onto H 0 (C) takes the constant value A t on each circle B e −t (0). The projection of h onto H † (C) is independent from the projection onto H 0 (C) and agrees in law with the corresponding projection of a whole-plane GFF.
We will typically be interested in quantum cones with α = 0 or α = γ. The case α = γ is special since a γ-LQG surface has a γ-log singularity at a typical point sampled from its γ-LQG measure (see, e.g., [DS11, Section 3.3]), so the γ-quantum cone describes the local behavior of such a surface near a quantum typical point. The γ-quantum cone is also the type of LQG surface appearing in the embedding of the mated-CRT map. Similarly, the 0-quantum cone describes the behavior of a γ-LQG surface near a Lebesgue typical point.
By the definition of an LQG surface, one can get another distribution describing the γ-quantum cone by replacing h by h(r·) + Q log r for some r > 0. But, we will almost always consider the particular choice of h appearing in Definition 2.3, which satisfies sup{r > 0 : h r (0) + Q log r = 0} = 1. This choice of h is called the circle average embedding. A useful property of the circle average embedding (which is essentially immediate from [DMS14, Definition 4.10]) is that h| D agrees in law with the corresponding restriction of a whole-plane GFF plus −α log | · |, normalized so that its circle average over ∂D is 0.

Space-filling SLE κ
The Schramm-Loewner evolution (SLE κ ) for κ > 0 is a one-parameter family of random fractal curves originally defined by Schramm in [Sch00]. SLE κ curves are simple for κ ∈ (0, 4], self-touching, but not space-filling or self-crossing, for κ ∈ (4, 8), and space-filling (but still not self-crossing) for κ ≥ 8 [RS05]. One can consider SLE κ curves between two marked boundary points of a simply connected domain (chordal), from a boundary point to an interior point (radial), or between two points in C ∪ {∞} (whole-plane). We refer to [Law05] or [Wer04] for an introduction to SLE. We will occasionally make reference to whole-plane SLE κ (ρ), a variant of whole-plane SLE κ where one keeps track of an extra marked "force point" which is defined in [MS17, Section 2.1]. However, we will not need many of its properties so we will not provide a detailed definition here.
Space-filling SLE κ is a variant of SLE κ for κ > 4 which was originally defined in [MS17, Section 1.2.3] (see also [DMS14, Section 1.4.1] for the whole-plane case). Here we will review the construction of whole-plane space-filling SLE κ from ∞ to ∞, which is the only version we will use in this paper.
The basic idea of the construction is that, by SLE duality [Zha08, Zha10, Dub09, MS16d, MS17], the outer boundary of an ordinary SLE κ curve stopped at any given time is a union of SLE κ -type curves for κ = 16/κ ∈ (0, 4). It is therefore natural to try to construct a space-filling SLE κ -type curve by specifying its outer boundary at each fixed time. To construct the needed boundary curves, we will use the theory of imaginary geometry, which allows us to couple many different SLE κ curves with a common GFF. Let . For our purposes we will only need the flow lines started from points z ∈ Q 2 with angles π/2 and −π/2, which we denote by η L z and η R z , respectively (the L and R stand for "left" and "right", for reasons which will become apparent momentarily). We define a total order on Q 2 by declaring that z comes before w if and only if w lies in a connected component of C \ (η L z ∪ η R z ) which lies to the right of η L z (equivalently, to the left of η R z ). The whole-plane analog of [MS17, Theorem 4.12] (which can be deduced from the chordal case; see [DMS14, Footnote 4]) shows that there is a well-defined continuous curve η : R → C which traces the points of Q 2 in the above order, is such that η −1 (Q 2 ) is a dense set of times, and is continuous when parameterized by Lebesgue measure, i.e., in such a way that area(η([a, b])) = b − a whenever a < b. The curve η is defined to be the whole-plane space-filling SLE κ from ∞ to ∞ associated with h IG .
The definition of η implies that for each z ∈ C, it is a.s. the case that the left and right boundaries of η stopped when it first hits z are equal to the flow lines η L z and η R z (which can be defined for a.e. z ∈ C simultaneously as the limits of the curves η L w and η R w as Q 2 w → z w.r.t., e.g., the local Hausdorff distance). See Figure 3. The topology of η is rather simple when κ ≥ 8. In this case, the left/right boundary curves η L z and η R z do not bounce off each other, so for a < b the set η([a, b]) has the topology of a disk. In the case when κ ∈ (4, 8), the curves η L z and η R z intersect in an uncountable fractal set and for a < b the interior of the set η([a, b]) a.s. has countably many connected components, each of which has the topology of a disk.
It is shown in [MS17, Theorem 1.16] that for chordal space-filling SLE, the curve η is a.s. determined by h IG . The analogous statement in the whole-plane case can be proven using the same argument or deduced from the chordal case and [DMS14, Footnote 4]. We will need the following refined version of this statement.
Lemma 2.4. Let D ⊂ C and let h IG and η be as above. Assume that η is parameterized by Lebesgue measure. Let U ⊂ D be an open set and for z ∈ U ∩ Q 2 , let T z (resp. S z ) be the last time η enters U before hitting z (resp. the first time z exits U after hitting z). Then for each δ > 0, h IG | B δ (U ) a.s. determines the collection of curve segments (2.7) Here B δ (U ) is the Euclidean δ-neighborhood of U , as in Section 1.3.
Proof. It is shown in [MS17, Theorem 1.2] that the left/right boundary curves η L z and η R z for z ∈ Q 2 are a.s. determined by h IG . For z ∈ U ∩ Q 2 , let τ L z (resp. τ R z ) be the exit time of η L z (resp. η R z ) from U . By [

The degree of the root vertex has an exponential tail
Most of the results in this paper make use of the embedding of the mated-CRT maps which comes from SLE-decorated LQG (see Section 1.2). However, the following result is proved directly from the "Brownian motion" definition of the mated-CRT maps in (1.1), and does not rely on this embedding.
Lemma 2.5. Let γ ∈ (0, 2) and let G ε for ε > 0 be a mated-CRT map. There are constants c 0 , c 1 > 0, depending only on γ, such that for n ∈ N, ε > 0, and x ∈ εZ, Proof. By Brownian scaling and translation invariance, the law of the pointed graph (G ε , x) does not depend on ε or x, so we can assume without loss of generality that ε = 1 and x = 0. By (1.1), the time reversal symmetry of (L, R), and the fact that (L, R) d = (R, L), it suffices to show that there exists constants c 0 , c 1 > 0 as in the statement of the lemma such that with From Lemma 2.5 and a union bound, we get the following upper bound for the maximal degree of the cells of G ε which intersect a specified Euclidean ball.
Lemma 2.6. Suppose we are in the setting of Section 1.4 with h equal to the circle-average embedding of a γ-quantum cone into (C, 0, ∞). If we define the vertex x ε z ∈ εZ as in (1.4), then for each ζ > 0, Combining the above estimates and sending M → ∞ shows that (2.8) holds.
In light of Lemma 2.5, we can deduce the recurrence of the simple random walk on G ε from the results of [GGN13] (we will give an independent proof in Section 3.3). Indeed, by [GGN13, Theorem 1.1], the random walk on an infinite rooted random planar map (G, v) is recurrent provided the law of the degree of the root vertex v has an exponential tail and (G, v) is the distributional limit of finite rooted planar maps in the local (Benjamini-Schramm) topology [BS01]. The first condition for (G ε , 0) follows from Lemma 2.5. To obtain the second condition, we observe that the law of G ε is invariant under the operation of translating its vertex set by ε, and consequently G ε is the distributional local limit as n → ∞ of the planar map whose vertex set is [−nε, nε] εZ , with two vertices connected by an edge if and only if they are connected by an edge in G ε , each rooted at a uniformly random vertex in [−nε, nε] Z .
We note that it is also known that the simple random walk on the adjacency graph of cells associated with a space-filling SLE κ on an independent 0-quantum cone is recurrent: indeed, this follows from [GMS18, Theorem 1.16] and [GMS17, Proposition 3.1].

Maximal cell diameter
In this brief subsection we establish a polynomial upper bound for the maximum size of the cells of G ε which intersect a fixed Euclidean ball. In other words, we prove an analog of Corollary 1.6 with the SLE/LQG embedding in place of the Tutte embedding, which we will eventually use to prove Corollary 1.6.
Lemma 2.7. Suppose we are in the setting of Section 1.4, with h either a whole-plane GFF normalized so that its circle average over ∂D is zero or the circle-average embedding of a 0-quantum cone or a γ-quantum cone. For each q ∈ 0, 2 (2+γ) 2 , each ρ ∈ (0, 1), and each ε ∈ (0, 1), where the rate of the o ε (1) depends only on q, ρ, and γ and Lemma 2.7 is an easy consequence of the following basic estimate for the γ-LQG measure.
Proof. If h is a circle-average embedding of an α-quantum cone, then h| D agrees in law with a whole-plane GFF normalized so that its circle average over ∂D is 0 plus −α log | · |. If h is a whole-plane GFF, then µ h−α log |·| (A) ≥ µ h (A) for each Borel set A ⊂ D. So, we can restrict attention to the case when h is a whole-plane GFF normalized so that its circle average over ∂D is 0. Let {h r } r≥0 be the circle average process of h and fix ζ ∈ (0, 1). By standard estimates for the γ-LQG measure (see, e.g., [GHM15, Lemma 3.12]), for z ∈ C, For z ∈ B ρ (0), the random variable h δ (z) is centered Gaussian with variance log δ −1 + O δ (1). Therefore, Combining these estimates and sending ζ → 0 shows that 2γ 2 +o δ (1) . (2.11) We obtain (2.10) by applying (2.11) with δ/2 in place of δ then taking a union bound over all z ∈ ( δ Proof of Lemma 2.7. Fix q ∈ q, 2 (2+γ) 2 . By Lemma 2.8 applied with 1/ q − 2 − γ 2 /2 > 2γ in place of p and ε q in place of δ, it holds with probability at least 1 − ε α( q,γ)+oε(1) that each Euclidean ball contained in B (1+ρ)/2 (0) with radius at least ε q has µ h -mass at least ε. By [GHM15, Proposition 3.4 and Remark 3.9], it holds except on an event of probability o ∞ ε (ε) that each segment of η contained in D with diameter at least ε q contains a Euclidean ball of radius at least ε q . Hence with probability 1 − ε α( q,γ)+oε(1) , each segment of η which intersects B ρ (0) and has Euclidean diameter at lest ε q has µ h -mass at least ε. Each cell H ε is a segment of η with µ h -mass ε, so with probability at least 1 − ε α( q,γ)+oε(1) each such cell which intersects B ρ (0) has diameter at most ε q . Sending q → q concludes the proof.

Estimates for integrals against structure graph cells
A key input in our estimates for harmonic functions on G ε (i.e., Theorem 1.3 and Theorem 1.5) is a bound for the integrals of Euclidean functions against quantities associated with the cells of G ε . We state this bound in this subsection, and postpone its proof until Section 4.
Suppose we are in the setting of Section 1.4 and that h is the circle-average embedding of a γ-quantum cone. For z ∈ C, let (2.12) Our main estimate for u ε (z) is a one-sided "law of large numbers" type estimate for integrals against u ε (z). The following is a simplified (but perhaps more intuitive) version of our result, which states in quantitative way that the mean value of u ε (z) tends to be smaller than a fixed constant.
Proposition 2.9 is an immediate consequence of Proposition 2.10 below; it can be derived from Proposition 2.10 by setting f = 1 D .
3 Estimates for harmonic functions on G ε Suppose we are in the setting of Section 1.4 with h equal to the circle-average embedding of a γ-quantum cone. Recall that η is a whole-plane space-filling SLE κ parameterized by γ-quantum mass with respect to h, and G ε for ε > 0 is the associated mated-CRT map. Recall also that for z ∈ C, x ε z is the smallest (and a.s. only) element of εZ = VG ε for which z is contained in the cell H ε . In this section, we assume Proposition 2.10 and use it to deduce various bounds for harmonic functions on the sub-graph G ε (D) defined as in (1.5), which will eventually lead to Theorems 1.3 and 1.5.
Many of the estimates in this subsection will include constants α, β, and A which are required to be independent of ε, but are allowed to be different in each lemma/proposition/theorem. Throughout this section, we fix ρ ∈ (0, 1) and work on the ball B ρ (0).

Comparing sums over cells and Lebesgue integrals
In this subsection we establish a variant of Proposition 2.10 which allows us to compare the weighted sum of the values of a function on C over all cells in the restricted structure graph VG ε (D) to its integral over D.
We note that the choice of boundary data for f ε in (3.1) (which will also show up in other places) is somewhat arbitrary-we just need boundary data which is close in some sense to the boundary data for f when ε is small.
Proof of Lemma 3.1. Let q ∈ 0, 2 (2+γ) 2 , chosen later in a manner depending only on γ, and let β > 0 be smaller than the minimum of q and the parameter β of Proposition 2.10. We define the event where here we recall that H ε x ε z is the cell of G ε containing z. By Lemmas 2.6 and 2.7, P[( E ε 0 ) c ] decays faster than some positive power of ε.

Dirichlet energy bounds
In this subsection, we will use Proposition 2.10 to prove bounds for the Dirichlet energy of discrete harmonic functions on subgraphs of G ε in terms of the Dirichlet energy of functions on subsets of C.
We will consider the following setup. Recall that we have fixed ρ ∈ (0, 1). For C ≥ 1 and δ ∈ (0, 1), let C C (δ) = C C (δ, ρ) be the set of pairs (D, f ) where D is an open subset of B ρ (0) and f : D → R is a differentiable function such that the following is true.
2. D has C-bounded convexity, i.e., for each z, w ∈ D there is a path from z to w contained in D which has length at most C|z − w|.
The main result of this subsection is the following more quantitative version of Theorem 1.3.
Theorem 3.2 will be an immediate consequence of the following estimate, which in turn is deduced from Lemma 3.1.
Proof. Fix q ∈ 0, 2 (2+γ) 2 , chosen in a manner depending only on γ, and let Also fix β > 0 to be chosen later in a manner depending only on γ and suppose (D, f ) ∈ C C (ε β ). By analogy with (3.11), define . By the C-convexity of D, for any z ∈ H ε x ∩ D and any w ∈ H ε y ∩ D, there is a path P z,w from z to w in D of Euclidean length at most C(diam(H ε x ) + diam(H ε y )). By the ε −β -Lipschitz continuity of ∇f , for each u ∈ P z,w we have and similarly with y in place of x. Therefore, Using the above estimate and the inequality (a + b) 2 ≤ 2(a 2 + b 2 ) and breaking up the sum over edges based on those edges which have a given vertex x as an endpoint, we obtain that on E ε 0 , with implicit constant depending only on C, where here we use that diam(H ε x ) ≤ ε q on E ε 0 . Since (D, f ) ∈ C C (ε β ), the function |∇f | 2 is 2ε −2β -Lipschitz and |∇f | 2 ∞ ≤ ε −2β . We can therefore apply Lemma 3.1 (with each of |∇f | 2 and 1 in place of f ) to see that if β is smaller than the minimum of q and 1/2 times the parameter β from Lemma 3.1, then the following is true. For appropriate constants A, α > 0 as in the statement of the lemma, it holds except on an event of probability decaying faster than some positive power of ε that the right side of (3.14) is bounded above by A Energy(f ; D) + ε α . Since P[(E ε 0 ) c ] decays like a positive power of ε, this concludes the proof.
Proof of Theorems 1.3 and 3.2. Since discrete harmonic functions minimize Dirichlet energy subject to specified boundary data, Theorem 3.2 is an immediate consequence of Lemma 3.3. Theorem 1.3, in turn, follows from Theorem 3.2.

Green's function and recurrence
We will now explain why Theorem 3.2 implies Theorem 1.4. The main step is the following upper bound for the Dirichlet energy of certain discrete harmonic functions on G ε .
Proof of Lemma 3.4. To lighten notation, define the open annulus A s := B ρ (0) \ B s (0). We will apply Theorem 3.2 to the function g s : A s, → [0, 1] which is equal to 0 on ∂B ρ (0), 1 on ∂B s (0), and is harmonic on the interior of A s . That is, g s (z) = log(ρ/|z|)/ log(ρ/s). A direct calculation shows that the Euclidean Dirichlet energy of g s on A s is 2π/ log(ρ/s). Furthermore, g s and each of its first and second order partial derivatives are bounded above by a universal constant times a universal 25 negative power of s on A s . Consequently, Theorem 3.2 implies that there exists β = β(γ) > 0 and an appropriate choice of α and A as in the statement of the lemma such that the statement of the lemma is true if we impose the additional requirement that s ≥ ε β .
To remove the restriction that s ≥ ε β , we extend f ε s to all of VG ε (B ρ (0)) by requiring it to be identically equal to 1 on VG ε (B s (0)). Then the total Dirichlet energy of f ε s is unchanged and if s > s, then f ε s and f ε s agree on VG ε (A s ). Since f ε s has the minimal Dirichlet energy among all functions on VG ε (A s ) with the same boundary data, we infer that s → Energy(f ε s ; G ε (A s )) is non-decreasing. Therefore, (3.15) for s = ε β implies (3.15) for s ∈ [0, ε β ] with A/β in place of A.
To deduce the recurrence of simple random walk on G ε from this bound, it suffices to consider the case when ε = 1 since the law of G ε (as a graph) does not depend on ε. We will use the scaling property of the γ-quantum cone (described just below) to produce an increasing sequence of sub-graphs of G 1 (each corresponding to an open ball of random radius), whose union is all of G 1 , with the property that the Green's function of the walk stopped upon exiting these subgraphs a.s. tends to ∞, which implies recurrence by a well-known criterion [LP16, Theorem 2.3]. For this purpose, for b > 0 let R b := sup{r > 0 : h r (0) + Q log r = 1 γ log b}, where h r (0) denotes the circle average. Note that R 0 = 1 since h is assumed to have the circle average embedding. By [DMS14,Proposition 4.13 where Q = 2/γ + γ/2 is as in (2.6). It is easily seen from the definition of h (Definition 2.3) that a.s. R b → ∞ as b → ∞. Since η is sampled independently from h and then parameterized by γ-LQG mass with respect to h, it follows that ( 0)). Applying this with b = 2 k for k ∈ N, using (1.8) with ε = 2 −k , and applying the Borel-Cantelli lemma now gives the desired recurrence.

Random walk on G ε stays close to a curve with positive probability
In this subsection, we will prove Proposition 3.6, which says that, roughly speaking, the simple random walk on G ε has positive probability to stay close to a fixed Euclidean curve for a long time, even if we condition on G ε . This estimate is the key input in the proof of our modulus of continuity bound in Section 3.5 below, but we expect it to also have other applications.
Definition 3.5. For x ∈ VG ε , we write P ε x for the conditional law given (h, η) (which determines G ε ) of the simple random walk X ε on G ε started from x.
(3.16) Then with probability at least 1 − N O ε (ε α ) (at a rate depending only on ρ and γ), min x∈VG ε (B r (P )) min z∈B r (P ) P ε x [X ε enters VG ε (B r (z)) before leaving VG ε (B r (P ))] ≥ 2 −N . (3.17) We note that Proposition 3.6 is not implied by the quenched convergence of X ε to Brownian motion modulo time parameterization (proven in [GMS17, Theorem 3.4]) since the latter convergence does not give a quantitative bound for the annealed probability that (3.17) holds.
To prove Proposition 3.6, we will show that a simple random walk on G ε started close to the inner boundary of a Euclidean annulus is likely to hit the inner boundary before the outer boundary (Lemma 3.8). This leads to Proposition 3.6 by considering N such annuli with the property that the union of their inner boundaries contains a path from η(x) to z. See Figure 7 for an illustration of the proof.
Br(P ) Figure 7: Left: Illustration of the proof of Lemma 3.8. We use Lemma 3.7 and Theorem 3.2 to find a circle P 1,ε s separating ∂B 5sr (z) and ∂B 10sr (z) and a radial line segment P 2,ε s from ∂B sr (z) to ∂B 10sr (z) over which the total variation of f ε s is at most a constant times 1/ log s −1 . By the maximum principle, this gives us a bound for the maximum value of f ε s on B 5sr (z) \ B sr (z). Right: Illustration of the proof of Proposition 3.6. We cover B r (P ) by N balls of radius s 0 (r − r), then use Lemma 3.8 to force a random walk to follow a "string" of such balls from η(x) to z.
Our desired bound for the probability of exiting an annulus at a point of its inner boundary can be re-phrased as a pointwise bound for a certain discrete harmonic function on G ε . The following technical lemma (which is a variant of [GMS18, Lemma 2.16]) enables us to transfer from the Dirichlet energy bounds of Section 3.2 to the needed pointwise bounds. The idea of the statement and proof of the lemma is to consider a collection of paths {P t } t∈ [a,b] , indexed by some finite interval, with the property that the Euclidean distance between P s and P t is bounded below by |s − t|. Due to the Cauchy-Schwarz inequality, the t-average of the total variation of a function on VG ε over the paths P t can be bounded above in terms of the Dirichlet energy of f ε . There must be one path P t over which the total variation of f ε is smaller than average, which (together with the maximum principle) will allow us to prove pointwise bounds for harmonic functions on G ε in Lemma 3.8 below.
Lemma 3.7. There exists α = α(γ) > 0 such that for each C ≥ 1 and each ρ ∈ (0, 1), we can find A = A(C, ρ, γ) > 0 such that the following is true. Let D ⊂ B ρ (0) be a domain such that area(B r (∂D)) ≤ Cr for each r ∈ (0, 1). For ε ∈ (0, 1), it holds with probability at least 1 − O ε (ε α ), at a rate depending only on C, ρ, and γ, that the following holds. Let {P t } t∈[a,b] be a collection of compact subsets of D such that dist(P s , P t ) ≥ C −1 |s − t| for each s, t ∈ [a, b]. Then for each function (3.18) Proof. By Lemma 3.1 (applied with f = 1 D ) we can find α = α(γ) > 0 and A 0 = A 0 (C, ρ, γ) > 0 such that with probability 1 − O ε (ε α ), and an edge {x, y} ∈ EG ε (D), let M ε (x, y) be the Lebesgue measure of the set of t ∈ [a, b] for which {x, y} ∈ EG ε (P t ). By interchanging the order of integration and summation, for any f ε : (3.20) Since the cells H ε x and H ε y intersect whenever {x, y} ∈ EG ε , our hypothesis on the paths P t implies that if {x, y} ∈ EG ε (P t ), then {x, y} / ∈ EG ε (P s ) whenever |s − t| ≥ C diam(H ε x ∪ H ε y ). Therefore, with universal implicit constants. Thus (3.18) holds for an appropriate choice of A.
The following lemma says that the random walk on G ε started close to the inner boundary of a Euclidean annulus is likely to hit the inner boundary before the outer boundary. The lemma will be a consequence of Lemma 3.7 applied to the discrete harmonic function which equals 0 on the inner boundary and 1 on the outer boundary.
We observe that for each t ∈ [0, sr], each of P 1 s,t and P 2 s,t is contained in A sr,10sr (z). Furthermore, there is a universal constant C > 0 such that for t, t ∈ [0, sr] and i ∈ {1, 2} we have dist(P i s,t , P i s,t ) ≥ C −1 |t − t |.
Step 3: existence of good paths. If (3.25) holds for each i ∈ {1, 2}, then since the average of any function is bounded below by its minimum value, there must exist t ε i ∈ [0, sr] such that with P i,ε s := P ε (3.26) The union P 1,ε s ∪ P 2,ε s is connected, intersects ∂B sr (z), and disconnects A sr,5sr (z) from ∞. Since f ε s vanishes on VG ε (∂B sr (z)), we infer from (3.26) and the maximum principle for the discrete harmonic function f ε s that with probability at least 1 − O ε (ε α 1 ), 29 which by the definition of f ε s implies min x∈VG ε (A 5sr (z)) P ε x [X ε hits VG ε (∂B sr (z)) before VG ε (∂B r (z))] ≥ 1 − 2A 2 log s −1 . (3.27) The statement of the lemma with α slightly smaller than α 1 , β = α 1 /2, and A = 4A 2 follows by applying (3.27) for a collection of O ε (log ε −1 ) different values of s ∈ [ε β , 1/10], chosen so that each interval [s, 4s] for s ∈ [ε β , 1/4] is contained in [s , 5s ] for some s in this collection, then taking a union bound (note that this last step is why we used 5s instead of 4s above).
Proof of Proposition 3.6. We will iteratively apply Lemma 3.8 and the Markov property of the random walk to a collection of balls which cover B r (P ). Let α = α(γ) > 0, β = β(γ) > 0, and A = A(ρ, γ) > 0 be as in Lemma 3.8 and let s 0 ∈ (0, 1/10] be chosen so that such that A/ log s −1 ≤ 1/2. To lighten notation, set r := s 0 (r − r). By (3.16), for each P, ε, r, r as in the statement of the lemma we can find a finite deterministic set W ⊂ B r+r (P ) such that #W ≤ N and B r (P ) ⊂ w∈W B r (w). (3.28) By Lemma 3.8 (applied with r − r in place of r) and a union bound over all w ∈ W , it holds with probability at least 1 Suppose now that (3.29) holds. By (3.28), for each x ∈ VG ε (B r (P )) and each z ∈ B r (P ) we can find distinct points w 0 , . . . , w m ∈ W such that H ε x ∩ B r (w 0 ) = ∅, z ∈ B r (w m ), and B r (w k−1 ) ∩ B r (w k ) = ∅ for each k ∈ [1, m] Z . Since s 0 ≤ 1/10 and each w ∈ W lies in B r+r (P ), each ball B 4r (w k ) for k ∈ [1, m] Z is contained in B r (P ). Moreover, each B r (w k−1 ) is contained in B 4r (w k ). By m applications of (3.29) and the Markov property of X ε , it holds with P ε x -probability at least 2 −m ≥ 2 −N that X ε enters each VG ε (B r (w k )) before leaving VG ε (B r (P )), in which case X ε enters VG ε (B r (z)) ⊃ VG ε (B r (w m )) before leaving VG ε (B r (P )). Thus (3.17) holds.

Hölder continuity for harmonic functions on G ε
In this subsection, we use Proposition 3.6 to deduce a uniform (ε-independent) Hölder continuity estimate for harmonic functions on G ε , which is a more quantitative version of Theorem 1.5 Theorem 3.9. For each ρ ∈ (0, 1), there exists α = α(γ) > 0, ξ = ξ(ρ, γ) > 0, and A = A(ρ, γ) > 0 such that for each ε ∈ (0, 1), the following holds with probability at least be a connected domain and let f ε : VG ε (D) → R be discrete harmonic on VG ε (D) \ VG ε (∂D). Then we have the interior continuity estimate where f ε ∞ denotes the L ∞ norm. If D is simply connected, then we also have the boundary continuity estimate min y∈VG ε (∂D∩Bt(η(x))) In particular, if f ε has Hölder continuous boundary data, in the sense that there exists χ ∈ (0, 1] and C > 0 such that We note that (3.33) immediately implies Theorem 1.5. The basic idea of the proof of Theorem 3.9 is to bound the total variation distance between the conditional laws given (h, η) of the positions where the simple random walks on G ε started at two nearby vertices of VG ε (D) first hit VG ε (∂D). This is sufficient to establish a modulus of continuity bound since we are working with discrete harmonic functions. Our bound for total variation distance will be established using the following lemma, which is an easy consequence of Wilson's algorithm (see [GMS18,Lemma 3.12] for a proof).
Lemma 3.10 ( [GMS18]). Let G be a connected graph and let A ⊂ V(G) be a set such that the simple random walk started from any vertex of G a.s. hits A in finite time. For x ∈ V(G), let X x be the simple random walk started from x and let τ x be the first time X x hits A. For x, y ∈ V(G) \ A, where d TV denotes the total variation distance.
Proof of Theorem 3.9. Let α, β, and p be chosen so that the conclusion of Lemma 3.11 is satisfied for ε > 0. We can assume without loss of generality that β < 1 50(2+γ) 2 , so that Lemma 2.7 applies with q = 100β. Let E ε be the event that (3.35) holds for each z ∈ B ρ (0) and each r ∈ [ε β , ρ] with B 2r (z) ⊂ B ρ (0) and that diam(H ε x ) ≤ ε 100β for each x ∈ VG ε (B ρ (0)), so that by Lemma 3.11 and Lemma 2.7, after possibly shrinking α we can arrange that P[E ε ] = 1 − O ε (ε α ). Throughout the proof we assume that E ε occurs and we let f ε : VG ε (D) → R be a discrete harmonic function as in the theorem statement.
Applying the Markov property of the walk X ε and the estimate (3.35) with r = e −k for each k ∈ [log dist(η(x), ∂D) −1 , log (2(ε β ∨ |η(x) − η(y)|)) −1 ] Z , then multiplying over all such k, shows that for each x, y ∈ VG ε (D), P ε x [X ε disconnects y from VG ε (∂D) before hitting VG ε (∂D)] for constants A > 0 and ξ > 0 depending only on p and β (and hence only on ρ and γ). Note that we have absorbed β into ξ. By (3.37) and Lemma 3.10, we find that the total variation distance between the P ε x -law of the first place where X ε hits VG ε (∂D) and the P ε y -law of the first place where X ε hits VG ε (∂D) is at most the right side of (3.37). Since f ε is discrete harmonic, this implies (3.30). Now assume that D is simply connected. To prove the boundary estimate (3.31), we observe that if x ∈ VG ε (D) and r ≥ 2 dist(η(x), ∂D), then since C \ D is connected, in order for a random walk started from x to disconnect VG ε (B r/2 (0)) from ∞, it must first hit VG ε (∂D). By applying the Markov property of X ε and (3.35) with r = e −k for k ∈ log t −1 , log 2 dist(η(x), ∂D) −1 , we get that for t > 2 dist(η(x), ∂D), P ε x [X ε first hits VG ε (∂D) at a point in VG ε (∂D ∩ B t (η(x)))] for a possibly larger choice of A and smaller choice of ξ. Again using that f ε is discrete harmonic, we obtain (3.31) (the when case t < 2 dist(η(x), ∂D) can be dealt with by increasing A).
4 Estimates for the area, diameter, and degree of a cell The goal of this section is to prove Proposition 2.10. Throughout most of this section, we restrict attention to the case when h is either the circle-average embedding of a 0-quantum cone or a whole-plane GFF normalized so that h 1 (0) = 0. Note that the structure graph corresponding to such a choice of h is not the same as the mated-CRT map. We will transfer to the γ-quantum cone case, which corresponds to the mated-CRT map, in Section 4.6. Throughout, we define the cell H ε x ε z containing a fixed point z ∈ C as in Section 1.4 and we define u ε (z) as in (2.12).
Most of this section is devoted to the proof of the following variant of Proposition 2.10 for a 0-quantum cone or whole-plane GFF which does not assume any continuity conditions for f or D. Proposition 2.10 (which we recall is a statement about the γ-quantum cone) will be deduced from this proposition and an absolute continuity argument in Section 4.6.
The reason why we first prove the statement for the 0-quantum cone given in Proposition 4.1 is as follows. For a 0-quantum cone, the origin is a "Lebesgue typical point"; in particular, there is no log singularity. Hence, estimates for u ε (0) can be transferred to estimates for u ε (z) for a deterministic point z ∈ B ρ (0) (or a point sampled uniformly from Lebesgue measure on B ρ (0)); see Lemma 4.9. This will allow us to apply the bounds for diam 2 (H ε 0 )/ area(H ε 0 ) and deg(H ε 0 ) in the case of the 0-quantum cone from [GMS17, Section 4] to estimate the integral appearing in (4.1). At one point in the proof of Proposition 4.1 (in particular, Lemma 4.6), we will need to prove an estimate for the γ-quantum cone, then transfer to the 0-quantum cone. The reason for this is that we want to use the degree bound of Lemma 2.5, which is proven using the Brownian motion definition of the mated-CRT map.

Outline of the proof
Here we give an outline of the content of the rest of this section. Throughout this section, we will work with "localized" versions of diam(H ε The reason why we need things to depend locally on h and η is that in Section 4.5, we will use long-range independence estimates for SLE and the GFF to get a second moment bound for the integral appearing in Proposition 4.1. The localized quantities appearing in (4.3) are defined in Section 4.2 and illustrated in Figure 8. The quantity DA ε (z) is the maximum of the ratio of square diameter to area over all ε-length segments of η contained in η([τ z − ε, τ z + ε]) (one of which is equal to H ε x ε z ). The localized degree is split into two parts: the "inner degree" deg ε in (z), which counts the number of ε-length segments of η contained in the ball centered at z with radius 4 diam(η([τ z − ε, τ z + ε])) (here we use the notation (4.2)); and the "outer degree" deg ε out (z), which counts the number of segments of η which intersect both η([τ z − ε, τ z + ε]) and the boundary of this ball.
In Section 4.3, we state ε-independent moment bounds for the above three quantities which were proven in [GMS17].
In Section 4.4, we prove a global regularity estimate (Proposition 4.5) which bounds the maximum over all z ∈ B ρ (0) of the localized versions of diam(H ε x ε z ) and deg(x ε z ; G ε ). The bound for diam(H ε x ε z ) follows from Lemma 2.7, but the bound for the localized degree will take a bit more work since Lemma 2.6 only provides a bound for the non-localized degree and only applies in the γ-quantum cone case.
In Section 4.5, we prove Proposition 4.1 by, roughly speaking, using the moment bounds of Section 4.3 to bound the expectation of D f (z)u ε (z) dz and using long-range independence results for the Gaussian free field from Appendix A.2 to show that the variance of D f (z)u ε (z) dz decays like a positive power of ε. The fact that we use long-range independence for the GFF is the reason why we need to replace u ε (z) by a localized version.
In Section 4.6, we deduce Proposition 2.10 from Proposition 4.1 using the fact that sampling a point z uniformly from the γ-LQG measure on some domain and re-centering so that z is mapped to 0 produces a field with a γ-log singularity at 0, which locally looks like a γ-quantum cone.

Localized versions of area, diameter, and degree
Suppose we are in the setting of Section 1.4 with h equal to the circle-average embedding of a 0-quantum cone or a γ-quantum cone, or a whole-plane GFF normalized so that h 1 (0) = 0. In this subsection we will define modified versions of the quantities diam(H ε x ε z ) 2 / area(H ε x ε z ) and deg(x ε z ; G ε ) appearing in the definition (2.12) of u ε (z) which are locally determined by h and η-in the sense of Lemma 4.3 just below-and which we will work with throughout most of this subsection (recall from Remark 4.2 that the original quantities are not locally determined by h and η). The definitions are illustrated in Figure 8.
We start with the ratio of squared diameter to area. For z ∈ C, let with τ z as in (4.2). In words, DA ε (z) is a.s. equal to the maximum ratio of the squared diameter to the area over all of the segments of η with quantum mass ε which contain z, so using DA ε (z) instead of diam(H ε x ε z ) 2 / area(H ε x ε z ) removes the arbitrariness coming from the choice to define G ε using elements of εZ rather than εZ + t for some t ∈ (0, ε). By definition, the cell H ε x ε z is one of these ε-length segments of η, so diam(H ε with the implicit constants depending only on p. Proof. The bounds (4.8), (4.9), and (4.10) in the case ε = 1 follow from [GMS17, Propositions 4.4 and 4.5]. We will now use the scaling property of the 0-quantum cone to argue that the laws of DA ε (0), deg ε in (0), and deg ε out (0) do not depend on ε. To this end, let R ε := sup r > 0 : h r (0) + Q log r = 1 γ log ε , where h r (0) denotes the circle average, and let h ε := h(R ε ·) + Q log R ε − 1 γ log ε and η ε (t) := η(εt)/R ε .
By [DMS14, Proposition 4.13(i)], h ε d = h. By the γ-LQG coordinate change formula (2.6), µ h (X) = εµ h ε (X/R ε ) for each Borel set X ⊂ C, hence η ε is parameterized by γ-quantum mass with respect to h ε . From this and the scale invariance of the law of space-filling SLE κ 4 , we get (h ε , η ε ) d = (h, η). On the other hand, since the definitions of DA ε (0), deg ε in (0), and deg ε out (0) are unaffected by spatial scaling, DA ε (0) is determined by (h ε , η ε ) in the same manner that DA 1 (0) is determined by (h, η), and similarly for deg ε in (0) and deg ε out (0). Hence the laws of these three quantities do not depend on ε and the theorem statement follows.

Global regularity event for area, diameter, and degree
Theorem 4.4 and the translation invariance of the law of the whole-plane GFF modulo additive constant allow us to control the quantities DA ε (z), deg ε in (z), and deg ε out (z) for one point z at a time (see Lemma 4.9), but to control various error terms in our estimates we will also need global regularity bounds for these quantities which hold for all points z in a fixed Euclidean ball simultaneously. One does not expect ε-independent global bounds (since the number of cells in a fixed ball tends to ∞ as ε → 0) but the following proposition will be sufficient for our purposes.
2. The circle average process of h satisfies and the same is true with h replaced by the whole-plane GFF h IG used to construct η in Section 2.1.3 (when it is normalized so that h IG 1 (0) = 0).
The hardest part of the proof of Proposition 4.5 is the upper bound for the localized degree (as defined just after (4.6)) which we treat in the following two lemmas. We note that the needed bound is not immediate from Lemma 2.6 since we are working with a 0-quantum cone rather than a γ-quantum cone and we need a bound for localized degree, rather than un-localized degree. We first consider the inner localized degree, in which case we get a polylogarithmic upper bound with extremely high probability thanks to Lemma 2.6.
Lemma 4.6. Suppose we are in the setting of Proposition 4.5. For ζ ∈ (0, 1), (4.12) Proof. Here we want to apply Lemma 2.6 (which is proven using the Brownian motion definition of the mated-CRT map), so we first consider the case when h is the R-circle-average embedding of a γ-quantum cone in (C, z 0 , ∞) for some R > 0 and z 0 ∈ C. This means that h is normalized so that R is the largest radius r > 0 for which h r (z 0 ) + Q log r = 0 and h| B R (z 0 ) agrees in law with the corresponding restriction of a whole-plane GFF plus −γ log | · −z 0 |, normalized so that its circle average over ∂B R (z 0 ) is 0. From here until (4.15), we work in this setting and define the objects involved in Proposition 4.5 with this choice of h (note that the estimates of Section 4.3 do not apply in this setting due to the log singularity of h at z 0 , but we will not need these estimates here). By Lemma 2.6 (applied to the structure graph generated from the field h(R · +z 0 ) + Q log R, which is the circle average embedding of a γ-quantum cone in (C, 0, ∞), and the SLE κ curve R −1 η − z 0 , which is parametrized by γ-LQG mass w.r.t. this field), (4.13) In fact, the proof of Lemma 2.6 shows that also P max where here DA ε (z), deg ε in (z), and deg ε out are defined as in Section 4.2. We note that (4.5) and (4.7) together imply that u ε (z) ≤ 2u ε * (z). (4.20) The remainder of this subsection is devoted to the proof of the following proposition, which (by (4.20)) immediately implies Proposition 4.1.
Proposition 4.8. The statement of Proposition 4.1 is true with u ε * (z) in place of u ε (z).
Fix ρ ∈ (0, 1). Also fix q ∈ 0, min 2 (2+γ) 2 , γ and ζ ∈ (0, q/100), (4.21) chosen in a manner depending only on γ, and define the regularity event E ε = E ε (ρ, q, ζ) as in Proposition 4.5 for this choice of ρ, q, and ζ. We note that the particular choice of q and ζ satisfying (4.21) does not matter for the proof. The idea of the proof of Proposition 4.1 is to show that (roughly speaking) the variance of D f (z)u ε * (z) dz on E ε decays like a positive power of ε using the local independence result Lemma A.3; and bound the expectation of this integral on E ε using Theorem 4.4.
Since we will be using long-range independence for (h, h IG ), we need to consider a localized version of the event E ε . To this end, for z ∈ C and ε ∈ (0, 1), let F ε (z) be the event that the following is true.

40
The larger range of possible values of p in (4.24) as compared to (4.25) is due to the difference in the range of possible p values in Theorem 4.4.
Proof of Lemma 4.9. We will prove (4.25); the estimate (4.24) is proven in an identical manner except that we only need to use (4.8) instead of all three of the estimates of Theorem 4.4.
Fix a 1 , a 2 ∈ [−2q log ε −1 , 2q log ε −1 ]. To prove (4.25), we will condition on {h ε q−4ζ (z) = a 1 , h IG ε q−4ζ (z) = a 2 }, apply an affine transformation sending B ε q−4ζ (z) to D and re-normalize to get a new pair of fields with the same law as (h, h IG )| D , then apply Theorem 4.4 to this new field/curve pair with ε replaced by a larger (a 1 -dependent) value determined by the γ-LQG coordinate change formula. Note that we can restrict attention to this range of a 1 , a 2 due to condition 3 in the definition of F ε (z).
By Lemma A.2 (applied to each of the independent fields h and h IG and with δ = ε q−4ζ ) there exists ε * ∈ (0, 1) as in the statement of the lemma such that for ε ∈ (0, ε * ], the conditional law of (h, h IG )| B ε q−4ζ (z) given {h ε q−4ζ (z) = a 1 , h IG ε q−4ζ (z) = a 2 } is absolutely continuous with respect to the law of the restriction to B ε q−4ζ (z) of a pair (h a 1 , h IG,a 2 ) of independent whole-plane GFFs normalized to have circle averages a 1 and a 2 , respectively, over ∂B ε q−4ζ (z). Furthermore, since a 1 , a 2 ∈ [−2q log ε −1 , 2q log ε −1 ], for each p > 0 the pth moment of the Radon-Nikodym derivative M a 1 ,a 2 of Lemma A.2 is bounded above by a constant depending only on ρ and p.
Let X a 1 ,a 2 be the random variable which is determined by (h a 1 , h IG,a 2 )| B ε q−4ζ (z) in the same manner that 1 F ε (z) u ε * (z) is determined by (h, h IG )| B ε q−4ζ (z) (c.f. (4.22)). The preceding paragraph together with Hölder's inequality shows that for ε ∈ (0, ε * ] and p, p ∈ (1, 4/γ 2 ) with p < p , with the implicit constant depending only on ρ, p, and p . We now estimate the right side of (4.26) using Theorem 4.4. If we compose (h a 1 , h IG,a 2 ) with an affine transformation which takes D to B ε q−4ζ (z) and subtract (a 1 , a 2 ), we obtain a new pair of fields ( h, h IG ) with the same law as (h, h IG )| D . Let ε := e −γa 1 ε 1−γQ(q−4ζ) , recall the ball B ε 0 from (4.6), and let X a 1 ,a 2 be the random variable which is determined by ( h, h IG ) in the same manner that u ε * (0)1 B ε 0 ⊂D is determined by (h, h IG )| D (c.f. Lemma 4.3).
Using the γ-LQG coordinate change formula (2.6), we find that u ε * (z) (resp. ε −(q−4ζ) (B ε z − z)) is determined by ( h, h IG ) in the same manner that u ε * (0) (resp. B ε 0 ) is determined by (h, h IG ). Since B ε z ⊂ B ε q−4ζ (z) on F ε (z), we infer that a.s. X a 1 ,a 2 ≤ X a 1 ,a 2 . Since the h| D agrees in law with the corresponding restriction of the circle-average embedding of a 0-quantum cone, we can apply Theorem 4.4 with ε in place of ε to get E X p a 1 ,a 2 ≤ E X p a 1 ,a 2 1 for each p ∈ (1, 4/γ 2 ). Combining this with (4.26) concludes the proof.
Proof of Proposition 4.8. For most of the proof we consider the case of a whole-plane GFF normalized so that h 1 (0) = 0; we transfer to the case of a 0-quantum cone only at the very end.
Step 1: the integral over W ε 1 . Let z, w ∈ B ρ (0). Using the definition (4.28) of U ε * (z), we make the following calculation, each line of which we justify just below.
Lemma A.5. Fix R > 1 and w ∈ C such that D ⊂ B R (w). Let h and h be whole-plane GFFs, normalized so that their circle averages over ∂B R (w) are 0, coupled together so that the circle averages h 1 (0) and h 1 (0) agree and h and h are conditionally independent given these circle averages (note that the circle averages of our GFFs over both ∂B R (w) and ∂D agree). For δ ∈ (0, 1), the conditional law of h| B δ (0) given h| C\D is a.s. absolutely continuous with respect to the conditional law of h| B δ (0) given h 1 (0). Let M δ = M δ h| C\D , h| B δ (0) be the Radon-Nikodym derivative. There is a universal constant a ∈ (0, 1) such that for δ ∈ (0, a], E M a/δ δ 1 and E M −a/δ δ 1 (A.7) with universal implicit constants.
Proof. Since ∂B R (w) ∩ D = ∅, by Lemma 2.2 (applied with h(R · +w) in place of h) we can write h| D = h +h, where h is a random harmonic function on D which is determined by h| C\D andh is a zero-boundary GFF on D which is independent from h| C\D . Decompose h| D = h +h analogously. and 0 on C \ B 2 (0). Let φ δ (z) := φ 1 (z/δ), so that φ is supported on B 2δ (0) and is identically equal to 1 on B δ (0). Also let g δ := (h − h)φ δ . If we condition on (h, h)| C\D , then the proof of [MS16d, Proposition 3.4] shows that the conditional laws of h| B δ (0) and h| B δ (0) are mutually absolutely continuous, and the Radon-Nikodym derivative of the former with respect to the latter is given by where (·, ·) ∇ denotes the Dirichlet inner product. Averaging over the possible realizations of h shows that that the Radon-Nikodym derivative of the conditional law of h| B δ (0) given h| C\D with respect to the conditional law of h| B δ (0) given h| C\D is equal to Note that M δ is also the Radon-Nikodym derivative of the conditional law of h| B δ (0) given h| C\D with respect to the conditional law of h| B δ (0) given h 1 (0) since h| C\D determines h 1 (0) = h 1 (0) and h is conditionally independent from h given h 1 (0). We now estimate M δ . By Jensen's inequality, for δ ∈ (0, a], If we condition on h and h, the conditional law of (h, g δ ) ∇ is centered Gaussian with variance (g δ , g δ ) ∇ (note that (h, g δ ) ∇ = 0 since h is harmonic in D and g δ is compactly supported in D).
In order to apply Lemma A.5, we let h = (h 1 , . . . , h N ) be another N -tuple of independent GFFs with the same law as h, coupled together with h in such a way that h 1 (0) = h 1 (0) and h and h are conditionally independent given this circle average. Let X = X(h| B δ (0) , h 1 (0)) be determined by h| B δ (0) and h 1 (0) in the same manner that X is determined by h| B δ (0) and h 1 (0). is the Radon-Nikodym derivative of the conditional law of h| B δ (0) given h| C\D w.r.t. the conditional law of h| B δ (0) given h 1 (0) (equivalently, by conditional independence, w.r.t. the conditional law of h B δ (0) given h| C\D ). Hence where in the last line we use that the conditional law of X given h| C\D is the same as the conditional law of X given h 1 (0) (by our choice of coupling). Similarly, (A.14) By (A.13), (A.14), and Jensen's inequality,