Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) $h$, there is a unique random metric $D_h = ``e^{\gamma h} (dx^2 + dy^2)"$ on $\mathbb C$ which is characterized by a certain list of axioms: it is locally determined by $h$ and it transforms appropriately when either adding a continuous function to $h$ or applying a conformal automorphism of $\mathbb C$ (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of \emph{Liouville first passage percolation} (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding, Dub\'edat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $\gamma \in (0,2)$, we conjecture that our metric is the Gromov-Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.


Overview
Fix γ ∈ (0, 2), let U ⊂ C be an open domain, and let h be the Gaussian free field (GFF) on U , or some minor variant thereof. The γ-Liouville quantum gravity (LQG) surface described by (U, h) is formally the random two-dimensional Riemannian manifold with metric tensor e γh (dx 2 + dy 2 ), (1.1) where dx 2 + dy 2 is the Euclidean Riemannian metric tensor. LQG surfaces were first introduced non-rigorously in the physics literature by Polyakov [Pol81a,Pol81b] as a canonical model of a random Riemannian metric on U . Another motivation to study LQG surfaces is that they describe the scaling limit of random planar maps. The special case when γ = 8/3 (called "pure gravity") corresponds to uniformly random planar maps, including uniform triangulations, quadrangulations, etc. Other values of γ (sometimes referred to as "gravity coupled to matter") correspond to random planar maps weighted by the partition function of an appropriate statistical mechanics model on the map, for example the uniform spanning tree for γ = √ 2 or the Ising model for γ = √ 3. The definition (1.1) of LQG does not make literal sense since h is only a distribution, not a function, so it does not have well-defined pointwise values and cannot be exponentiated. Nevertheless, it is known that one can make sense of the associated volume form µ h = e γh(z) dz (where dz denotes Lebesgue measure) as a random measure on U via various regularization procedures [Kah85,DS11,RV14]. One such regularization procedure is as follows. For s > 0 and z, w ∈ C, let p s (z, w) = 1 2πs exp − |z−w| 2 2s be the heat kernel, and note that p s (z, ·) approximates a point mass at z when s is small. For ε > 0, we define a mollified version of the GFF by h * ε (z) := (h * p ε 2 /2 )(z) = U h(w)p ε 2 /2 (z, w) dw, ∀z ∈ U, where the integral is interpreted in the sense of distributional pairing. One can then define the γ-LQG measure µ h as the a.s. weak limit [Kah85,DS11,RV14,Ber17,Sha16] lim ε→0 ε γ 2 /2 e γh * ε (z) dz.  4). We think of two equivalent pairs as representing different parameterizations of the same random surface. The conformal covariance property of µ h says that this measure is intrinsic to the quantum surface -it does not depend on the particular equivalence class representative.
In order for γ-LQG to be a reasonable model of a "random two-dimensional Riemannian manifold", one also needs a random metric D h on U which is in some sense obtained by exponentiating h and which satisfies a conformal covariance property analogous to that of the γ-LQG area measure. Moreover, this metric should be the scaling limit of the graph distance on random planar maps with respect to the Gromov-Hausdorff topology. Constructing a metric on γ-LQG is a much more difficult problem than constructing the measure µ h . Indeed, any natural regularization schemes for LQG distances involves minimizing over a large collection of paths, which results in a substantial degree of non-linearity.
Prior to this work, a γ-LQG metric has only been constructed in the special case when γ = 8/3 in a series of works by Miller and Sheffield [MS15b,MS16b,MS16c]. In this case, for certain special choices of the pair (U, h), the random metric space (U, D h ) agrees in law with a Brownian surface, such as the Brownian map [Le 13, Mie13] or the Brownian disk [BM17]. These Brownian surfaces are continuum random metric spaces which arise as the scaling limits of uniform random planar maps with respect to the Gromov-Hausdorff topology. Miller and Sheffield's construction of the 8/3-LQG metric does not use a direct regularization of the field h. Instead, they first construct a candidate for 8/3-LQG metric balls using a process called quantum Loewner evolution, which is built out of the Schramm-Loewner evolution with parameter κ = 6 (SLE 6 ), then show that there is a metric with the desired balls.
In this paper, we will construct a γ-LQG metric for all γ ∈ (0, 2) via an explicit regularization procedure analogous to (1.3). We will also show that this metric is uniquely characterized by a list of natural properties that any reasonable notion of a metric on γ-LQG should satisfy, so is in some sense the only "correct" metric on γ-LQG. For simplicity, we will mostly restrict attention to the whole-plane case, but metrics associated with GFF's on other domains can be easily constructed via restriction and/or absolute continuity (see Remark 1.5). In contrast to [MS15b,MS16b,MS16c], the present work will make no use of SLE. Furthermore, we do not a priori have an ambient metric space to compare to (such as the Brownian map in the case γ = 8/3) and we do not have any sort of exact solvability, i.e., we do not know the exact laws of any observables related to the metric.
We now describe how our metric is constructed. It is shown in [DG18], building on [DZZ18,GHS17], that for each γ ∈ (0, 2), there is an exponent d γ > 2 which describes distances in various discrete approximations of γ-LQG. A posteriori, once the γ-LQG metric is constructed, one can show that d γ is its Hausdorff dimension [GP19b]. The value of d γ is not known explicitly except in the case when γ = 8/3, in which case we know that d √ 8/3 = 4 (see Problem 7.1). We refer to [DG18,GP19a] for 3 bounds for d γ and some speculation about its possible value. For γ ∈ (0, 2), we define ξ = ξ γ := γ d γ .
(1. 5) We say that a random distribution h on C is a whole plane GFF plus a continuous function if there exists a coupling of h with a random continuous function f : C → R such that the law of h − f is that of a whole-plane GFF. We similarly define a whole-plane GFF plus a bounded continuous function, except we require that f is bounded. 1 Note that the whole-plane GFF is defined only modulo a global additive constant, but these definitions does not depend on the choice of additive constant.
If h is a whole-plane GFF plus a bounded continuous function, we define h * ε (z) for ε > 0 and z ∈ C as in (1.2). For z, w ∈ C and ε > 0, we define the ε-LFPP metric by where the infimum is over all piecewise continuously differentiable paths from z to w. One should think of LFPP as the metric analog of the approximations of the LQG measure in (1.3). 2 The intuitive reason why we look at e ξh * ε (z) instead of e γh * ε (z) to define the metric is as follows. By (1.3), we can scale LQG areas by a factor of C > 0 by adding γ −1 log C to the field. By (1.6), this results in scaling distances by C ξ/γ = C 1/dγ , which is consistent with the fact that the "dimension" should be the exponent relating the scaling of areas and distances.
Let a ε be the median of the D ε h -distance between the left and right boundaries of the unit square in the case when h is a whole-plane GFF normalized so that its circle average 3 over ∂D is zero. We do not know the value of a ε explicitly, but see Corollary 1.11. It was shown by Ding, Dubédat, Dunlap, and Falconet [DDDF19] that the laws of the metrics a −1 ε D ε h are tight w.r.t. the local uniform topology on C × C, and every possible subsequential limit induces the Euclidean topology on C (see also the earlier tightness results for small γ > 0 [ DD19,DF18] and for Liouville graph distance, a related model, for all γ ∈ (0, 2) [DD18]). Subsequently, it was shown by Dubedát, Falconet, Gwynne, Pfeffer, and Sun [DFG + 19], using a general theorem from [GM19c], that every subsequential limit can be realized as a measurable function of h, so in fact the metrics a −1 ε D ε h admit subsequential limits in probability. One of the main results of this paper gives the uniqueness of this subsequential limit.
the local uniform topology on C × C to a random metric on C which is a.s. determined by h.
It is natural to define the limiting metric from Theorem 1.1 to be the γ-LQG metric associated with h. However, this definition is not entirely satisfactory since it is a priori possible that there are other natural ways to construct a metric on γ-LQG which do not yield the same result as the one in

Axiomatic characterization of the γ-LQG metric
To state our list of axioms precisely, we will need some preliminary definitions concerning metric spaces. In what follows, we let (X, D) be a metric space. where the supremum is over all partitions T : a = t 0 < · · · < t #T = b of [a, b]. Note that the D-length of a curve may be infinite. where the infimum is over all paths P in Y from x to y. Then D(·, ·; Y ) is a metric on Y , except that it is allowed to take infinite values.
We say that (X, D) is a length space if for each x, y ∈ X and each ε > 0, there exists a curve of D-length at most D(x, y) + ε from x to y.
A continuous metric on an open domain U ⊂ C is a metric D on U which induces the Euclidean topology on U , i.e., the identity map (U, | · |) → (U, D) is a homeomorphism. We equip the space of continuous metrics on U with the local uniform topology for functions from U × U to [0, ∞) and the associated Borel σ-algebra. We allow a continuous metric to satisfy D(u, v) = ∞ if u and v are in different connected components of U . In this case, in order to have D n → D w.r.t. the local uniform topology we require that for large enough n, D n (u, v) = ∞ if and only if D(u, v) = ∞.
Let D (C) be the space of distributions (generalized functions) on C, equipped with the usual weak topology. For γ ∈ (0, 2), a (strong) γ-Liouville quantum gravity (LQG) metric is a measurable 5 function h → D h from D (C) to the space of continuous metrics on C such that the following is true whenever h is a whole-plane GFF plus a continuous function.
I. Length space. Almost surely, (C, D h ) is a length space, i.e., the D h -distance between any two points of C is the infimum of the D h -lengths of D h -continuous paths (equivalently, Euclidean continuous paths) between the two points. IV. Coordinate change for translation and scaling. For each fixed deterministic r > 0 and z ∈ C, a.s. (1.9) Let us briefly discuss why the above axioms are natural. Recall that γ-LQG should be the random Riemannian metric with metric tensor e γh (dx 2 + dy 2 ). Axiom I is simply the LQG analog of the statement that for a true Riemannian metric, the distance between two points can be defined as the infima of the lengths of paths connecting them. In a similar vein, Axiom II corresponds to the fact that for a smooth Riemannian metric, the lengths of paths are determined locally by the Riemannian metric tensor. Axiom III is just expressing the fact that the metric is obtained by exponentiating ξh, so adding a continuous function f to h results in re-scaling the metric length measure on paths by e ξf .
Axiom IV is the metric analog of the conformal coordinate change formula (1.4) for the γ-LQG area measure, but restricted to translations and scalings. This axiom together with Corollary 1.3 says that D h depends only on the LQG surface (C, h), not on the particular choice of parameterization. We will prove a conformal covariance property for the γ-LQG metric w.r.t. conformal automorphisms between arbitrary domains, directly analogous to the conformal covariance of the γ-LQG area measure, in [GM19b].
Theorem 1.2 (Existence and uniqueness of the LQG metric). Fix γ ∈ (0, 2). There is a γ-LQG metric D such that the limiting metric of Theorem 1.1 is a.s. equal to D h whenever h is a whole-plane GFF plus a bounded continuous function. Furthermore, the γ-LQG metric is unique in the following sense. If D and D are two γ-LQG metrics, then there is a deterministic constant C > 0 such that if h is a whole-plane GFF plus a continuous function, then a.s. D h = C D h . Theorem 1.2 justifies us in referring to the γ-LQG metric. Technically speaking there is a one-parameter family of such metrics, which differ by a global deterministic multiplicative constant. But, one can fix the constant in various ways to get a single canonically defined metric. For example, we can require that the median distance between the left and right boundaries of the unit square is 1 for the metric associated with a whole-plane GFF normalized so that its circle average over ∂D is zero (the limiting metric in Theorem 1.1 has this normalization).

6
In Axiom IV in the definition of a strong γ-LQG metric, we did not require that the metric is invariant under rotations of C. It turns out that rotational invariance is implied by the other axioms. See Remark 1.6 below for an intuitive explanation of why this is the case. Corollary 1.3 (Rotational invariance). If γ ∈ (0, 2) and D is a γ-LQG metric then D is rotationally invariant, i.e., if ω ∈ C with |ω| = 1 and h is a whole-plane GFF plus a continuous function, then a.s. D h (u, v) = D h(ω·) (ω −1 u, ω −1 v) for all u, v ∈ C.
Proof. Define D (ω) h (u, v) := D h(ω·) (ω −1 u, ω −1 v). It is easily verified that D (ω) is a strong LQG metric, so Theorem 1.2 implies that there is a deterministic constant C > 0 such that a.s. D (ω) h = CD h whenever h is a whole-plane GFF plus a continuous function. To check that C = 1, consider a whole-plane GFF h normalized so that its circle average over ∂D is 0. Then the law of h is rotationally invariant, so P[D h (0, ∂D) > R] = P[D (ω) h (0, ∂D) > R] for every R > 0. Therefore C = 1.
define the metric D h in the case when h is a zero-boundary GFF plus a continuous function on U . It is shown in [GM19b] that this metric satisfies a conformal coordinate change relation analogous to the one satisfied by the γ-LQG measure (as discussed just below (1.4)).
Remark 1.6 (Why rotational invariance is unnecessary). At a first glance, it may seem surprising that one does not need rotational invariance to uniquely characterize the LQG metric in Theorem 1.2. Indeed, one can define variants of LFPP which are not rotationally invariant by working with a stretched version of the Euclidean metric. For example, for a given A > 1 one can replace (1.6) by D ε h,A (z, w) := inf P :z→w 1 0 e ξh * ε (P (t)) P 1 (t) 2 + AP 2 (t) 2 dt (1.11) where the infimum is over all piecewise continuously differentiable paths P = (P 1 , P 2 ) from z to w. The arguments of this paper and its predecessors apply verbatim with D ε h,A in place of D ε h . In particular, D ε h,A converges in probability to (a deterministic constant times) the γ-LQG metric and hence satisfies the rotational invariance property of Corollary 1.3. This is despite the fact that the metrics (1.11) do not satisfy this rotational invariance property.
Here is an intuitive explanation for this phenomenon. First, we note that D ε h,A is bi-Lipschitz equivalent with respect to D ε h,1 = D ε h for each ε > 0, with a deterministic bi-Lipschitz constants. Therefore in a subsequential limit as ε → 0, we obtain two metrics D h,A and D h = D h,1 which are bi-Lipschitz equivalent with deterministic bi-Lipschitz constants. Suppose that P is a D h -geodesic connecting z and w. Using the confluence of geodesics results from [GM19a], one can show that (very roughly speaking) for distinct times s, t ∈ [0, D h (z, w)], the restrictions of h to small neighborhoods of P (s) and P (t) are approximately independent; see the outline of Section 4 in Section 1.5 below for details. Moreover, since P is a fractal type curve, it has no local notion of direction, so one expects that the law of h restricted to a small neighborhood of P (t) does not depend very strongly on t or on the endpoints z, w of P . If we fix n ∈ N and let 0 = t 0 < · · · t n = D h (z, w) be equally spaced times, we can approximate the D h,A -length of P by n j=1 D h,A (P (t j−1 ), P (t j )).
The above considerations suggest that each of the random variables D h,A (P (t j−1 ), P (t j )) has approximately the same distribution and is bounded above and below by deterministic constants times t j − t j−1 . From law of large numbers type considerations, it follows that the D h,A -length of P is a deterministic constant times the D h -length of P , where the constant does not depend on the endpoints of P .
Knowing that the D h,A -length of every D h geodesic is a constant times its D h -length (and vice-versa) does not immediately imply that D h is equal to a constant times D h,A . This is because if P n is a sequence of paths which converge uniformly to P , then it is not necessarily true that len(P n ; D h,A ) converges to len(P ; D h,A ). For this and other reasons, we will argue in a somewhat different manner than we have indicated above, though our arguments will still be based on the bi-Lipschitz equivalence of metrics and approximate independence statements for the local behavior of a geodesic at different times. We will explain the general strategy in Section 1.5 in more detail.

Conjectured random planar map connection
As noted above, the γ-LQG metric should describe the large scale behavior of the graph metric for random planar maps. Since our γ-LQG metric is in some sense canonical, it is natural to make the following conjecture.
Conjecture 1.7. For each γ ∈ (0, 2), random planar maps in the γ-LQG universality class, equipped with their graph distance, converge in the scaling limit with respect to the Gromov-Hausdorff topology to γ-LQG surfaces equipped with the γ-LQG metric constructed in Theorem 1.1 (see also Remark 1.5).
For γ = 8/3, Conjecture 1.7 has already been proven for many different uniform-type random planar maps. The reason for this is that we know that our 8/3-LQG metric is equivalent to the metric of [MS15b,MS16b,MS16c] (Corollary 1.4); which in turn is equivalent to a Brownian surface, such as the Brownian map, for certain special 8/3-LQG surfaces [MS16b, Corollary 1.5]; which in turn is the scaling limit of uniform random planar maps of various types [Le 13, Mie13].
Conjecture 1.7 has not been proven for any random planar map model for γ = 8/3. However, we already have a relationship between the continuum LQG metric and graph distances in random planar maps at the level of exponents for all γ ∈ (0, 2). Indeed, the quantity d γ appearing in (1.5) describes several exponents associated with random planar maps, such as the ball volume exponent [GHS17,DG18] and the displacement exponent for simple random walk on the map [GM17,GH18]. It is proven in [GP19b] that d γ is the Hausdorff dimension of D h . Conjecture 1.7 can be made somewhat more precise by specifying exactly what type of γ-LQG surface should arise in the scaling limit. For random planar maps with the topology of the sphere (resp. disk, plane, half-plane) this surface should be the quantum sphere (resp. quantum disk, γ-quantum cone, γ-quantum wedge). See [DMS14] for precise definitions of these quantum surfaces. Equivalent definitions of the quantum sphere and quantum disk, respectively, can be found in [DKRV16,HRV18] (see [AHS17] for a proof of the equivalence in the sphere case). Some planar map models have been proven to converge to these quantum surfaces, for general γ ∈ (0, 2), with respect to topologies which do not encode the metric structure explicitly. Examples of such topologies include convergence in the so-called peanosphere sense [She16b,DMS14] and convergence of the counting measure on vertices to the γ-LQG measure when the planar map is embedded appropriately into the plane [GMS17].

Weak LQG metrics and a stronger uniqueness statement
We will prove Theorem 1.1 and 1.2 simultaneously by establishing a uniqueness statement for metrics under a weaker list of axioms, which are satisfied for both the strong LQG metrics considered in Section 1.2 and for subsequential limits of LFPP (as is shown in [DDDF19, DFG + 19]).
Let D (C) be the space of distributions as in Section 1.2. A weak γ-LQG metric is a measurable IV. Translation invariance. For each fixed deterministic z ∈ C, a.s. D h(·+z) = D h (· + z, · + z).
V. Tightness across scales. Suppose h is a whole-plane GFF and for z ∈ C and r > 0 let h r (z) be the average of h over the circle ∂B r (z). For each r > 0, there is a deterministic constant c r > 0 such that the set of laws of the metrics c −1 r e −ξhr(0) D h (r·, r·) for r > 0 is tight (w.r.t. the local uniform topology). Furthermore, the closure of this set of laws w.r.t. the Prokhorov topology is contained in the set of laws on continuous metrics on C (i.e., every subsequential limit of the laws of the metrics c −1 r e −ξhr(0) D h (r·, r·) is supported on metrics which induce the Euclidean topology on C). Finally, there exists Λ > 1 such that for each δ ∈ (0, 1), (1.12) Axioms I through III for a weak LQG metric are identical to the corresponding axioms for a strong LQG metric. Axiom IV for a weak LQG metric is equivalent to Axiom IV for a strong LQG metric with r = 1. Axiom V for a weak γ-LQG metric is a substitute for the exact scale invariance property given by Axiom IV for a strong LQG metric. This axiom implies the tightness of various functionals of D h . For example, if U ⊂ C is open and K ⊂ C is compact, then the laws of as r varies are tight. It is shown in [DFG + 19, Theorem 1.5] that for any weak γ-LQG metric, one in fact has the following stronger version of (1.12): c δr c r = δ ξQ+o δ (1) , uniformly over all r > 0. (1.14) By the scale invariance of the law of the whole-plane GFF, modulo additive constant, Axiom IV for a strong LQG metric immediately implies Axiom V for a weak γ-LQG metric with c r = r ξQ , for Q as in (1.4). Indeed, using Axiom IV and then Axiom III for a strong γ-LQG metric shows that (1.15) Hence every strong γ-LQG metric is a weak γ-LQG metric. It is shown in [DFG + 19, Theorem 1.2] that every subsequential limit in probability of the LFPP metrics D ε h of (1.6) is of the form D h where D is a weak γ-LQG metric. Consequently, the following theorem contains both Theorem 1.1 and Theorem 1.2.
Theorem 1.8 (Strong uniqueness of weak LQG metrics). Let γ ∈ (0, 2). Every weak γ-LQG metric is a strong γ-LQG metric. In particular, by Theorem 1.2, such a metric exists for each γ ∈ (0, 2) and if D and D are two weak γ-LQG metrics, then there is a deterministic constant C > 0 such that if h is a whole-plane GFF plus a continuous function, then a.s.
It turns out that all of our main results are easy consequences of the following statement, which superficially seems to be weaker that Theorem 1.8. Theorem 1.9 (Weak uniqueness of weak LQG metrics). Let γ ∈ (0, 2) and let D and D be two weak γ-LQG metrics which have the same values of c r in Axiom V. There is a deterministic constant C > 0 such that if h is a whole-plane GFF plus a continuous function, then a.s.
Most of the paper is devoted to the proof of Theorem 1.9. Let us now explain how Theorem 1.9 implies the other main theorems stated above. We first establish the first statement of Theorem 1.8. Lemma 1.10. Every weak γ-LQG metric is a strong γ-LQG metric.
Proof of Lemma 1.10 assuming Theorem 1.9. Suppose that D is a weak γ-LQG metric. For b > 0, we define D (1.16) We claim that D (b) is a weak γ-LQG metric with the same scaling constants c r as D. It is easily verified that D (b) satisfies Axioms I through IV in the definition of a weak γ-LQG metric. To check Axiom V, we compute for r > 0: In the case when h is a whole-plane GFF, the random variable h r (0) − h br (0) is centered Gaussian with variance log b −1 [DS11, Section 3.1]. By (1.12), c br /c r is bounded above by a constant depending only on b (not on r). Axiom V for D applied with h(·/b) in place of h and br in place of r therefore implies that the laws of the metrics c −1 h (r·, r·) are tight in the case when h is a wholeplane GFF, and that every subsequential limit of the laws of these metrics is supported on metrics (not pseudometrics).
Hence we can apply Theorem 1.9 with D = D (b) to get that for each b > 0, there is a deterministic constant k b > 0 such that whenever h is a whole-plane GFF plus a continuous function, a.s. D It is also easy to see that k b depends continuously on b. Indeed, by Axiom III and since h(·/b) − By the continuity of (z, w) → D h (z, w) and h → D h in law as b → 1. This gives the continuity of b → k b at b = 1. Using (1.17) then gives the desired continuity in general.
The relation (1.17) and the continuity of b → k b (actually, just Lebesgue measurability is enough) Therefore, Axiom V holds for D with c r = r −α . By (1.14), we get that α = −ξQ. Hence for b > 0, we have (using Axiom III in the first equality) (1.20) Therefore, D is a strong LQG metric.
Proof of Theorems 1.1, 1.2, and 1.8 assuming Theorem 1.9. By Lemma 1.10, every weak γ-LQG metric is a strong γ-LQG metric. By (1.15), every strong LQG metric satisfies the axioms in the definition of a weak γ-LQG metric with c r = r ξQ . We can therefore apply Theorem 1.9 to get that there is at most one strong LQG metric. This completes the proof of the uniqueness parts of Theorems 1.2 and 1.8. As for existence, we recall that [DFG + 19, Theorem 1.2] (building on [DDDF19]) shows that for every sequence of ε's tending to zero, there is a weak γ-LQG metric D and a subsequence along which the re-scaled LFPP metrics a −1 ε D ε h converge in probability to D h , whenever h is a whole-plane GFF plus a bounded continuous function. By the uniqueness part of Theorem 1.8, D is in fact a strong γ-LQG metric and any two different subsequential limiting metrics differ by a deterministic multiplicative constant factor. Recall that a ε is the median D ε h -distance between the left and right boundaries of the unit square in the case when h is a whole-plane GFF normalized so that h 1 (0) = 0. Hence for any subsequential limiting metric the median D h -distance between the left and right boundaries of the unit square is 1. Therefore, the multiplicative constant factor is 1, so the subsequential limit of D ε h in probability is unique. This gives Theorem 1.1 and the existence parts of Theorems 1.2 and 1.8.
Finally, we note that our results give non-trivial information about the approximating LFPP metrics from (1.6). Indeed, let {a ε } ε>0 be the scaling constants from Theorem 1.1. It is shown in [DG18, Theorem 1.5] that a ε = ε 1−ξQ+oε(1) . Using Theorem 1.1, we obtain the following stronger form of this relation.
We expect, but do not prove here, that in fact Theorem 1.1 holds with a ε = ε 1−ξQ .
Proof of Corollary 1.11. It is shown in [DFG + 19, Lemma 2.14] that for any sequence of ε's tending to zero along which the re-scaled LFPP metrics a −1 ε D ε h converge in law, also a Cε /a ε converges (the limit is Cc 1/C , with c 1/C as in Axiom V for the limiting weak γ-LQG metric). By Theorem 1.1, a −1 ε D ε h converges in probability as ε → 0, so in fact a Cε /a ε converges, not just subsequentially. This means that a Cε is regularly varying with some exponent α > 0. Since a ε = ε 1−ξQ+oε(1) , we must have α = 1 − ξQ.

Outline
As explained above, to prove our main results it remains only to prove Theorem 1.9. We emphasize that unlike many results in the theory of LQG, this paper does not build on a large amount of external input. Rather, we will only use some results from the papers [DDDF19, GM19c, DFG + 19, GM19a], which can be taken as black boxes. All of the externally proven results which we will use are reviewed in Section 2.
Throughout this outline and the rest of the paper, we will use (without comment) the following two basic facts about D h -geodesics when D is a weak γ-LQG metric and h is a whole-plane GFF.
• Almost surely, for every z, w ∈ C, there is at least one D h -geodesic from z to w. This follows from [BBI01, Corollary 2.5.20] and the fact that (C, D h ) is a boundedly compact length space (i.e., closed bounded subsets are compact; see [DFG + 19, Lemma 3.8]).
• For each fixed z, w ∈ C, the D h -geodesic from z to w is a.s. unique. This follows from, e.g., the proof of [MQ18, Theorem 1.2] (see also [GM19a,Lemma 2.2]).
In the remainder of this section we give a very rough idea of the proof of Theorem 1.9. There are a number of technicalities involved, which we will gloss over in order to make the central ideas as transparent as possible. Consequently, some of the statements in this subsection are not exactly accurate without additional caveats. More detailed (and more precise) outlines can be found at the beginnings of the individual sections and subsections.
Main idea of the proof. Suppose D and D are two weak γ-LQG metrics as in Theorem 1.9 and let h be a whole-plane GFF. As explained in Proposition 2.2, it follows from a general theorem for local metrics of the Gaussian free field [GM19c, Theorem 1.6] that D h and D h are bi-Lipschitz equivalent, i.e., (1.21) It is easily seen that c * and C * are a.s. equal to deterministic constants (Lemma 3.1). We identify c * and C * with these constants (which amounts to re-defining c * and C * on an event of probability zero). To prove Theorem 1.9 we will show that c * = C * .
The basic idea of the proof of this fact is as follows. Suppose by way of contradiction that c * < C * .
Then for any c ∈ (c * , C * ) there a.s. exist distinct points In Section 3 (see outline below), using translation invariance of the GFF, modulo additive constant, and the local independence properties of the GFF, we will deduce from this that the following is true. There exists β, p ∈ (0, 1), depending only on the laws of D h and D h , such that for each c ∈ (c * , C * ) there are many small values of r > 0 (how small depends on c ) for which where B r (0) is the Euclidean ball of radius r centered at 0. By interchanging the roles of D h and D h , we can similarly find β, p ∈ (0, 1), depending only on the laws of D h and D h , such that for each C ∈ (c * , C * ), there are many small values of r > 0 (how small depends on C ) for which (1.23) See Section 3 for precise statements. The reason why the bounds only hold for "many" choices of r > 0, instead of for all r > 0, is that we only have tightness across scales (Axiom V), not exact scale invariance. We will use (1.22) to deduce a contradiction to (1.23).
Consider a D h -geodesic P between two fixed points z, w ∈ C. Using (1.22) and a local independence argument for different segments of P (which is explained in the outlines of Sections 4 and 5 below), one can show that it holds with superpolynomially high probability as δ → 0 (i.e., except on an event of probability decaying faster than any positive power of δ), at a rate which is uniform over the choice of z and w, that the following is true. There are times 0 < s < t < D h (z, w) such that D h (P (s), P (t)) ≤ c (t − s) and D h (P (s), P (t)) ≥ δD h (z, w). By the definition (1.21) of C * , the D h -distance from z to P (s) is at most C * s and the D h -distance from P (t) to w is at most Combining these facts shows that with superpolynomially high probability as (1.24) We now let β be as in (1.23) and fix a large constant q > 1. For any r > 0, we can take a union bound to get that with probability tending to 1 as δ → 0, at a rate which is uniform in r, the bound (1.24) holds simultaneously for all z, w ∈ δ q rZ 2 ∩ B r (0). Now consider an arbitrary pair of points z, w ∈ B r (0) with |z − w| ≥ βr. Let z , w ∈ rδ q Z 2 ∩ B r (0) be the points closest to z and w, respectively. By the bi-Hölder continuity of D h and D h w.r.t. the Euclidean metric [DFG + 19, Theorem 1.7], if we choose q sufficiently large, in a manner depending only on the Hölder exponents (i.e., only on γ), then |D h (z, w) − D h (z , w )| and | D h (z, w) − D h (z , w )| are much smaller than δD h (z, w). From this, we infer that with probability tending to 1 as δ → 0, at a rate which is uniform in r, the bound (1.24) holds simultaneously for all z, w ∈ B r (0) with |z − w| ≥ βr. If δ is chosen sufficiently small so that this probability is at least 1 − p/2, we get a contradiction to (1.23) with C = C * − (C * − c )δ.
The purpose of Sections 3, 4, and 5 is to fill in the details of the above argument. These three sections are mostly independent from one another: only the main theorem/proposition statements at the beginning of each section are used in later sections.
Section 3: bounds for ratios of distances at many scales. The purpose of Section 3 is to prove (more quantitative versions of) the bounds (1.22) and (1.23) stated above. Since we are only working with a weak γ-LQG metric, not a strong γ-LQG metric, we do not have exact scale invariance, just tightness across scales (Axiom V). Consequently, if c ∈ (c * , C * ), then we cannot necessarily say that pairs of points u, v for which D h (u, v) ≤ c D h (u, v) exist with uniformly positive probability over different Euclidean scales. That is, it could in principle be that for every small fixed β > 0, the probability that there exists u, v ∈ B r (0) with D h (u, v) ≤ c D h (u, v) and |u − v| ≥ βr is very small for some values of r > 0. However, we can say that such pairs of points exist with uniformly positive probability for a suitably "dense" set of scales r via an argument which proceeds (very roughly) as follows.
Let β, p ∈ (0, 1) be small and suppose by way of contradiction that there is a sequence r k → 0 such that r k+1 /r k is bounded above and below by deterministic constants and the following is true. For each k, it holds with probability at least 1 − p that D h (u, v) ≥ c D h (u, v) for every pair of points u, v ∈ B r k (0) for which |u − v| ≥ βr k . Using the translation invariance of the metric (Axiom IV) and the local independence properties of the GFF (in particular, Lemma 2.6 below), we see that if β, p are sufficiently small (how small depends only on the laws of D h and D h , not on c or r k ), then the following is true. We can cover any fixed compact subset of C by Euclidean balls of the form B r k (z) with the property that D h (u, v) ≥ c D h (u, v) for every pair of points u ∈ ∂B (1−β)r k (z) and v ∈ ∂B r k (z). By considering the times when a D h -geodesic between two fixed points of C crosses an annulus B r k (z) \ B (1−β)r k (z) for z as above, we get that a.s. inf z,w∈C D h (z, w)/D h (z, w) ≥ c for a constant c ∈ (c * , c ). This contradicts the definition (1.21) of c * .
Hence the set of "bad" scales r for which points u, v ∈ B r (0) with |u − v| ≥ βr and D h (u, v) ≤ c D h (u, v) are unlikely to exist cannot be too large, which means that the complementary set of "good" scales for which such points exist with probability at least p has to be reasonably dense. This leads to (1.22). The bound (1.23) follows by interchanging the roles of D h and D h .
Section 4: independence along an LQG geodesic. Once we know that there are many pairs of points u, v with D h (u, v) ≤ c D h (u, v), we want to use some sort of local independence to say that a D h -geodesic P is extremely likely to get close to at least one such pair of points (i.e., we need the D h -distance from P to each of u and v to be much smaller than D h (u, v)). However, D h -geodesics are highly non-local functionals of the field and do not satisfy any reasonable Markov property. So, techniques for obtaining local independence which may be familiar from the theory of SLE/GFF couplings [SS13, Dub09, MS16d, MS16e, MS16a, MS17, She16a, DMS14] do not apply in our setting.
Instead we need to develop a new set of techniques to obtain local independence at different points of D h -geodesics. See Figure 1 for an illustration. In fact, we will prove a general theorem (Theorem 4.1) which roughly speaking says the following. Suppose we are given events E z,w r (z) for z, z, w ∈ C and r > 0 with the following properties. The event E z,w r (z) is determined by h| Br(z) and the part of the D h -geodesic P z,w from z to w which is contained in B r (z). Moreover, for each z, z, w ∈ C, the conditional probability of E z,w r (z) given h| C\Br(z) and the event {P z,w ∩ B r (z) = ∅} z w P u v Figure 1: Illustration of the main ideas in Section 4. Using results on confluence of geodesics from [GM19a], we can show that there are many points along the D h -geodesic P at which it is stable, in the sense that changing the behavior of the field in a small Euclidean ball does not result in a macroscopic change to the D h -geodesic (the precise condition is given in (4.11)). In fact, using the results of Section 3, we can arrange that there are many such points whose corresponding balls contain a pair of points u, v such that D h (u, v) ≤ c D h (u, v) and |u − v| is comparable to the Euclidean radius of the ball. These pairs of points and the D h -geodesics between them are shown in blue. Using the results of Section 5, we can show that for each of these stable times, it holds with positive conditional probability given the past that P gets close to the corresponding pair of points u, v. By a standard concentration inequality for Bernoulli sums, applied at the stable radii, this shows that P has to get close to at least one such pair of points u, v with extremely high probability.
is a.s. bounded below by a deterministic constant. Then when r is small it is very likely that for nearly every choice of z, w ∈ C, the event E z,w r (z) occurs for at least one ball B r (z) hit by P z,w . We will eventually apply this theorem with E z,w r (z) given by, roughly speaking, the event that P z,w gets close to a pair of points This together with the triangle inequality and the bi-Hölder continuity of D h and D h w.r.t. the Euclidean metric (to transfer from |u − v| ≥ const ×r to a lower bound for D h (u, v)) will lead to (1.24).
We will prove the above "independence along a geodesic" theorem using the results on confluence of D h -geodesics established in [GM19a]. These results tell us that if z ∈ C is fixed and w 1 , w 2 ∈ C are close together, then the D h -geodesics P 1 from z to w 1 and P 2 from z to w 2 typically agree until they get close to w 1 and w 2 , i.e., P 1 | [0,τ ] = P 2 | [0,τ ] for a time τ which is close to D h (z, w 1 ) (equivalently, to D h (z, w 2 )) when D h (w 1 , w 2 ) is small. Note that this property is not true for geodesics for a smooth Riemannian metric, but it is true for geodesics in the Brownian map [Le 10]. Now fix z, w and consider the D h -geodesic P = P z,w from z to w. The above confluence property applied with w 1 = P (t) for a typical time t ∈ [0, D h (z, w)] and w 2 a point near P (t) will allow us to show that with extremely high probability, there are many times t ∈ [0, D h (z, w)] at which P is "stable" in the following sense. If we make a small modification to h in a neighborhood of P (t), then we will not change P | [0,τ ] for a time τ a little bit less than t. This allows us to say that events depending on the field in a small neighborhood of P (t) have positive conditional probability given an initial segment of P . Applying this at a large number of evenly spaced times t ∈ [0, D h (z, w)] will show that it is extremely likely that the event E z,w r (z) discussed above occurs for at least one Euclidean ball B r (z) hit by P .
Section 5: an LQG geodesic gets close to a shortcut with positive probability. Fix z, w ∈ C and let P = P z,w be the D h -geodesic from z to w. By (1.22) and translation invariance (Axiom IV) we know that there exists β, p ∈ (0, 1) such that if c ∈ (c * , C * ), then there are many values of r > 0 such that (1.22) holds with z in place of 0 (actually, we will use a variant of (1.22) which gives more precise information about the locations of u and v; see Proposition 3.5). In light of the results of Section 4, we want to show that if we condition on {P ∩ B r (z) = ∅}, then the conditional probability that P gets close to a pair of points u, v as in (1.22) (with z in place of 0) is bounded below by a positive deterministic constant which does not depend on r or z.
For a deterministic open set U ⊂ C, one can prove that the D h -geodesic P enters U with positive probability as follows. Consider a deterministic path from z to w and let φ be a smooth bump function which takes large values in a narrow "tube" around this path and which vanishes outside a slightly larger tube. By Weyl scaling (Axiom III), D h−φ distances in the tube are much shorter than distances anywhere else. Hence the D h−φ -geodesic from z to w has to stay in the tube and hence has to enter U . Since the laws of h and h − φ are absolutely continuous, we get that the D h -geodesic enters U with positive probability.
We will use a similar strategy to show that P has positive conditional probability given {P ∩ B r (z) = ∅} to get near a pair of points u, v ∈ B r (z) with D h (u, v) ≤ c D h (u, v) and |u − v| ≥ βr. However, additional complications arise. For example, the region we want P to enter (a small neighborhood of either u or v) is random, which will be resolved by choosing a deterministic region with contains the D h -geodesic between u and v with positive probability. We also need to ensure that the condition is not destroyed when we add our bump function. To do this, we will need to make sure that the D h -geodesic between u and v is contained in the region where the bump function attains its largest possible value. Another issue is that we need the bump function φ to be supported on a region of diameter of order r ≈ |u − v|, so that its Dirichlet energy is bounded independently of r. In particular, this support cannot contain the starting and ending points z and w of the D h -geodesic. This will be resolved by growing the D h -metric balls from z and w until they hit B 3r (z) and choosing a bump function whose support approximates a path between the hitting points.
In Section 6, we combine all of the above ingredients to conclude the proof of Theorem 1.9, following the argument in the "main ideas" section above. Section 7 contains the a list of open problems.

Preliminaries
In this subsection, we first introduce some basic (mostly standard) notation. We then review all of the results from [GM19c, DFG + 19, GM19a] which we will need for the proof of Theorem 1.9. On a first read, the reader may wish to read only Sections 2.1 (which introduces notation) and 2.2 (which proves the bi-Lipschitz equivalence of the metrics D h and D h in Theorem 2.5) then refer back to the other subsections as needed.
We often specify requirements on the dependencies on rates of convergence in O(·) and o(·) errors, implicit constants in , etc., in the statements of lemmas/propositions/theorems, in which case we implicitly require that errors, implicit constants, etc., in the proof satisfy the same dependencies.

Balls and annuli
For z ∈ C and r > 0, we write B r (z) for the Euclidean ball of radius r centered at z. We also define the open annulus A r 1 ,r 2 (z) := B r 2 (z) \ B r 1 (z), ∀0 < r r < r 2 < ∞. For a metric D on C, r > 0, and z ∈ C we write B r (z; D) for the filled metric ball which is the union of B r (z; D) and the bounded connected components of C \ B r (z; D). Lemma 2.1. Let D be a weak γ-LQG metric and let h be a whole-plane GFF. Also let z ∈ C and let τ be a stopping time for the filtration generated by The same is true with closures of ordinary D h -metric balls in place of filled D h -metric balls.

Bi-Lipschitz equivalence of weak LQG metrics
In this subsection we explain why the results of [GM19c] imply that any two weak γ-LQG metrics with the same scaling constants are bi-Lipschitz equivalent.
Proposition 2.2. Let h be a whole-plane GFF, let γ ∈ (0, 2), and let D and D be two weak γ-LQG metrics, with the same scaling constants c r . There is a deterministic constant C > 0 such that a.s.
(2.3) Proposition 2.2 is a special case of a general theorem from [GM19c] which tells us when two random metrics coupled with the same GFF are bi-Lipschitz equivalent. To state the theorem, we first recall some definitions. In the setting of Proposition 2.2, the metrics D h and D h are each local for h due to Axiom II. Since these metrics are each determined by h, they are conditionally independent given h. Therefore, we can apply [GM19c, Lemma 1.4] to get that D h and D h are jointly local for h.
Definition 2.4 (Additive local metrics). Let (h, D 1 , . . . , D n ) be a coupling of h with n random continuous length metric which are jointly local for h. For ξ ∈ R, we say that D 1 , . . . , D n are ξ-additive for h if for each z ∈ C and each r > 0 such that B r (z) ⊂ U , the metrics (e −ξhr(z) D 1 , . . . , e −ξhr(z) D n ) are jointly local metrics for h − h r (z).
By Axiom III, it follows that our metrics D h and D h are jointly local for h. The following theorem is a special case of [GM19c, Theorem 1.6].
Theorem 2.5 ( [GM19c]). Let ξ ∈ R, let h be a whole-plane GFF normalized so that h 1 (0) = 0, and let (h, D h , D h ) be a coupling of h with two random continuous metrics on C which are jointly local and ξ-additive for h. There is a universal constant p ∈ (0, 1) such that the following is true. Suppose there is a constant C > 0 such that (using the notation for annuli from (2.2)), we have P sup u,v∈∂Br(z) D h u, v; A r/2,2r (z) ≤ CD h (∂B r/2 (z), ∂B r (z)) ≥ p, ∀z ∈ C, ∀r > 0.
Proof of Proposition 2.2. By Axioms IV and V for each of D h and D h , for any p ∈ (0, 1) we can find a constant C p > 1 such that for each z ∈ C and each r > 0, it holds with probability at least p that (2.5) and the same is true with D h in place of h. Therefore, (2.4) holds with C = C 2 p for each of the pairs (D h , D h ) and ( D h , D h ). Theorem 2.5 therefore implies Proposition 2.2 with C = C 2 p , where p is as in Theorem 2.5.

Local independence for the GFF
In many places throughout the paper, we will estimate various probabilities using the local independence properties of the GFF. We will do this using two different lemmas, which we state in this section. The first is a restatement of part of [GM19c, Lemma 3.1].
Lemma 2.6 (Iterating events in nested annuli). Fix 0 < s 1 < s 2 < 1. Let {r k } k∈N be a decreasing sequence of positive numbers such that r k+1 /r k ≤ s 1 for each k ∈ N and let {E r k } k∈N be events such that E r k ∈ σ (h − h r k (0))| As 1 r k ,s 2 r k (0) for each k ∈ N. For K ∈ N, let N (K) be the number of k ∈ [1, K] Z for which E r k occurs. For each a > 0 and each b ∈ (0, 1), there exists p = p(a, b, s 1 , s 2 ) ∈ (0, 1) and c = c(a, b, s 1 , s 2 ) > 0 such that if (2.7) We will only ever apply Lemma 2.6 to say that N (K) ≥ 1 with high probability, i.e., the choice of b in (2.7) will not matter for our purposes.
Lemma 2.7 (Iterating events in disjoint balls). Let h be a whole-plane GFF and fix s > 0. Let n ∈ N and let Z be a collection of #Z = n points in C such that |z − w| ≥ 2(1 + s) for each distinct z, w ∈ C. For z ∈ Z, let E z be an event which is determined by (h − h 1+s (z))| B 1 (z) . For each p, q ∈ (0, 1), there exists n * = n * (s, p, q) ∈ N such that if P[E z ] ≥ p for each z ∈ Z, then P z∈Z E z ≥ q, ∀n ≥ n * .
Proof. Let U := z∈Z B 1+s (z) and let h be the harmonic part of h| U . Since the balls B 1+s (z) for z ∈ Z are disjoint, the Markov property of h implies that the fields (h − h 1+s (z))| B 1+s (z) for z ∈ Z, and hence also the events E z , are conditionally independent given h| C\U (equivalently, given h).
We will now compare the conditional law given h| C\U to the unconditional law. For z ∈ Z, let By a standard Radon-Nikodym derivative calculation for the GFF (see, e.g., [MQ18,Lemma 4.1]) and the translation and scale invariance of the law of h, modulo additive constant, for each α > 0 there is a constant C = C(α, s) > 0 such that the following is true. The conditional law given of (h − h 1+s (z))| B 1 (z) given h| C\U is absolutely continuous with respect to its marginal law and if H z denotes the Radon-Nikodym derivative of the conditional law with respect to the marginal law, then a.s.
Each M z is an a.s. finite random variable. By the translation invariance of the law of h, modulo additive constant, the law of M z does not depend on z. So, we can find a constant A = A(s, q) > 0 (2.10) Since E z is determined by (h − h 1+s (z))| B 1 (z) and P[E z ] ≥ p for each z ∈ Z, (2.9) implies that there exists p = p(p, A) > 0 such that on the event {M z ≤ A} (which is determined by h| C\U ), a.s. P E z | h| C\U ≥ p. (2.11) Since the E z 's are conditionally independent given h| C\U , we see that a.s. (2.12) We now choose n * large enough that 1 − p n * /2 ≥ 1 − (1 − q)/2 and combine (2.10) with (2.12).

Estimates for weak LQG metrics
In this subsection we review from results from [DFG + 19] which we will need for the proofs of our main theorems. Throughout, D denotes a weak γ-LQG metric and h denotes a whole-plane GFF. All of the results which we state in this subsection involve a parameter r, which controls the "Euclidean scale" at which we are working. This parameter is necessary since we are only assuming tightness across scales (Axiom V) instead of exact scale invariance. All estimates are required to be uniform in the choice of r. Our first result, which follows from [DFG + 19, Lemmas 3.20 and 3.22], is a form of local Hölder continuity for the identity map (C, | · |) → (C, D h ) and its inverse.
Lemma 2.8 (Hölder continuity). Fix a compact set K ⊂ C and exponents χ ∈ (0, ξ(Q − 2)) and χ > ξ(Q + 2). For each r > 0, it holds with probability tending to 1 as a → 0, at a rate which is uniform in r, that for each u, v ∈ rK with |u − v| ≤ ar, (2.14) 19 We note that (2.14) gives an upper bound for the D h -distance from u to v along paths which stay in B 2|u−v| (u). This is slightly stronger than just an upper bound for D h (u, v). In Section 5, we will also need the following variant of (2.14) which gives an upper bound for the D h -internal diameters of Euclidean squares and is proven in [DFG + 19, Lemma 3.20].
Lemma 2.9 (Internal diameters of Euclidean squares). Let K and χ be as in Lemma 2.8. For each χ ∈ (0, ξ(Q − 2)) and each r > 0, it holds with probability tending to 1 as ε → 0, at a rate which is uniform in r, that for each k ∈ N 0 and each 2 −k εr × 2 −k εr square S with corners in 2 −k εrZ 2 which intersects rK, In several places throughout the paper, we will want to prevent a D h -geodesic from staying in small neighborhood of a fixed Euclidean path. The following lemma, which is a restatement of [DFG + 19, Proposition 4.1], will allow us to do this.
Lemma 2.10 (Lower bound for distances in a narrow tube). Let L ⊂ C be a compact set which is either a line segment or an arc of a circle and fix b > 0. For each r > 0 and each q > 0, it holds with probability at least where the rate of the o ε (1) depends on L, b, q but not on r.
By [Ang19, Theorem 1.9], for each γ ∈ (0, 2) we have 1 − ξQ ≥ 0, and hence ξQ − 1 − ξ 2 /2 < 0. Therefore, the power of ε on the right side of (2.16) is negative for small enough q. Hence, Lemma 2.10 implies that when ε is small and u, v ∈ B εr (rL) with |u − v| ≥ br, it holds with high probability that D h (u, v; B εr (rL)) is much larger than D h (u, v). In particular, a D h -geodesic from u to v cannot stay in B εr (L). Lemma 2.10 has the following useful corollary. For the statement, we recall the notation for Euclidean annuli from (2.2).
Lemma 2.11 (Lower bound for distances in a narrow annulus). For each S > s > 0 and each p ∈ (0, 1), there exists α * = α * (s, S, p) ∈ (1/2, 1) such that for each α ∈ [α * , 1), each z ∈ C, and each r > 0, (2.17) Proof. By Weyl scaling (Axiom III), the event in (2.17) does not depend on the choice of additive constant for h. By Axiom IV and the translation invariance of the law of h modulo additive constant, the probability of this event does not depend on z. By Axiom V, we can find b = b(s) > 0 such that Combining with Lemma 2.10 (with ε = 1 − α and L = ∂D) concludes the proof.
Finally, we record a lemma which prevents D h -geodesics from spending a long time near the boundary of a D h -metric ball which is needed in Section 4.2. The lemma is a re-statement of [DFG + 19, Proposition 4.3].

Confluence of geodesics
In this subsection we will review some facts about D h -geodesics which are proven in [GM19a]. These facts are used only in Section 4.4. For z ∈ C, r > 0, and n ∈ N we define the radii ρ n r (z) as in [GM19a,Equation (3.13)]. The radius ρ n r (z) is the smallest t ∈ {2 k r} k∈N for which a certain event in σ((h − h 6r (z))| A 2r,5r (z) ) occurs. We will not need the precise definition of ρ n r (z) here, only a few facts which we will review in this subsection.
We have ρ n r (z) ≥ 6r and ρ n r (z) is a stopping time for the filtration generated by h| B 6t (z) for t ≥ r. The following is immediate from [GM19a, Lemma 3.4], the translation invariance of the law of h, modulo additive constant, and Axiom IV.
Lemma 2.13 (Bounds for radii used to control geodesics). There is a constant η > 0 depending only on the choice of metric such that the following is true. If we abbreviate then for each compact set K ⊂ C, each r > 0, and each z ∈ C, it holds with probability 1 − O ε (ε 2 ) (at a rate depending on K, but not on r or z) that (2.20) Henceforth fix η as in Lemma 2.13 and let ρ r,ε (z) be as in (2.19). For r > 0, ε > 0, and a (2.21) Since ρ r,ε (z) is a stopping time for the filtration generated by h| B 6t (z) for t ≥ r, each ρ r,ε (z) for z ∈ εr 4 Z 2 ∩B εr (K) is a.s. determined by R ε r (K) and the restriction of h to B R ε r (K) (K). Lemma 2.13 shows that for each fixed choice of K, P[R ε r (rK + z) ≤ (6ε 1/2 + ε)r] tends to 1 as ε → 0, uniformly over all z ∈ C and r > 0.
Recall from Section 2.1 that B • s (z; D h ) for z ∈ C and s > 0 denotes the filled D h -ball of radius s centered at z. Throughout the rest of this subsection we fix z ∈ C and abbreviate B • , then so is σ ε τ,r . The following lemma is used to prevent D h -geodesics from getting near a specified boundary point of a D h -metric ball. It is an immediate consequence of [GM19a, Lemma 3.6] (which is the case when z = 0) together with the translation invariance of the law of h, modulo additive constant, and Axiom IV.
Lemma 2.14 (Geodesics are unlikely to get near a specified point of ∂B • τ ). There exists α > 0, depending only on the choice of metric, such that the following is true. Let r > 0, let τ be a stopping time for the filtration generated by B • s , h| B • s s≥0 , and let x ∈ ∂B • τ and ε ∈ (0, 1) be chosen in a manner depending only on with the following properties.
B. There is a deterministic constant C 0 > 1 depending only on the choice of metric such that a.s.

21
We will now state a confluence property for LQG geodesics started from z. Each point x ∈ ∂B • s lies at D h -distance exactly s from z, so every D h -geodesic from z to x stays in B • s (z; D h ). For some atypical points x there might be many such D h -geodesics. But, it is shown in [GM19a,Lemma 2.4] that there is always a distinguished D h -geodesic from z to x, called the leftmost geodesic, which lies (weakly) to the left of every other D h -geodesic from 0 to x if we stand at x and look outward from Theorem 2.15 (Confluence of geodesics across a metric annulus). Almost surely, for each 0 < t < s < ∞ there is a finite set of D h -geodesics from 0 to ∂B • t such that every leftmost D h -geodesic from 0 to ∂B • s coincides with one of these D h -geodesics on the time interval [0, t]. In particular, there are a.s. only finitely many points of ∂B • t which are hit by leftmost D h -geodesics from 0 to ∂B • s .
Combined with [GM19a, Lemma 2.7], Theorem 2.15 tells us that we can decompose ∂B • s into a finite union of boundary arcs such that for any points x, y ∈ ∂B • s which lie in the same arc, the leftmost D h -geodesics from z to x and from z to y coincide in the time interval [0, t]. We will need a more quantitative version of Theorem 2.15 which gives us stretched exponential concentration for the number of such arcs if we truncate on a certain high-probability regularity event. To this end, we define We also fix χ ∈ (0, ξ(Q − 2)), chosen in a manner depending only on ξ and Q, so that by Lemma 2.8 D h is a.s. locally χ-Hölder continuous w.r.t. the Euclidean metric. For r > 0 and a ∈ (0, 1), we define E z r (a) to be the event that the following is true.

(Comparison of D h -balls and Euclidean balls)
3. (Bounds for radii used to control geodesics) The radii of Lemma 2.13 satisfy ρ r,ε (z) ≤ ε 1/2 r for each z ∈ εr 4 Z 2 ∩ B 4r (z) and each dyadic ε ∈ (0, a]. It is easy to see that P[E z r (a)] → 1 as a → 0, uniformly over the choice of r and z: in particular, this follows from [GM19a, Lemma 3.8] (which is the case when z = 0) and Axiom IV. We will in fact show in Section 4.3 that with high probability, E z r Theorem 2.16 (Quantitative confluence of geodesics). For each a ∈ (0, 1), there is a constant b 0 > 0 depending only on a and constants b 1 , β > 0 depending only on the choice of metric D such that the following is true. For each z ∈ C, each r > 0, each N ∈ N, and each stopping time τ for s., the probability that E z r (a) occurs and there are more

The optimal bi-Lipschitz constant
Throughout this section, we assume that we are in the setting of Theorem 1.9, so that D and D are two weak γ-LQG metrics with the same scaling constants. We also let h be a whole-plane GFF. We know from Proposition 2.2 that D h and D h are a.s. bi-Lipschitz equivalent. We define the optimal bi-Lipschitz constants c * and C * as in (1.21). Since D h and D h are a.s. bi-Lipschitz equivalent (Proposition 2.2), a.s. 0 < c * ≤ C * < ∞.
Lemma 3.1. Each of c * and C * is a.s. equal to a deterministic constant.
Proof. We will prove the statement for C * ; the statement for c * is proven in an identical manner.
Suppose C > 0 is such that P[C * > C] > 0. We will show that in fact P[C * > C] = 1.
There is some large deterministic R > 0 such that with positive probability, there are points Since each of D h and D h induces the Euclidean topology on C, after possibly increasing R, we can arrange that with positive probability, there are . In fact, by Axiom III this event is determined by h| B R (0) viewed modulo additive constant, since adding a constant to h results in scaling D h and D h by the same constant factor.
For z ∈ C, let E(z) be the event that there exist points u, v ∈ B R (z) such that (3.1) holds with , viewed modulo additive constant. By Axiom IV and the translation invariance of the law of h, modulo additive constant, the probability of E(z) does not depend on z. The event that E(z) occurs for infinitely many z ∈ Z 2 is determined by the tail σ-algebra generated by h| C\Br(z) , viewed modulo additive constant, as r → ∞. This tail σ-algebra is trivial, so we get that a.s. E(z) occurs for infinitely many z ∈ C. This means that in fact P[C * > C] = 1, so C * is a.s. equal to a deterministic constant.
We henceforth re-define each of c * and C * on an event of probability zero so that they are deterministic. The main goal of this section is to show that there are many values of r > 0 for which it holds with uniformly positive probability that there are points z, w ∈ C such that |z|, |w|, and |z − w| are all of order r and D h (z, w)/D h (z, w) is close to C * (resp. c * ). To quantify this, we introduce the following events. For r > 0, C ∈ (0, C * ], and β ∈ (0, 1), define It is easy to see from the definition (1.21) of C * that for each fixed r > 0 and C ∈ (0, C * ), there exists p, β ∈ (0, 1) (allowed to depend on C * and r) such that P[G r (C , β)] ≥ p. 4 Since we are working with weak LQG metrics, which are not known to be exactly invariant under spatial scaling, it is not clear a priori that p and β can be taken to be uniform in the choice of r. It is also not clear a priori that p and β can be chosen independently of C . Similar considerations apply for G r (c , β). We will establish that one can choose p and β independently of C and r provided r is restricted to lie in a suitably "dense" subset of (0, 1), in the following sense.

Quantitative versions of Propositions 3.2 and 3.3
We will need more quantitative versions of Propositions 3.2 and 3.3 which differ from the original proposition statements in two important respects. First, instead of starting at a constant-order scale, we will start at some given scale r > 0 for which we have an a priori lower bound on P[G r (C , β)] for some C ∈ (0, C * ) and β ∈ (0, 1) (or P[G r (c , β)] for some c > c * and β ∈ (0, 1)). We will then produce many radii in [ε 1+ν r, εr] instead of in [ε 1+ν , ε]. The reason for introducing r is that we only have tightness across scales (Axiom V) instead of true scale invariance. Second, instead of just lower bounding the probability of G r (C , β) or G r (c , β), we will obtain a lower bound for the probability of a smaller event which is more complicated, but also more useful. Let us begin by stating a more quantitative version of Proposition 3.2.
By the symmetry between out hypotheses on D h and D h , Proposition 3.4 implies the analogous statement with the roles of D h and D h interchanged, which reads as follows.
As in the case of Proposition 3.4, Proposition 3.5 immediately implies Proposition 3.3. To prove Proposition 3.4, we will (roughly speaking) prove the contrapositive.
If (A) does not hold for some r > 0 and ε ∈ (0, ε 0 ], then there are fewer than µ log 8 ε −1 values of k ∈ K ε r for which the last sentence of (A) holds with probability at least p. For such a choice of r and ε, there are at least (ν − µ) log 8 ε −1 values of k ∈ K ε r for which the last sentence of (B) holds with probability at least 1 − p. That is, (B) holds for the pair (r, ε). By Proposition 3.6, this means that P[G r (C , β)] < β. Hence we have proven the contrapositive of Proposition 3.4.

Proof of Proposition 3.6
As explained in Section 3.1, to prove all of the propositions statements from earlier in this section it remains only to prove Proposition 3.6. The basic idea of the proof is as follows. If we assume that (B) holds for a small enough choice of p ∈ (0, 1) (depending only on µ and ν), then we can use Lemma 2.6 to cover space by Euclidean balls of the form B r/2 (z) for r ∈ [ε 1+ν r, εr] with the following property. For each u ∈ ∂B αr (z) and each v ∈ ∂B r (z) such that the D h -geodesic from u to v is unique and is contained in . By considering the times when a D h -geodesic between two fixed points z, w ∈ C crosses the annulus A αr,r (z) for such a z and r, we will be able to show that D h (z, w) ≤ C D h (z, w) for a suitable constant C ∈ (C , C * ). Applying this to an appropriate β-dependent collection of pairs of points (z, w) will show that P[G r (C , β)] < β. The reason why we need to make α close to 1 is to ensure that the events we consider depend on h in a sufficiently "local" manner (see the discussion just after the definition of E r (z) below).
Let us now define the events to which we will apply Lemma 2.6. For z ∈ C, r > 0, and parameters α ∈ (1/2, 1), A > 1 and C ∈ (0, C * ), let E r (z) = E r (z; α, A, C ) be the event that the following is true.
1. For each u ∈ ∂B αr (z) and each v ∈ ∂B r (z) such that the D h -geodesic from u to v is unique 3. There is a path in A αr,r (z) which disconnects the inner and outer boundaries of A αr,r (z) and has D h -length at most AD h (∂B αr (z), ∂B r (z)).
Condition 1 is the main point of the event E r (z), as discussed just above. The purpose of condition 2 is to ensure that E r (z) is determined by h| A r/2,2r (z) . Without this condition, we would not necessarily be able to tell whether a path in A αr,r (z) is a D h -geodesic without seeing the field outside of A r/2,2r (z) (see Lemma 3.7). The purpose of condition 3 is as follows. If a D h -geodesic between two points outside of B r (z) enters B αr (z), then it must cross the path from condition 3 twice. This means that it can spend at most AD h (∂B αr (z), ∂B r (z)) units of time in B αr (z) since otherwise the path from condition 3 would provide a shortcut, which would contradict the definition of a geodesic. If we assume (B), this fact will eventually allow us to force a D h -geodesic to spend a positive fraction of its time tracing segments between points We want to use Lemma 2.6 to argue that if (B) holds, then with high probability there are many values of z ∈ C such that E r (z) occurs for some r ∈ [ε 1+ν r, εr]. We first check the measurability condition in Lemma 2.6 Lemma 3.7. For each z ∈ C and r > 0, (3.7) Proof. By Axiom III subtracting h 4r (0) from h results in scaling D h and D h by the same factor, so does not affect the occurrence of E r (z). Hence it suffices to prove (3.7) with h| A r/2,2r (z) in place of (h − h 4r (z))| A r/2,2r (z) . From Axiom III, it is obvious that condition 3 in the definition of E r (z) is determined by h| A r/2,2r (z) .
For u ∈ ∂B αr (z) and v ∈ ∂B r (z), we can determine whether D h (u, v) > D h (u, ∂A r/2,2r (z)) from the internal metric D h ·, ·; A r/2,2r (z) : indeed, D h (u, ∂A r/2,2r (z)) is clearly determined by this , in which case we can tell whether P is a D h -geodesic from the restriction of h to the D h -metric ball of radius D h (u, ∂A r/2,2r (z)) centered at u, which in turn is determined by h| A r/2,2r (z) . Furthermore, on the event that D h (u, v) ≤ D h (u, ∂A r/2,2r (z)) and are determined by h| A r/2,2r (z) . Therefore, the intersection of conditions 1 and 2 in the definition of E r (z) is determined by h| A r/2,2r (z) . Hence we have proven (3.7).
We now show that (B) implies a lower bound for P[E r (z)] for some values of r ∈ [ε 1+ν r, εr].
Proof. Assume (B) is satisfied for some choice of r, ε, p, α, C and let r 1 , . . . , r K ∈ [ε 1+µ r, εr]∩{8 −k r : k ∈ N} be the values of r from (B), enumerated in decreasing order. Note that K ≥ (ν − µ) log 8 ε −1 by assumption. By Lemma 3.7, we can apply Lemma 2.6 to find that there exists p = p(q, µ, ν) ∈ (0, 1) such that if (3.9) then (3.8) holds. It therefore suffices to choose p, α * , and A in an appropriate manner depending on p so that if (B) holds, then (3.9) holds. By tightness across scales (Axiom V), we can find S > s > 0 depending on p such that for each z ∈ C and r > 0, it holds with probability at least 1 Lemma 2.11 with the above choice of s and S gives an α * ∈ [3/4, 1) depending on p such that for each α ∈ [α * , 1), z ∈ C, and r > 0, condition 2 in the definition of E r (z) holds with probability at least 1 − (1 − p)/3. Now suppose α ∈ [α * , 1). We can again apply Axiom V to find that there exists A > 1 depending on α and p such that for each z ∈ C and r > 0, condition 3 in the definition of E r (z) occurs with probability at least 1 − (1 − p)/3.
If (B) holds for the above choice of α and with p < (1 − p)/3, then for each z ∈ C and each k ∈ [1, K] Z , condition 1 in the definition of E r k (z) holds with probability at least 1 − (1 − p)/3.
Combining the three preceding paragraphs shows that (3.9) holds.
Lemma 3.9. There is a q > 1 depending only on µ, ν such that if p, α * , α ∈ [α * , 1), and A is chosen as in Lemma 3.8 for this choice of q, then the following is true for each C ∈ (0, C * ). If (B) holds for some r > 0 and ε ∈ (0, 1) and for this choice of p, α, C , then for each open set U ⊂ C, it holds with probability tending to 1 as ε → 0 (at a rate which is uniform in r) that for z ∈ rU, there occurs.
Proof. Upon choosing q sufficiently large, this follows from Lemma 3.8 and a union bound over all Proof of Proposition 3.6. See Figure 2 for an illustration of the proof.
Step 1: setup. Let p, α * , α ∈ [α * , 1), and A > 1 be chosen as in Lemma 3.9. Also fix Figure 2: Illustration of the proof of Proposition 3.6. The D h -geodesic P from z to w along with one of the balls B r j (w j ) hit by P for which E r j (w j ) occurs are shown. The time t j is the first time that P exits B r j (w j ) after time t j−1 and the time s j is the last time before t j at which P The orange path comes from condition 3 in the definition of E r j (w j ), and its D h -length is at most Since P crosses this orange path both before time t j−1 and after time s j and P is a D h -geodesic, we have that s j − t j−1 ≤ A(t j − s j ). This shows that the intervals [s j , t j ] occupy a uniformly positive fraction of the total D h -length of P , which in turn and note that we can choose C in a manner depending only on α, C , µ, ν (since A depends only on α, µ, ν).
By Axiom V, there is some large bounded open set U ⊂ C depending only on β such that for each r > 0, it holds with probability at least 1 − β/2, the D h -diameter of B r (0) is smaller than the D h -distance from B r (0) to ∂(rU ), in which case every D h -geodesic between points of B r (0) is contained in rU. Henceforth fix such a choice of U . Let F ε r be the event that every D h -geodesic between points of B r (0) is contained in rU and the event of Lemma 3.9 with the above choices of α, A, C , and U , so that P[F ε ] ≥ 1 − β/2 − o ε (1), uniformly in r, under the assumption (B).
Step 2: covering a D h -geodesic with paths of short D h -length. To prove (3.12), we consider points z, w ∈ B r (0) ∩ Q 2 with |z − w| ≥ βr and let P : [0, D h (z, w)] → C be the (a.s. unique) D h -geodesic from z to w. Let t 0 = 0 and inductively let t j for j ∈ N be the smallest time t ≥ t j−1 at which P exits a Euclidean ball of the form B r (w) for w ∈ ε 1+ν r 100 Z 2 ∩(rU ) and r ∈ [ε 1+ν r, εr]∩{8 −k r : k ∈ N} such that P (t j−1 ) ∈ B r/2 (w) and E r (w) occurs; or let t j = D h (z, w) if no such t exists. If t j < D h (z, w), let w j and r j be the corresponding values of w and r. Also let s j be the last time before t j at which P hits ∂B αr j (w), so that s j ∈ [t j−1 , t j ] and P ([s j , t j ]) ⊂ A αr j ,r j (w j ). Finally, define J := max{j ∈ N : |z − P (t j−1 )| < 2εr} and J := min{j ∈ N : |w − P (t j+1 )| < 2εr}. (3.13) The reason for the definitions of J and J is that z, w / ∈ B r j (w j ) for j ∈ [J, J] Z (since r j ≤ εr and P (t j ) ∈ B r j (w j )). By the definition of F ε r , on this event we have t j < D h (z, w) and 28 (3.14) is the only D h -geodesic from P (s j ) to P (t j ) since otherwise we could re-route P along another such D h -geodesic to contradict the uniqueness of the D h -geodesic from z to w. Combining this with condition 1 in the definition of E r j (w j ) (applied with u = P (s j ) and v = P (t j )) and the definition (1.21) of C * , we find that We will now argue that s j − t j−1 is not too much larger than t j − s j . If j ∈ [J, J] Z , then since at least once before time t j−1 and at least once after time s j . By condition 3 in the definition of E r j (w j ), there is a path disconnecting the inner and outer boundaries of this annulus with D h -length at most AD h ∂B αr j (w j ), ∂B r j (w j ) . The geodesic P must hit this path at least once before time t j−1 and at least once after time s j . Since P is a geodesic and P (s j ) ∈ ∂B αr j (w j ), Adding A(s j − t j−1 ) to both sides of this inequality, then dividing by A + 1, gives Step 3: upper bound for D h . By combining the above relations, we get that on F ε r , (3.17) By (3.11), Axiom V for D and D, and the triangle inequality, it holds with probability tending to 1 as ε → 0, uniformly in r, that simultaneously for all z, w ∈ B r (0) with |z − w| ≥ βr. By combining this with (3.17) and recalling holds, we get that if ε 0 is chosen to be sufficiently small, in a manner which does not depend on r, then if (B) holds for r > 0 and ε ∈ (0, ε 0 ], then it holds with probability at least 1 − β that (3.12) holds simultaneously for each z, w ∈ B r (0) ∩ Q 2 with |z − w| ≥ βr. By the continuity of D h and D h , we can remove the requirement that z, w ∈ Q 2 (which was only used to get the uniqueness of the D h -geodesic from z to w).

Independence along a geodesic
Let h be a whole-plane GFF and let D be a weak γ-LQG metric. The goal of this section is to prove the following general "local independence" type result for events depending on a small segment of a D h -geodesic. We will first state a simplified version of our result which is easier to parse (Theorem 4.1), then state the full version (Theorem 4.2).
Theorem 4.1. Suppose we are given events E z,w r (z) ∈ σ(h) for z ∈ C, r > 0, and z, w ∈ C and a deterministic constant Λ > 1 which satisfy the following properties, where here P = P z,w denotes the (a.s. unique) D h -geodesic from z to w.

(Measurability)
The event E z,w r (z) is determined by h| Br(z) and the geodesic P stopped at the last time it exists B r (z).
Intuitively, the reason why Theorem 4.1 is true is as follows. The geodesic segments P ∩ B r (z) and P ∩ B r (w) are approximately independent from one another when |z − w| is much larger than r.
When r is small, we can cover P by a large number of balls B r (z) whose corresponding center points z lie at Euclidean distance much further than r from one another. Using (4.1) and a general concentration inequality for independent random variables, one gets that for each fixed pair (z, w), with high probability there exists z ∈ C such that P z,w ∩ B r (z) = ∅ and E z,w r (z) occurs. One then takes a union bound over all pairs z, w ∈ r q Z 2 ∩ U .
The above heuristic is not quite right since D h -geodesics do not depend locally on the field, so P ∩ B r (z) and P ∩ B r (w) are not approximately independent when |z − w| is much greater than r. Indeed, it is possible that changing what happens in B r (z) could affect the behavior of P macroscopically even when r is very small. As a substitute for this lack of long-range independence, we will use the confluence of geodesics results from [GM19a], as discussed in Section 1.5, and only make changes to the field at places where the geodesics are "stable" in the sense that a microscopic change does not lead to macroscopic changes to P . The reason why we only get a statement which holds with probability tending to 1 as r → 0 at the end of Theorem 4.1 is that we need to truncate on a global regularity event in order to make confluence hold with high probability.
We will actually prove (and use) a more general version of Theorem 4.1 which differs from Theorem 4.1 in the following respects.
• We allow for more flexibility in the Euclidean radii involved in the various conditions, which is represented by constants {λ i } i=1,...,5 .
• We introduce events E r (z) which are determined by the restriction of h to an annulus A λ 1 r,λ 4 r (z) (for constants λ 1 < λ 4 ) and which are required to have probability close to 1. We replace (4.1) by a comparison between the conditional probabilities of E z,w r (z) and E r (z) given h| C\B λ 3 r (z) , for another constant λ 3 . The occurrence of E r (z) can be thought of as the statement that "h| A λ 1 r,λ 4 r (z) is sufficiently well behaved that E z,w r (z) has a chance to occur".
• We do not require our events to be defined for all r > 0, but rather only for values of r in a suitably "dense" set R ⊂ (0, ∞). The reason why we need to allow for this is that the results of Section 3 only hold for values of r in a suitably dense set.
• We work with a given "base scale" r > 0 (e.g., we consider points in rU instead of in U ) and we require our estimates to be uniform in the choice of r. The reason for this is that we have only assumed tightness across scales (Axiom V) instead of exact scale invariance.
2. (Measurability) For each z ∈ C and r ∈ R, E r (z) is determined by (h − h λ 5 r (z))| A λ 1 r,λ 4 r (z) for each z, w ∈ C, and E z,w r (z) is determined by h| B λ 4 r (z) and the (a.s. unique) D h -geodesic from z to w stopped at the last time it exists B λ 4 r (z).

(Lower bound for P[E r (z)])
For each z ∈ C and r ∈ R, we have P[E r (z)] ≥ p.
Theorem 4.1 is the special case of Theorem 4.2 where R = (0, ∞); λ 2 = λ 3 = λ 4 = 1; E r (z) is the whole probability space; and r = 1. The parameter p in Theorem 4.2 will eventually be chosen to be sufficiently close to 1 that we can apply Lemma 2.6 to cover a large region of space by balls B λ 1 r (z) for pairs (z, r) such that E r (z) occurs (see Lemma 4.10). The statement of Theorem 4.2 is easier to understand if one thinks of the particular setting in which we will apply it. Recall the optimal bi-Lipschitz constants from (1.21). For us, E r (z) will be the event that there exists a pair of points u, v ∈ A λ 1 r,λ 2 r (z) at Euclidean distance of order r from each other for which D h (u, v) ≤ c 2 D h (u, v) for a constant c 2 ∈ (c * , C * ); and some regularity conditions hold which are needed to ensure that conditions 2 and 4 in the theorem 31 statement are satisfied. We will only be able to show that P[E r (z)] is bounded below for a "dense" set of scales R as in condition 1 due to the results in Section 3. The event E z,w r (z) will be the event that, roughly speaking, the D h -geodesic P z,w gets close to u, v and hence (by the triangle inequality) hits a pair of points P (s), P (t) at Euclidean distance of order r from each other for which D h (P (s), P (t)) ≤ c 2 D h (P (s), P (t)). More precisely, we will prove the following statement in Section 5.
Proposition 4.3. Assume (by way of eventual contradiction) that c * < C * . Let 0 < µ < ν ≤ ν * and c * < c 1 < c 2 < C * . There exist universal constants {λ i } i=1,...,5 and parameters b, ρ ∈ (0, 1) depending only on µ, ν such that the following is true. Let p be as in Theorem 4.2 for the above choice of µ, ν, {λ i } i=1,...,5 and let c = c (c 1 , µ, ν) > c * be as in Proposition 3.5 with c = c 1 . If β ∈ (0, 1) and r > 0 are such that P[G r (c , β)] ≥ β (in the notation (3.3)), then there exists ε 0 = ε 0 (β, c 1 , c 2 , µ, ν) ∈ (0, 1), a deterministic set of radii R ⊂ (0, ε 0 ], events E r (z) and E z,w r (z), and a deterministic constant Λ = Λ(c 1 , c 2 , µ, ν) > 1 which satisfy the hypotheses of Theorem 4.2 with ρ −1 r in place of r and have the following additional property. Suppose z ∈ C, r ∈ R, and z, w ∈ C \ B λ 4 r (z), and let P = P z,w be the D h -geodesic from z to w. If E z,w r (z) occurs, then there are times 0 < s < t < |P | such that P ([s, t]) ⊂ B λ 2 r (z), |P (s) − P (t)| ≥ br, and D h (P (s), P (t)) ≤ c 2 D h (P (s), P (t)). (4.3) Roughly speaking, Proposition 4.3 combined with Theorem 4.2 implies that the pairs of points and |u − v| is not too small are sufficiently dense that a typical D h -geodesic is extremely likely to get close to such a pair of points. This will be applied in Section 6 to derive a contradiction to the definition (1.21) of C * if we assume that c * < C * , and thereby to show that c * = C * .

Setup and outline
Assume that we are in the setting of Theorem 4.2 for some r > 0. To lighten notation, we will also impose the assumption that λ 3 = 1 (the proof when λ 3 = 1 is identical, just with extra factors of λ 3 in various subscripts). Let U ⊂ C be open and bounded and let > 0, as in the conclusion of Theorem 4.2. Also fix ε ∈ (0, ε 0 ] and distinct points z, w ∈ rU with |z − w| ≥ 4 r (the reason for the factor of 4 here is to reduce factors of 4 elsewhere). Let P = P z,w := (D h -geodesic from z to w). (4.4) To lighten notation, throughout the rest of this section we will not include the parameters r, ε, z, w in the notation. But, we will always require that all estimates are uniform in the choice of r, z, and w (we will typically be sending ε → 0). Since we will commonly be growing metric balls starting from z, we also introduce the following abbreviations for z ∈ C and r, s > 0: where here we recall that B • s (z; D h ) is the filled metric ball. We now define several objects which we will work with throughout the rest of this section. See Figure 3 for an illustration. Fix β ∈ (0, 1) to be chosen later, in a manner depending only on D. Define s k := τ r + kε β c r e ξhr(z) and t k := s k + ε 2β c r e ξhr(z) ∈ [s k , s k+1 ], ∀k ∈ N 0 . , P (s k )). Moreover, P | [0,t k ] is a.s. the unique (hence also leftmost) D h -geodesic from z to P (t k ), hence P (s k ) is one of the points of Conf k . By the definition of I k , this point is determined by which arc of I k contains P (t k ).
We now introduce the set of Euclidean balls B r (z) which we will consider when trying to produce a ball for which E r (z) occurs. With r ε 1 , . . . , r ε µ log ε −1 ∈ [ε 1+ν r, εr] ∩ R as in condition 1 from Theorem 4.2, let Z k for k ∈ N be the set of pairs (z, r) such that (4.10) We want to say that with extremely high probability, there are many values of k ∈ N 0 for which the event E r (z) occurs for some (z, r) ∈ Z k such that P ∩ B λ 2 r (z) = ∅. We will do this by lower-bounding the conditional probability given F k that E r (z) occurs for at least one (z, r) ∈ Z k , then considering a polynomial (in ε) number of values of k and applying a standard concentration inequality for binomial random variables.
In order to say something useful about the conditional law given F k of what happens in one of the balls B r (z) for (z, r) ∈ Z k , we need to know that making a small change to what happens in B r (z) does not affect which arc of I k contains P (t k ). For z ∈ C and r > 0, we therefore let Stab k,r (z) be the event that (z, r) ∈ Z k and Each D h (·, ·; C \ B r (z))-geodesic from z to a point of ∂B r (z) hits ∂B • t k in the same arc of I k . (4.11) Here, by a D h (·, ·; C \ B r (z))-geodesic from z to a point x ∈ ∂B r (z) we mean a path from z to x in C \ B r (z) which has minimal D h -length among all such paths and which does not hit ∂B r (z) except at x. Note that such a geodesic need not exist for every point of ∂B r (z). However, if P is a D h geodesic started from z which enters B r (z), then P , stopped at the first time when it enters B r (z), is a D h (·, ·; C \ B r (z))-geodesic from z to a point of ∂B r (z).
In Section 4.4, we will use various quantitative results on confluence of geodesics from [GM19a] to show that with high probability Stab k,r (z) occurs for most of the pairs (z, r) ∈ Z k such that P enters B λ 2 r (z). The reason why the events Stab k,r (z) are useful is the following lemma.
Lemma 4.5. For each z ∈ C, r > 0, and k ∈ N 0 the event Stab k,r (z) of (4.11) is a.s. determined by h| C\Br(z) . Furthermore, on the event Stab k, ) and the arc of I k which contains P (t k ) are a.s. determined by h| C\Br(z) and the indicator 1 Stab k,r (z)∩{P ∩Br(z) =∅} .
In particular, for any event F ∈ F k the event F ∩ Stab k,r (z) ∩ {P ∩ B r (z) = ∅} is a.s. determined by h| C\Br(z) and the indicator 1 Stab k,r (z)∩{P ∩Br(z) =∅} . Section 2.5: confluence of geodesics. Figure 4: Schematic outline of Section 4. An arrow between two sections/results means that the first is used in the proof of the second. Note that Proposition 4.3 is proven in Section 5 and Theorem 1.9 is proven in Section 6. see which arc this is by observing h| C\Br(z) . Therefore, on Stab k,r (z) ∩ {P ∩ B r (z) = ∅}, the arc of I k which contains P (t k ) is a.s. determined by h| C\Br(z) and 1 Stab k,r (z)∩{P ∩Br(z) =∅} .
The last statement of the lemma follows from the second statement and Lemma 4.4.
We define the set of "good" pairs and the set of "very good" pairs The proof of Theorem 4.1 is based on lower-bounding the conditional probability that Z E k = ∅ given F k , which allows us to say that the number of k for which Z E k = ∅ stochastically dominates a binomial random variable. To lower-bound P[Z E k = ∅ | F k ], we will first establish a lower bound for using condition 4 in Theorem 4.2 (Section 4.2). We will then show that it is very likely that Z E k = ∅ for many values of k (Section 4.4). This will imply that it is very likely that there are many values of k for which P[Z E k = ∅ | F k ] is bounded below, and hence there are many values of k for which P[Z E k = ∅ | F k ] is bounded below (Section 4.5). We will now outline the rest of the proof of Theorem 4.2. See Figure 4 for a schematic illustration of how the various results in this section fit together.
In Section 4.2, we show that for each k ∈ N, Here, one should think of ν as being small (relative to β), so that ε 2ν+oε(1) is not too much different from ε oε(1) .
In Section 4.3, we define a global regularity event E r which we will truncate on for most of the rest of the proof and show that it occurs with high probability. This event includes various bounds for D h -distances (e.g., Hölder continuity), but the most important condition is that E r (z) occurs for many pairs (z, r). To make the latter condition occur with high probability, we will make p sufficiently close to 1 to allow us to apply Lemma 2.6 and a union bound.
In Section 4.4, we show that if we truncate on E r , then with very high probability there are many values of k for which Z E k = ∅. Since the definition of E r already includes the condition that E r (z) occurs for many pairs (z, r) ∈ Z k , the main difficulty here is showing that Stab k,r (z) occurs for most of the pairs (z, r) such that P ∩ B λ 2 r (z) = ∅. This will be accomplished by applying the results on confluence of geodesics from [GM19a], as reviewed in Section 2.5, and multiplying over k to get concentration. We will choose the parameter β from (4.6) to be small so that we have enough "room" between ∂B • s k and ∂B • t k for various confluence effects to occur. In Section 4.5, we will transfer from the statement that "Z E k = ∅ for many values of k" to the statement that "Z E k = ∅ for many values of k". This will be accomplished using the result of Section 4.2 and an elementary probabilistic lemma (Lemma 4.17) which allows us to convert between conditional and unconditional probabilities. We will then complete the proof of Theorem 4.2 by truncating on E r and then taking a union bound over many pairs of initial and terminal points z, w.
In Section 4.6, we collect the proofs of some geometric lemmas which are stated in Sections 4.4 and 4.5, but whose proofs are postponed to avoid distracting from the core of the argument. These geometric lemmas are used to control the behavior of D h -geodesics on the regularity event E r .

Comparison of E r (z) and E r (z)
Recall the definitions of the filtration {F k } k≥0 from (4.8), the set of "good" pairs Z E k from (4.12), and the set of "very good" pairs Z E k from (4.13). The events E r (z) are easier to work with than the events E r (z) since E r (z) has high probability and is determined locally by h. The goal of this subsection is to prove the following lemma, which will eventually allow us to transfer from a lower bound for the probability that Z E k = ∅ to a lower bound for the probability that Z E k = ∅. Lemma 4.6. Let M > 0. On the event where the rates of the o ε (1) and o ∞ ε (ε) are deterministic and depend only on M, µ, ν, {λ i } i=1,...,5 . Nothing from this section besides Lemma 4.6 is used in subsequent subsections. Lemma 4.6 will eventually be a consequence of condition 4 of Theorem 4.2, which together with Lemma 4.5 allows us to compare the conditional expectations of #Z E k and #Z E k given F k . To transfer from a lower bound for the conditional expectation of #Z E k to a lower bound for the probability that Z E k = ∅, we will use a Payley-Zygmund type argument. For this purpose we need the following upper bound for #Z E k , which comes from Lemma 2.12 and Markov's inequality (to transfer from unconditional to conditional probability).
We will also need the following elementary probabilistic lemma which will be used in conjunction with Lemma 4.5 to transfer from conditional probabilities given h| C\Br(z) to conditional probabilities given F k .
Lemma 4.8. Let (Ω, M, P) be a probability space. Let F, G ⊂ M be sub-σ-algebras. Let E ∈ M be an event such that F ∩ E ∈ G ∨ σ(E) for each F ∈ F. Also let G ∈ F ∩ G. Suppose H 1 , H 2 ∈ M are events and Λ > 0 is a deterministic constant such that a.s. (4.20) Then a.s. (4.21) Proof. Let G := G ∨ σ(E). On the event that P[E | G] > 0, for any H ∈ M, (4.23) We claim that for any H ∈ M, a.s. (4.24) Once (4.24) is proven, we can take the conditional expectations given F of both sides of (4.23) to get (4.21). To prove (4.24), let F ∈ F. By hypothesis, F ∩ E ∈ G . Therefore, (4.25) By the definition of conditional expectation, this implies (4.24).
Proof of Lemma 4.6. Recall that we are assuming that λ 3 = 1, so that our hypothesis (4.2) says that for (z, r) ∈ C × R such that z, w / ∈ B λ 4 r (z), Since Stab k,r (z) ∈ σ h| C\Br(z) (Lemma 4.5), we infer from (4.26) and the definitions (4.12) and (4.13) of Z E k and Z E k that for each (z, r) ∈ C × R such that z, w / ∈ B λ 4 r (z), a.s. (4.27) We will now deduce from (4.27) and Lemma 4.8 that on (4.28) In particular, we will apply Lemma 4.8 with We check the hypotheses of Lemma 4.8 with the above choice of parameters, starting with the requirement that the event G defined above belongs to F k ∩ σ h| C\Br(z) . Indeed, it is clear ). By the definition (4.9) of F k , we have G ∈ F k . By Lemma 4.5, also G ∈ σ h| C\Br(z) . By the definition (4.10) of Z k , if G occurs with positive probability then we must have we have z, w / ∈ B λ 4 r (z), so in particular (4.27) holds a.s. on G. By Lemma 4.5, the intersection of any event in F k with Stab k,r (z) ∩ {P ∩ B r (z) = ∅} is a.s. determined by h| C\Br(z) and 1 Stab k,r (z)∩{P ∩Br(z) =∅} . We may therefore apply Lemma 4.8 to deduce (4.28) from (4.27).

Global regularity event
Throughout most of the rest of the proof of Theorem 4.2, we will truncate on a global regularity event which we define in this subsection. The parameter p ∈ (0, 1) of Theorem 4.2 has to be chosen sufficiently close to 1 to allow us to apply Lemma 2.6 to make the probability of one of the conditions in the event as close to 1 as we like. We emphasize that our global regularity event does not depend on the particular choice of z, w in (4.4).

38
Fix bounded, connected open sets U ⊂ V ⊂ C and parameters ν, > 0 (ν, U , and are the parameters from Theorem 4.2). Also fix, once and for all, parameters χ ∈ (0, ξ(Q − 2)) and χ > ξ(Q + 2) as in Lemma 2.8, chosen in a manner which depends only on γ (we will not make the dependence on these parameters explicit). For r > 0 and a ∈ (0, 1), let E r = E r (a, ν, , U, V ) be the event that the following is true.

(Comparison of D h -balls and Euclidean balls)
For each z ∈ C and r > 0, let τ r (z) be the smallest t > 0 for which the filled D h -metric ball B • t (z; D h ) intersects ∂B r (z), as in (2.23).
We note that the upper bound in (4.33) uses D h z, w; B 2|z−w| (z) ≥ D h (z, w) instead of D h (z, w). We will need this slightly stronger upper bound for D h -distances in the proof of Lemma 4.18 below.
Remark 4.9. Due to conditions 2, 3, and 6, and since ∈ (0, 1), for each z ∈ rU the event E r defined just above is contained in the event E z r (a) as defined just above Theorem 2.16 with r in place of r. Proof. By Axiom V, we can find a bounded open set V ⊃ U , depending only on U , such that condition 1 in the definition of E holds with probability at least 1 − (1 − p)/6. Again using Axiom V, we can find a small enough a ∈ (0, 1), depending on , V, p, such that condition 2 holds with probability at least 1 − (1 − p)/6. By Lemma 2.8, after possibly shrinking a we can further arrange that condition 3 holds with probability at least 1 − (1 − p)/6. By the continuity of the circle average process and the scale invariance of the law of h, modulo additive constant, after possibly further shrinking a we can arrange that condition 4 holds with probability at least 1 − (1 − p)/6. By Lemma 2.6, conditions 2 and 3 of Theorem 4.2, and a union bound over all z ∈ λ 1 ε 1+ν r 4 Z 2 ∩ V , if p is chosen sufficiently close to 1, in a manner depending only on µ, ν, and {λ i } i=1,...,5 , then the probability of condition 5 in the definition of E r tends to 1 as a → 0, uniformly over the choice of r. Therefore, after possibly further shrinking a, we can arrange that condition 5 in the definition of E r holds with probability at least 1 − (1 − p)/6. By Lemma 2.13 and a union bound over values of ε ∈ (0, a] ∩ {2 −n } n∈N , after possibly further shrinking a we can also arrange that condition 6 in the definition of E r holds with probability at least 1 − (1 − p)/6.

Geodesic stability event occurs at many times
Henceforth fix p ∈ (0, 1) (which we will eventually send to 1), a bounded open set U ⊂ C, and ∈ (0, 1) and let V, a be as in Lemma 4.10 for this choice of p, U, and the given values of µ, ν from Theorem 4.2. Let E r be the event of Section 4.3 with this choice of parameters, so that P[E r ] ≥ p.

Define
K := aε −β − 1. (4.35) The significance of the value K is that condition 2 in the definition of E r implies that, in the notation (4.6), (4.36) Recalling the parameter β from (4.6) and the parameters χ < χ as in condition 3 in the definition of E r , we henceforth impose the requirement that β ∈ (0, χ/χ ). (4.37) We will make our final choice of β in Proposition 4.11 just below.
Let Z E k be as in (4.12) and let K be as in (4.35). The goal of this section is to show that with high probability there are many values of k ∈ [0, K] Z for which Z E k = ∅. In the next subsection, we will combine this with Lemma 4.6 to show that there are many values of k ∈ [0, K] Z for which The following proposition is the main result of this subsection and is the only statement from this subsection which is referenced in Section 4.5.
Proposition 4.11. There are small constants β, θ ∈ (0, 1) depending only on the choice of metric D such that if we use this choice of β in (4.6), then on E r it holds except on an event of probability decaying faster than any positive power of ε, at a rate which is uniform in r, z, w, that there are at By condition 5 in the definition of E r , we already know that on this event, for each k ∈ [0, K] Z there are many pairs (z, r) ∈ Z k for which P ∩ B λ 2 r (z) = ∅ and E r (z) occurs. The main point of this subsection is to show that there are many such pairs for which also the event Stab k,r (z) of (4.11) occurs. Roughly speaking, the idea of the proof is as follows; see Figure 5 for an illustration. If P enters B r (z) but Stab k,r (z) fails to occur, then P has to get "close" in some sense to one of the endpoints of one of the arcs in I k . 5 Indeed, otherwise Hölder continuity allows us to force all of the D h (·, ·; C \ B r (z))-geodesics from z to points of ∂B r (z) to hit the same arc of I k as P . This is explained in Lemma 4.12.
On the other hand, if we choose β sufficiently small then results from [GM19a] (in particular, Theorem 2.16) show that #I k is extremely likely to be of smaller order than ε −α , where α is the r (z) Figure 5: Illustration of the proof of Proposition 4.11. The points in the set EndPts k of endpoints of arcs in I k are shown in red. We first use Theorem 2.16 to bound #EndPts k = #Conf k . Lemma 4.13 allows us to choose for each y ∈ EndPts k a point z y ∈ ∂B • t k (not shown) such that an arc of B 16ε κ r (z) disconnects the set C ε κ r y (defined just after Lemma 4.13) from ∞ in C \ B • t k . The set which this arc disconnects from ∞, which contains C ε κ r y , is shown in pink. Note that the sets C ε κ r y for different choices of y are allowed to overlap. Lemma 2.14 and a union bound over y ∈ EndPts k shows that with high probability, for each y ∈ EndPts k , no D h -geodesic from z to ∂B • s k+1 can enter any of the C ε κ r y 's. This together with Lemma 4.12 allows us to show that Stab k,r (z) occurs for each (z, r) ∈ Z k such that P ∩ B r (z) = ∅. One such ball B r (z) is shown in green and several segments of D h (·, ·; C \ B r (z))-geodesics from 0 to points of ∂B r (z) is shown in red. exponent from Lemma 2.14. We can therefore apply that lemma once for each of the endpoints of the I k 's and take a union bound to say that with polynomially high probability given ( , no D h -geodesic from z to a point at macroscopic distance from ∂B • s k can get near any of the endpoints of the I k 's (Lemma 4.14). The claimed superpolynomial concentration when we truncate on E r comes from a standard concentration bound for independent Bernoulli random variables, provided we choose θ to be sufficiently small relative to α. In order to quantify how close D h -geodesics get to the endpoints of the I k 's, we will need some deterministic definitions. Let U ⊂ C be a connected domain such that C \ U is compact and connected. View ∂U as a collection of prime ends. If X ⊂ U , we define the prime end closure Cl (X) to be the set of points in z ∈ U ∪ ∂U with the following property: if φ : where here diam denotes the Euclidean diameter. Then d U is a metric on U ∪ ∂U which is bounded below by the Euclidean metric on C restricted to U ∪ ∂U and bounded above by the internal Euclidean metric on U ∪ ∂U . Note that d U is not a length metric. Proof. See Figure 6 for an illustration of the statement and proof of the lemma. Assume that E r occurs and let s, I, ε, and P be as in the lemma statement. By hypothesis, for each δ ∈ (0, 1) there is a connected set X 0 ⊂ C \ B • s which has Euclidean diameter at most (ε + δ)r and which satisfies P ∩ X 0 = ∅ and Cl (X 0 ) ∩ (∂B • s \ I) = ∅. By possibly shrinking X 0 , we can assume without loss of generality that Cl (X 0 ) ∩ (∂B • s \ I) is a single prime end, which is necessarily in B • s \ I. Let t be the first time after s at which P hits X 0 . By the upper bound in condition 3 in the definition of E r , the D h -diameter of X 0 is at most (ε + δ) χ c r e ξhr(0) . Since P is a D h -geodesic, P (t) ∈ X 0 , and Cl (X 0 ) contains a point of ∂B has Euclidean diameter at most ((ε + δ) χ/χ + ε + δ)r and its prime end closure contains both the point P (s) ∈ I and a point of ∂B • s \ I. Hence one of the connected components V of C \ (B • s ∪ X) is bounded and contains an endpoint of I. Since Cl (X) intersects I only at P (s) (here we use that Cl (X 0 ) ∩ ∂B • s is a single point), it follows that also P (s) ∈ ∂V . We now conclude the proof by choosing δ to be sufficiently small (depending on ε) so that (ε + δ) χ/χ + ε + δ ≤ 2ε χ/χ . We will eventually apply the contrapositive of Lemma 4.12, i.e., we will say that if P does not enter a region which contains one of the endpoints of I and which is disconnected from ∞ in C \ B • s by a set of small diameter, then d C\B • s (P ; ∂B • s \ I) is bounded below. The following elementary deterministic lemma will be used in conjunction with Lemma 2.14 to prevent P from entering such a region (we will apply the lemma with K = B • t k ).
Lemma 4.13. Let K ⊂ C be a compact connected set such that C \ K is connected and view ∂K as a collection of prime ends. For y ∈ ∂K and ε > 0, let C ε y be the set of points in z ∈ C \ K such that the following is true. There is a connected set X ⊂ C \ K (allowed to depend on z and y, ε) with Euclidean diameter at most ε such that z and y lie in the prime end closure of the same bounded connected component of C \ (X ∪ K). Then there is a single compact connected set Y ε y ⊂ C \ K of Euclidean diameter at most 16ε (depending only on y, ε) such that C ε y is contained in the prime end closure of a single bounded connected component of C \ (Y ε y ∪ K).
The proof of Lemma 4.13 is straightforward, but it takes a few paragraphs so we postpone it until Section 4.6 to avoid interrupting the proof of Theorem 4.2.
Returning now to the setting of Proposition 4.11, for k ∈ [0, K] Z let EndPts k be the set of endpoints of the arcs in I k . As in Lemma 4.13, for δ > 0 we let C δ y be the set of points in (C \ B • t k ) ∪ ∂B • t k with the following property: there is a compact connected set X ⊂ C \ B • t k with Euclidean diameter at most δ such that z and y lie in the closure of the same bounded connected Lemma 4.14. Fix κ ∈ (0, 1). If β, θ ∈ (0, 1) are chosen sufficiently small, in a manner depending only on κ and the choice of metric D, then on E r it holds except on an event of probability decaying faster than any positive power of ε, at a rate which is uniform in r, that there are at least (1 − ε θ )K values of k ∈ [0, K] Z for which the following is true. In the notation introduced just above, no Proof. Fix parameters θ, ω ∈ (0, 1) to be chosen later, in a manner depending only on D. We will first choose β in a manner depending on ω, D and then choose ω in a manner depending on κ, D, and then choose θ in a manner depending on β, ω. The parameter κ will be chosen in a manner depending only on D in the proof of Proposition 4.11 below.
We will first show, using Theorem 2.16, that if β is chosen to be sufficiently small (depending on ω, D) then with extremely high probability on E r one has for each k ∈ [0, K] Z that #Conf k ≤ ε −ω , which implies that #EndPts k ≤ ε −ω . We then show using Lemma 2.14 and a union bound over at most ε −ω elements of EndPts that if ω is chosen to be sufficiently small (depending on the parameter α of Lemma 2.14, which depends only on D), then for each k it holds with conditional probability at Finally, we will use a standard concentration inequality for Bernoulli random variables to show that if θ is chosen to be sufficiently small then with extremely high probability this happens for at least Step 1: bounding the number of confluence points. Recall from Section 4.1 that t k = s k + ε 2β c r e ξhr(0) and Conf k is the set of points of ∂B • s k which are hit by leftmost D h -geodesics from z to ∂B • t k . Due to Remark 4.9, we can apply Theorem 2.16 (with N = ε −ω and τ = s k ) to get that if β is chosen sufficiently small, in a manner depending only on ω and D, then for each k ∈ [0, K] Z , the probability that E r occurs and #Conf k > ε −ω decays faster than any positive power of ε. By a union bound over k, (4.39) Step 2: bounding the parameters from Lemma 2.14. Recall the radii ρ r,ε (z), which appear in Lemma 2.13 and condition 6 in the definition of E r (the precise definition of these radii is not needed here, only their role in Lemma 2.14). To lighten notation, for k ∈ [0, K] Z we define + εr, as in (2.21) and On E r , we have B • for each k ∈ [0, K] Z . By combining this with the upper bound for D h -distances from condition 3 in the definition of E r , we get that B R k (B • t k ) ⊂ B • t k +7 χ ε κχ/2 cre ξhr(0) . By this together with the definition of σ k and condition 4 in the definition of E r (to replace h r (0) with h r (z)), on E r we have σ k ≤ t k + Aε κχ/2 c r e ξhr(z) , where A = 7 χ e ξ/a is an unimportant constant.
We henceforth assume that β < κχ/2, so that by the conclusion of the preceding paragraph and the definition (4.6) of t k and s k+1 , for small enough ε ∈ (0, 1) (how small depends only on a, β, κ), (4.41) Step 3: killing off geodesics near the endpoints with polynomially high probability. Recall that EndPts k denotes the set of endpoints of arcs in I k . We have #EndPts k = #I k . By Lemma 4.13, each of the sets C ε κ r y can be disconnected from ∞ in C \ B • t k by a connected subset of C \ B • Henceforth assume that ω ∈ (0, ακ/2). On the event ) and has high probability by (4.39)), we can take a union bound over at most ε −ω elements of EndPts k to get (4.42) Since ω < ακ/2, the right side of (4.42) is at least 1 − ε ω for small enough ε ∈ (0, 1) (how small depends only on α, κ, ω, C 0 ). This implies that for each such ε, (4.43) Note that we have added the additional event {σ k ≤ s k+1 }, for reasons which will become apparent just below.
Step 4: independence across radii to get concentration. The radius σ k is a stopping time for .
By (4.41), on E r we have σ k ≤ s k+1 for each k ∈ [0, K] Z . By (4.39), if E r occurs then except on an event of probability o ∞ ε (ε) we have #Conf k ≤ ε −ω for each K ∈ [0, K] Z . Combining these observations with the preceding paragraph shows that Recall that R k ≤ 7ε κ/2 r on E r (see this discussion just after (4.40)). As t k ≥ τ r we have that A D h -geodesic started from z can hit ∂B • t k at most once. Therefore, the defining property ( * ) of z y , applied to the path P | (t k ,|P |] , shows that for each k as in the preceding paragraph, no D h -geodesic from z to ∂B • s k+1 can enter y∈EndPts k C ε κ r y . To deduce Proposition 4.11 from Lemma 4.14, we need some quantitative control on the D h (·, ·; C \ B r (z))-geodesics appearing in the definition 4.11 of Stab k,r (z). The needed control is provided by the following lemma.
By Lemma 4.14, we only need to prove that if E r occurs and ε ∈ (0, 1) is chosen to be sufficiently small (in a deterministic manner which does not depend on k or r), then the following is true. If Henceforth assume that E r occurs and k is as above. Recall the definition (4.10) of Z k . By condition 5 in the definition of E r , each point of ∂B 2λ 4 εr (B • if ε is sufficiently small then the union of these Euclidean balls disconnects ∂B • t k from w. Therefore, P must enter B λ 2 r (z) for some (z, r) ∈ Z k such that E r (z) occurs.
We will now conclude the proof by showing that, in the notation (4.11), Stab r,k (z) occurs for every (z, r) ∈ Z k with P ∩ B r (z) = ∅. (4.46) Recall that we are assuming that k is such that no D h -geodesic from z to ∂B • s k+1 can enter Hence P | [0,s k+1 ] cannot enter y∈EndPts k C ε κ r y . Since P does not re-enter B • s k+1 after time s k+1 and y∈EndPts k C ε κ r y ⊂ B • s k+1 , also P cannot enter y∈EndPts k C ε κ r y . From this and Lemma 4.12 (applied in the contrapositive direction with (ε κ /2) χ /χ in place of ε and I k in place of I), we infer that if I k ∈ I k is chosen so that P (t k ) ∈ I k , then Now let (z, r) ∈ Z k with P ∩ B r (z) = ∅ and let P : [0, |P |] → C \ B r (z) be a D h (·, ·; C \ B r (z))geodesic from z to a point of ∂B r (z). We will show that P (t k ) ∈ I k for any possible choice of P , which by definition implies that Stab r,k (z) occurs. By Lemma 4.15, with a deterministic implicit constant depending only on a and λ 4 .

Transferring from E r (z) to E r (z)
We now want to combine Lemma 4.6 and Proposition 4.11 to say that with high probability, there are many pairs (z, r) ∈ Z k for which E r (z) occurs. In particular, we will establish the following statement.
We want to apply Lemma 4.17 to the events {Z E k = ∅} and {Z E k = ∅}. However, these events are not F k+1 -measurable. To get around this, we need to instead work with a slightly modified event which is F k+1 -measurable. In particular, we will intersect each of {Z E k = ∅} and {Z E k = ∅} with the event F k of the following lemma.
with the following properties. If ε is sufficiently small (how small depends only on a, λ 4 ), then whenever E r occurs also K k=0 F k occurs. Moreover, if F k occurs then s k+1 ≤ τ 2 r and for each (z, r) ∈ Z k we have ).
Lemma 4.18 is a relatively straightforward consequence of the definition of E r . The proof is postponed until Section 4.6. The event F k is defined explicitly in Lemma 4.21 below, but only the properties of the event given in Lemma 4.18 are important for our purposes.
Lemma 4.19. Let F k for k ∈ [0, K] Z be the event of Lemma 4.18 and let F k be the σ-algebra from (4.8). Then for k ∈ N, (4.54) Proof. By Lemma 4.18, we have F k ⊂ F k+1 . By the definition (4.10) we also have Z k ∈ F k ⊂ F k+1 . We now argue that on F k , the set Z E k is determined by F k+1 . Since there are only countably many pairs (z, r) ∈ C × (0, ∞) which can possibly belong to Z E k , it suffices to show that the event {(z, r) ∈ Z E k }∩F k is F k+1 -measurable for each such pair (z, r). Recall from (4.12) that Z E k is the set of (z, r) ∈ Z k for which E r (z)∩Stab r,k (z)∩{P ∩B λ 2 r (z) = ∅} occurs. By Lemma 4.18, if F k occurs then and P does not re-enter B • s k+1 after time s k+1 , we have F k ∩ {P ∩ B λ 2 r (z) = ∅} ∩ {(z, r) ∈ Z k } ∈ F k+1 for each (z, r). By (4.11), each of the events Stab r,k (z) for (z, r) ∈ Z k is determined by F k and the set of D h (·, ·; C \ B r (z))-geodesics from z to points of ∂B r (z). By Lemma 4.18, it therefore follows that F k ∩ Stab r,k (z) ∈ F k+1 for each (z, r) ∈ Z k . Combining these statements shows that Using condition 2 from Theorem 4.2, we similarly obtain that and where the rate of the o ε (1) in (4.56) is deterministic and depends only on ν and the choice of metric D.
Proof. For k ∈ N, let E k := Z E k = ∅ ∩ F k . By Lemma 4.19, we have E k ∈ F k+1 . We may therefore apply Lemma 4.17 with m = 4ε θ K , α = 1/2, and δ = 1/4 to get that (4.57) By Proposition 4.11 and Lemma 4.18, on E r it holds except on an event of probability o ∞ ε (ε) that K k=1 1 E k ≥ (1 − ε θ )K. Combining this with (4.57) shows that if E r occurs, then except on an event of probability o ∞ ε (ε) there are at least (1 − 4ε θ )K values of k ∈ [0, K] Z for which (4.55) holds. On E r , for each k ∈ [0, K] Z we have B • t k ⊂ B 3 r (z) (by (4.36)) and w / ∈ B 3λ 4 εr (B • t k ) (since |z − w| ≥ 4 r). By Lemma 4.6, whenever these latter conditions hold it holds except on an event of By Lemma 4.17 applied with E k = Z E k = ∅ ∩ F k , m = (1 − 4ε θ )K , α = 1 − ε 2ν+ζ/2 , and δ = ε 2ν+ζ/2 /2, it follows that if E r occurs and ε is sufficiently small, then except on an event of probability at most there are at most values of k ∈ [0, K] Z for which E k occurs. Equivalently, there are at least ε 2ν+ζ K values of k ∈ [0, K] Z for which either Z E = ∅ or F k does not occur. By Lemma 4.18, on E r the event F k occurs for every k ∈ [0, K] Z . Since K ε −β (by (4.35)), if 4ν < β then for a small enough choice of ζ ∈ (0, 1), the quantity (4.59) is of order o ∞ ε (ε). The proposition now follows.
Proof of Theorem 4.2. Assume we are in the setting of the theorem statement with ν * = 1 8 (β ∧ θ).
Fix q > 0. Recall that we have been fixing z, w ∈ rU with |z − w| ≥ 4 r throughout this section.
Proposition 4.16 implies that if E r occurs, then for each fixed choice of z, w ∈ ε q rZ 2 ∩ (rU ) with |z − w| ≥ 4 r, it holds except on an event of probability o ∞ ε (ε), at a rate which does not depend on z, w, or r, that there exists k ∈ [0, K] Z for which the corresponding set Z E k of (4.13) is non-empty.
Since the definition of E r does not depend on z, w, we can truncate on E r then take a union bound over all pairs z, w ∈ ε q rZ 2 ∩ (rU ) with |z − w| ≥ 4 r to get that if E r occurs, then except on an event of probability o ∞ ε (ε), it holds for each such pair z, w that there exists z ∈ C and r ∈ [ε 1+ν r, εr] ∩ R such that P z,w ∩ B λ 2 r (z) = ∅ and E z,w r (z) occurs.
Since the parameters in the definition of E r can be chosen so as to make P[E r ] as close to 1 as we like (Lemma 4.10), we obtain the theorem statement with 4 in place of , which is sufficient since is arbitrary.

Proofs of geometric lemmas
In this section we prove the geometric lemmas stated in Sections 4.4 and 4.5 whose proofs were postponed to avoid distracting from the main argument, namely Lemmas 4.13, 4.15, and 4.18. The arguments in this section use only the definitions in Sections 4.1 and 4.3. In particular, we do not use any of the results in Sections 4.4 or 4.5.
Proof of Lemma 4.13. See Figure 7 for an illustration. The proof consists of two main steps.
1. We show that there is a finite collection of connected sets X ⊂ C \ K with Euclidean diameter at most 4ε such that each point of C ε y is contained in the bounded connected component of C \ (K ∪ X) for one of these sets X. The sets X can be taken to be appropriate boundary arcs of Euclidean balls of radius 2ε.
2. We consider the maximal elements of our finite collection, i.e., those which do not lie in a bounded connected component of any other set in the collection. We show that any two maximal elements have to intersect, so the union of the maximal elements has Euclidean diameter at most 8ε. We then choose a single connected set (which can be taken to be an arc of a Euclidean ball of radius 8ε) which disconnects the union of the maximal elements from ∞ in C \ K.
Step 1: reducing to finitely arcs of Euclidean balls. We will first reduce to considering only a finite collection of sets X as in the statement of the lemma by looking at arcs of Euclidean balls. Let B be the set of closed Euclidean balls of the form B = B 2ε (z) for z ∈ ε 4 Z 2 with the following properties: B ∩ ∂K = ∅ and every unbounded connected subset of C \ K whose prime end closure contains y has to intersect B. Since K is compact, B is a finite set.
For B ∈ B, the set ∂B \ K is a countable union of open arcs of ∂B. Each such arc divides C \ K into a bounded connected component and an unbounded connected component. There is one such arc Y B with the property that y lies on the boundary of the bounded connected component of and Y B is not contained in the bounded connected component of C \ (K ∪ X) for any other such arc X = Y B . Note that since B has radius 2ε, the arc Y B is connected and has Euclidean diameter at most 4ε. (4.60) Indeed, let X be as in the definition of C ε y for our given z and let V X be the bounded connected component of C \ X with y on its boundary. Since X has Euclidean diameter at most ε, we can find B ∈ B such that X is contained in the interior of B. We claim that V X ⊂ U B , and hence z ∈ U B . Since X is connected and X ∩ Y B ⊂ X ∩ ∂B = ∅, it follows that X is either entirely contained in U B or X is entirely contained in the unbounded connected component of C \ (K ∪ Y B ). We claim that X cannot be entirely contained in the unbounded connected component of C \ (K ∪ U B ). Indeed, by the definition of X, each unbounded connected subset of C \ K with y on its boundary must Each of these balls is disjoint from B εr (z) and is contained in B • s k+1 , which leads to (4.61). Using Lemma 4.21 we get an upper bound for the D h -length of the segment of a D h (·, ·; C \ B r (z))-geodesic from z to a point of ∂B r (z) (such as the one shown in red) stopped at the last time it hits ∂B 2λ 4 εr (z).
This upper bound allows us to prevent such a D h -geodesic from exiting B • s k+1 . These considerations lead to the proofs of Lemmas 4.15 and 4.18. intersect X. Since X ∩ U B = ∅ and y ∈ ∂U B , each unbounded connected subset of C \ K which intersects U B must intersect X. This implies that U B ⊂ V X , but this cannot happen since X ⊂ B and by the definition of Y B . Therefore X ⊂ U B , so V X ⊂ U B , so (4.60) holds.
Step 2: maximal elements of B. We define a partial order on B by declaring that B B if and only if U B ⊂ U B . Let B * be the set of maximal elements of B, i.e., B * ∈ B * if and only if there is no B ∈ B \ {B * } such that B * B. Since B is a finite set, for every B ∈ B there exists B * ∈ B * satisfying B B * .
We claim that if B 1 , B 2 ∈ B * , then . By the maximality of B 1 , Y B 1 must be contained in the unbounded connected component of C \ (K ∪ Y B 2 ). We will now argue that U B 2 ⊂ U B 1 , which will contradict the maximality of B 2 . Indeed, by the definition of Y B 1 , every unbounded connected subset of C \ K whose prime end closure contains y has to intersect Y B 1 .
Since Y B 1 is disjoint from U B 2 and y ∈ Cl (U B 2 ), it follows that every unbounded connected subset of C \ K which intersects U B 2 has to intersect Y B 1 . Therefore, U B 2 ⊂ U B 1 , which gives the desired contradiction.
Since each set Y B for B ∈ B has Euclidean diameter at most 4ε, the preceding paragraph implies that the set Y ε y := B * ∈B * Y B * is connected and has Euclidean diameter at most 8ε. Choose a Euclidean ball B of radius at most 8ε which contains Y ε y . As in Step 1, there is a unique connected arc Y ε y of ∂ B \ K with the property that y lies on the boundary of the bounded connected component and Y ε y is not contained in the bounded connected component of C \ (K ∪ X) for any other such arc X. This arc Y ε y has Euclidean diameter at most 16ε. Then each Y B * for B * ∈ B * , and hence also each U B * for B * ∈ B * , is contained in the bounded connected component of C \ (K ∪ Y ε y ). Since each z ∈ C ε y is contained in U B for some B ∈ B, and hence in U B * for some B * ∈ B * , we get that Y ε y satisfies the desired property.
We now turn our attention to Lemmas 4.15 and 4.18. Both lemmas will be proven using the following statement, which in particular gives an explicit definition of the event F k of Lemma 4.18.
Lemma 4.21. For k ∈ [0, K] Z , let F k be the event that the following is true. We have s k+1 ≤ τ 2 r and for each z ∈ B 2λ 4 εr ( where λ 4 is the constant from Theorem 4.2, χ is as in condition 3 in the definition of E r , and c > 0 is constant depending only on a, λ 4 (which we do not make explicit). If E r occurs and ε is sufficiently small (how small depends only on a, λ 4 ), then F k occurs for each k ∈ [0, K] Z .
The reason why we use internal distances in B • s k+1 \ B εr (z) in (4.61) is as follows. Such distances are bounded above by D h (·, ·; C \ B r (z))-distances if r ≤ εr (which is the case if (z, r) ∈ Z k ), which will be important for controlling D h (·, ·; C \ B r (z))-geodesics. Furthermore, such distances are ) by Axiom II, which will be important for the proof of Lemma 4.18.
We also emphasize that the right side of (4.61) is smaller than Proof of Lemma 4.21. See Figure 8 for an illustration of the statement and proof. Assume that E r occurs. By (4.36), we have s k+1 ≤ τ 2 r . Hence we just need to check (4.61). By the definition (4.6) of t k and s k+1 and since β < χ/χ (by (4.37)), it holds for small enough ε ∈ (0, 1) that (4.62) We can cover ∂B 2λ 4 εr (z) by a λ 4 -dependent constant number of Euclidean balls of the form B εr/2 (w) for w ∈ ∂B 2λ 4 εr (z). Note that since λ 4 ≥ 1, the corresponding balls B εr (w) are disjoint from B λ 4 εr (z) ⊃ B εr (z). By the upper bound for D h -distances from condition 3 in the definition of E r and then condition 4 in the definition of E r , each such ball satisfies sup u,v∈B εr/2 (w) D h (u, v; B εr (w)) ≤ 2(ε/2) χ c r e ξhr(0) ε χ c r e ξhr(z) , (4.63) with the implicit constant depending only on a.
By summing (4.63) over all such balls B εr/2 (w), using that ∂B 2λ 4 εr (z) ∩ ∂B • t k = ∅, and comparing to (4.62), we get that for small enough ε each such ball B εr (w) is contained in B • s k+1 \ B εr (z). We deduce that the D h ·, ·; B • s k+1 \ B εr (z) -diameter of ∂B 2λ 4 εr (z) is at most a a, λ 4 -dependent constant times ε χ c r e ξhr(z) . Since ∂B 2λ 4 εr (z) ∩ ∂B • Proof of Lemma 4.18. Define F k as in Lemma 4.21. That lemma tells us that E r ⊂ K k=0 F k for small enough ε ∈ (0, 1) (depending only on a, λ 4 ). Furthermore, it is clear from the definition of F k and Axiom II that . Now assume that F k occurs. By definition, we have s k ≤ τ 2 r .
We consider (z, r) ∈ Z k and check that if ε ∈ (0, 1) is small enough, then B λ 4 r (z) ⊂ B • s k+1 and the set of D h (·, ·; C \ B r (z))-geodesics from z to points of ∂B r (z) is determined by ).
Note that the right side of (4.61) satisfies t k + cε χ c r e ξhr(z) ≤ s k+1 . Since the left side of (4.61) is an upper bound for sup u∈∂B 2λ 4 εr (z) D h (z, u), it follows that ∂B 2λ 4 εr (z) ⊂ B • s k+1 . Since B λ 4 r (z) ⊂ B λ 4 εr (z) (by (4.10)) and B • s k+1 contains every point which it disconnects from ∞, we therefore have Finally, we claim that a D h (·, ·; C \ B r (z))-geodesic from z to a point of ∂B r (z) is the same as a D h (·, ·; B • s k+1 \ B r (z))-geodesic from z to a point of ∂B r (z), which gives the desired measurability statement due to Axiom II for D h . To see this, it suffices to show that if P is a D h (·, ·; C \ B r (z))geodesic from z to a point of ∂B r (z), then P ⊂ B • s k+1 . To this end, let t be the last time that P hits ∂B 2λ 4 εr (z). By (4.61) and since P is a D h (·, ·; C \ B r (z))-geodesic, it follows that the D h -length of P | [0,t] (which equals t) is at most t k + cε χ c r e ξhr(z) < s k+1 . Consequently, P cannot exit B • s k+1 before time t. Since t is the last time that P hits ∂B 2λ 4 εr (z) and the terminal point of P is contained in ∂B r (z) ⊂ B εr (z), P cannot exit B • s k+1 after time t, either.

Forcing a geodesic to take a shortcut
The goal of this section is to prove Proposition 4.3. Throughout, we assume that we are in the setting of Theorem 1.9, so D and D are two weak γ-LQG metrics with the same scaling constants. We also let h be a whole-plane GFF and we implicitly assume (by way of eventual contradiction) that the optimal bi-Lipschitz constants c * and C * of (1.21) satisfy c * < C * .
Let R 0 be the set of r > 0 for which it holds with probability at least p 0 that the following is true. There exists u ∈ ∂B αr (0) and v ∈ ∂B r (0) such that and the D h -geodesic from u to v is unique and is contained in A αr,r (0). We note that Proposition 3.4 implies in particular that for each r > 0 one has #(R 0 ∩ [ε 1+ν r, εr] ∩ {8 −k r} k∈N ) ≥ µ log ε −1 for small enough ε ∈ (0, 1).

Outline of the proof of Proposition 4.3
The main task in the proof of Proposition 4.3 is to define the event E r (0) (which we abbreviate as E r throughout most of this section). The other events E r (z) for z ∈ C will be defined by translation.
Main ideas. The basic idea to define E r is as follows. We will define for each pair of points x , y ∈ ∂B 3r (0) a deterministic smooth bump function φ which takes a large (but independent of r, x , y ) value in a long, narrow "tube" contained in B 3r (0) which (almost) contains a path from x to y and which vanishes outside of a small neighborhood of this tube. Roughly speaking, E r will be the event that, simultaneously for every choice of x and y , this tube contains a pair of points u, v such that D h (u, v) ≤ c 1 D h (u, v) and |u − v| r; and several regularity conditions hold. We will show using Proposition 3.2 and basic estimates for LQG distances that when ρ ∈ (0, 1) is small (but independent of r), P[E r ] is close to 1 for all r ∈ ρ −1 R 0 (Lemma 5.9). We will then consider a fixed pair of points z, w ∈ C \ B 4r (0) and let x and y be the first points of ∂B 3r (0) hit by the D h -metric balls grown from z and w, respectively. This choice of x and y (and hence also the corresponding bump function φ) are random, but are determined by h| C\B 3r (0) . We will show that if E r occurs and the D h -geodesic between z and w enters B 2r (0), then the D h−φ -geodesic between z and w has to stay close to the long narrow tube where φ is large, and hence has to get close to points u, v with D h (u, v) ≤ c 1 D h (u, v) and |u − v| r. Essentially, this is because Axiom III implies that subtracting φ makes distances inside the tube much shorter than distances outside. If we let E z,w r (0) be the event that the D h -geodesic gets close to such points u, v, then since the conditional laws of h − φ and h given h| C\B 3r (0) are mutually absolutely continuous (and we can add regularity conditions to E r to control the Radon-Nikodym derivative), we get condition 4 in Theorem 4.2 (with λ 3 = 3).
Section 5.2. We give a precise statement of the properties that we need the event E r and the bump function φ described above to satisfy. We then assume the existence of these objects and deduce Proposition 4.3. Condition 1 of Theorem 4.2 (with R = ρ −1 R 0 ) is true in our framework by the definition of R 0 and Proposition 3.4. Conditions 2 and 3 are true by assumption (these conditions will be clear from the construction of E r and φ). Condition 4 is proven by comparing the conditional laws of h and h − φ given h| C\B 3r (0) , as discussed above. The rest of the section is devoted to construction the event E r and the bump functions φ. Section 5.3. We first show that for any z ∈ C and r ∈ R 0 , we can find a deterministic open "tube" V r (z) ⊂ B 3r (z) such that with uniformly positive probability over the choice of z and r, there are points u, v ∈ V r (z) with the following properties. We have D h (u, v) ≤ c 1 D h (u, v), |u − v| r, the D h -geodesic from u to v is contained in V r (z), and any path in V r (z) between z − 2r and z + 2r has to get close to each of u and v (Lemma 5.5). This is illustrated in Figure 10.
To do this, we start with a pair of points u, v as in the definition of R 0 , but with z in place of 0. Such a pair of points exists with probability at least p 0 by Axiom IV. We then extend the D h -geodesic P from u to v to a path P from z − 2r to z + 2r by concatenating P with smooth paths.
For this purpose, the fact that P is contained in A αr,r (z) is useful to ensure that the extra smooth paths intersect P only at u and v. We consider the set of squares in a fine grid which intersect P . Since there are only finitely many possibilities for this set of squares, there has to be a deterministic set of squares which equals the set of squares which intersect P with uniformly positive probability. We define V r (z) to be the interior of the union of the squares in this set.
Section 5.4. We now have an event which satisfies many of the conditions which we are interested in, but it holds only with uniformly positive probability, not with probability close to 1. To get an event which holds with probability close to 1, we consider a small but fixed ρ ∈ (0, 1) and a radius r ∈ ρ −1 R 0 . We can find a large number of disjoint balls of the form B ρr (z) contained in B 2r (0) (note that ρr ∈ R 0 ). By the spatial independence properties of the GFF (Lemma 2.7), if we make ρ sufficiently small then it holds with high probability that the event of the preceding subsection occurs for a large number of these balls B ρr (z). We then link up the corresponding sets V ρr (z) by deterministic paths of squares to find a deterministic open "tube" U x,y r joining any two given points of x, y ∈ ∂B 2r (0) with the following property. With probability close to 1, there are points u, v ∈ U x,y r such that D h (u, v) ≤ c 1 D h (u, v), |u − v| r, the D h -geodesic from u to v is contained in U x,y r , and any path in U x,y r between x and y has to get close to each of u and v (Lemma 5.7). See Figure 11 for an illustration of this part of the argument.
Section 5.5. Taking Lemma 5.7 as our starting point, we then build the high-probability event E r in Proposition 4.3 for r ∈ ρ −1 R 0 . In addition to the aforementioned conditions on the tube U x,y r , we also include extra regularity conditions which will eventually be used to prevent D h -geodesics from staying close to the boundary of U x,y r without entering it, to get geodesics from ∂B 3r (0) to ∂B 2r (0), and to control the Radon-Nikodym derivative between the conditional law of h and h − φ (where φ is the bump function mentioned above) given h| C\B 3r (0) . We also give a precise definition of the bump function φ which we will subtract from the field: it is equal to a large positive constant on the long narrow tube U x,y r , it is equal to an even larger constant on even narrower tubes which approximate each of the segments [x, 3x/2] and [y, 3y/2], and it vanishes outside of a small neighborhood of the union of U x,y r and these two narrower tubes. The definitions of these objects are illustrated in Figure 12.
Section 5.6. We prove that a D h−φ -geodesic is likely to get near points u, v satisfying (5.1), using the definition of E r (0) and deterministic arguments to compare various distances. A key point here is that we have set things up so that on E r , the D h -geodesic from u to v is contained in U x,y r and is far away from the narrow tubes where φ is larger than it is on U x,y r . This allows us to show that subtracting φ does not change the fact that D h (u, v) ≤ c 1 D h (u, v).

Proof of Proposition 4.3 assuming the existence of events and functions
In this subsection, we assume the existence of an event E r = E r (0) and a collection G r of smooth bump functions φ which satisfy a few simple properties and deduce Proposition 4.3 from the existence of these objects. The later subsections are devoted to constructing these objects exist. In particular, we will deduce Proposition 4.3 from the following proposition.
Proposition 5.1. Let 0 < µ < ν ≤ ν * be as above and let p ∈ (0, 1). There exists ρ ∈ (0, 1), depending only on p, µ, ν, such that for each r ∈ ρ −1 R 0 , there is an event E r and a finite collection G r of smooth bump functions, each of which is supported on a compact subset of A r/4,3r (0), with the following properties.
(B) (Bound for Dirichlet inner products) There is a deterministic constant Λ 0 > 0 depending only on p, µ, ν such that, writing (·, ·) ∇ for the Dirichlet inner product, it holds on E r that (C) (Subtracting a bump function forces a geodesic to take a shortcut) Suppose we are given points z, w ∈ C \ B 4r (0). There is a random φ ∈ G r depending only on z, w, and h| C\B 3r (0) such that the following is true. Let P (resp. P φ ) be the a.s. unique D h -(resp. D h−φ -) geodesic from z to w. There is a deterministic constant b 0 > 0 depending only on p, µ, ν such that if P ∩ B 2r (0) = ∅ and E r occurs, then there are times 0 < s < t < D h−φ (z, w) and such that 3) The event E r and the collection of functions G r will be defined explicitly in Section 5.5; see Section 5.1 for an overview of the definitions. The reason why we are able to restrict to a finite collection G r of bump functions φ is that we will break up space into a fine grid and require that the "tube" where φ is very large (as referred to in Section 5.1) is a finite union of squares in the grid. As explained in Lemma 5.3 just below, Properties (B) and (C) are used to check condition 4 in Theorem 4.2. The purpose of Property (B)is to control the Radon-Nikodym derivative between the conditional laws of h and h − φ given h| C\B 3r (0) .
We now explain how to conclude the proof of Proposition 4.3 assuming Proposition 5.1. Fix points z, w ∈ C \ B 4r (0) and let P be the D h -geodesic from z to w, as in Property (C). We first define the event E z,w r (0) appearing in Proposition 4.3. Let E r = E z,w r (0) be the event that there are times 0 < s < t < D h (z, w) such that D h (P (s), P (t)) ≤ c 2 D h (P (s), P (t)), and where Λ 0 is the constant from Property B. We note that (5.4) is the same as (5.3) from Property (C), but with h instead of h − φ. This condition is the main point of the definition of E r . The extra condition (5.5) is only included to control a Radon-Nikodym derivative when we compare the conditional probabilities of E r and E r given h| C\B 3r (0) .
Lemma 5.2. The event E r is a.s. determined by h| B 3r (0) and the D h -geodesic P stopped at its last exit time from B 3r (0).

Proof.
Recall that each of the functions φ ∈ G r is supported on A r/4,3r (0). Since G r is a finite set, it is clear that the condition 5.5 is determined by h| B 3r (0) .
To deal with (5.4), we first observe that the set of pairs of times s, t ∈ [0, D h (z, w)] satisfying D h (P (s), P (t)) ≤ (c * /C * ) D h (P (s), ∂B 3r (0)) is determined by P stopped at its last exit time from B 3r (0) and the internal metric D h (·, ·; B 3r (0)). Indeed, a pair (s, t) belongs to this set if and only if P (t) is contained in the D h -metric ball of radius (c * /C * ) D h (P (s), ∂B 3r (0)) centered at P (s).
For each such pair of times s, t, we have D h (P (s), P (t)) = D h (P (s), P (t); B 3r (0)). Since P is a D h -geodesic, the points P (s), P (t) and the distance D h (P (s), P (t)) = t − s for each such pair of points s, t is determined by P stopped at its last exit time from B 3r (0). Since D h (·, ·; B 3r (0)) is determined by h| B 3r (0) (Axiom II) we get that the event that there exists times s, t ∈ [0, D h (z, w)] satisfying (5.4) is determined by h| B 3r (0) and P stopped at its last exit time from B 3r (0).
We can now check condition 4 of Theorem 4.2 for the above definitions of E r = E r (0) and E r = E z,w r (0) using the mutual absolute continuity of the laws of h and h − φ. Proof. The occurrence of the events E r and E r is unaffected by adding a constant to h, so we can assume without loss of generality that h is normalized so that its circle average over ∂B 4r (0), say, is zero. By the Markov property of h, under the conditional law given h| C\B 3r (0) , we can decompose h| B 3r (0) as the sum of a harmonic function which is determined by h| C\B 3r (0) and a zero-boundary GFF on B 3r (0) which is independent from h| B 3r (0) .
Let φ ∈ G r be the smooth bump function from Property (C), which is determined by h| C\B 3r (0) . By a standard Radon-Nikodym derivative calculation for the GFF, if we condition h| C\B 3r (0) then the conditional law of h − φ is a.s. absolutely continuous with respect to the conditional law of h, and the Radon-Nikodym derivative of the former w.r.t. the latter is Note that since φ is supported on B 3r (0), the Radon-Nikodym derivative M h depends only on the zero-boundary part of h| B 3r (0) .
Define the D h−φ -geodesic P φ from z to w and the event E φ r in the same manner as P and E r but with h − φ in place of h. By (5.5), on E r , we have M h ≤ Λ. Therefore, We now claim that Indeed, Property (C) says that the main condition (5.4) in the definition of E r is satisfied with h − φ in place of h whenever E r ∩ {P ∩ B 2r (0) = ∅} occurs, which implies in particular that P φ ∩ B 2r (0) = ∅ whenever E r ∩ {P ∩ B 2r (0) = ∅} occurs. Furthermore, Property (B)implies that the Dirichlet energy condition (5.5) in the definition of E r holds with h − φ in place of h whenever E r occurs. Thus (5.9) holds.
As an immediate consequence of (5.9), a.s. For z ∈ C, r ∈ ρ −1 R 0 , and z, w ∈ C \ B 4r (z), let E r (z) (resp. E z,w r (z)) be the event E r of Proposition 5.1 (resp. the event and E z+z,w+z Let c = c (α, c 1 , µ, ν) ∈ (c * , c 1 ) be chosen as in Proposition 3.5 with α as in Lemma 5.4 and c 1 in place of c . Also let R 0 be defined as in the discussion surrounding (5.1) and let R := ρ −1 R 0 . By the definition of E z,w r (z) (in particular, (5.4)), the conditions (4.3) hold on E z,w If r > 0 such that P[G r (c , β)] ≥ β, then Proposition 3.4 implies that there exists ε 0 = ε 0 (β, c 1 , µ, ν) > 0 such that for each ε ∈ (0, ε 0 ], ∂B αr (z) ∂B r (z) Figure 9: Illustration of the statement of Lemma 5.4. The lemma asserts that with probability at least p 0 /8, there is a D h -geodesic (red) between points u and v in the inner and outer boundaries, resp., of the pink half-annulus H r (z) which is contained in H r (z) and satisfies D h (u, v) ≤ c 1 D h (u, v). The main task of Section 5 is to force a D h -geodesic between two far away points to get near a D h -geodesic like the red one in the picture. equivalently, This shows that condition 1 of Theorem 4.2 is satisfied with ρ −1 r in place of r. By Property (A) and Lemma 5.2, conditions 2 and 3 of Theorem 4.2 are satisfied for the events E r (z) and E z,w r (z) above. By Lemma 5.3, condition 4 of Theorem 4.2 is also satisfied.

Building a tube which contains a shortcut with positive probability
We now turn our attention to constructing the event E r and the collection of functions G r of Proposition 5.1, following the strategy outlined in Section 5.1. Recall that α * ∈ (1/2, 1) and p 0 ∈ (0, 1) are the parameters from Proposition 3.5 with µ and ν as in Proposition 4.3.
In the following lemma, we define a half-annulus of an annulus A to be the intersection of A with a half-plane whose boundary passes through the center of A. It is easier for us to work with a D h -geodesic which is constrained to stay in a half-annulus rather than a whole annulus. The reason for this is that it allows us to easily find paths from each of the endpoints of the geodesic to points far away from the half-annulus which do not get near the geodesic except at their endpoints (this might be trickier if the geodesic wraps around the whole annulus). The following lemma, which is a slight improvement on the condition in the definition of R 0 , will allow us to work with a half-annulus rather than a whole annulus.
Lemma 5.4. There exists α ∈ [α * , 1) depending only on µ, ν such that for each r ∈ R 0 and each z ∈ C, there is a deterministic half-annulus H r (z) ⊂ A αr,r (z) such that with probability at least p 0 /8, there exists u ∈ ∂B αr (z) and v ∈ ∂B r (z) with the following properties.
2. The D h -geodesic from u to v is unique and is contained in H r (z).
Proof. By Axioms IV and V, we can find S > s > 0 depending only on p 0 (and hence only on µ, ν) such that for each r > 0, it holds with probability at least 1 − p 0 /2 that the following is true.
• Any two points of A 3r/4,r (z) which are not contained in a single quarter-annulus of A 3r/4,r (z) lie at D h -length at least sc r e ξhr(z) from each other.
Since A αr,r (z) ⊂ A 3r/4,r (z) for each α ∈ [3/4, 1), Lemma 2.11 applied with the above choice of s and S gives an α ∈ [(3/4) ∨ α * , 1) depending on p 0 and α * such that for each r > 0 it holds with probability at least 1 − p 0 /2 that the following is true. For each pair of points u, v ∈ A αr,r (z) such that D h (u, v; A αr,r (0)) = D h (u, v), it holds that u and v are contained in a single quarter-annulus of A αr,r (z) and D(u, v) ≤ D h (A αr,r (z), ∂B 2r (z)). This happens in particular if there is a D h -geodesic from u to v contained in A αr,r (z).
Combining this with translation invariance (Axiom IV) and the definition of R 0 shows that for r ∈ R 0 , it holds with probability at least p 0 /2 that the conditions in the lemma statement hold but with a random quarter-annulus in place of a deterministic half-annulus. This random quarter annulus is a.s. contained in one of four possible deterministic half-annuli, so must be contained in one of these four half-annuli with probability at least p 0 /8. We therefore obtain that for an appropriate choice of H r (z), it holds with probability at least p 0 /8 that all of the conditions in the lemma statement hold.
The particular choice of η in (5.15) will not be used until (5.49) below. For now, the reader should just think of it as a small constant depending on c 1 , c 2 . The following lemma gives us the basic "building blocks" which will be used to construct E r in the next two subsections.
Lemma 5.5. There exist small parameters b 1 , p 1 ∈ (0, 1/100) depending only on µ, ν and a parameter ε 1 ∈ (0, b 1 /100) depending only on c 1 , c 2 , µ, ν such that for each z ∈ C and each r ∈ R 0 , there exists a deterministic connected open set V r (z) ⊂ B (2+2ε 1 )r (z) with the following properties. The set V r (z) is the interior of a finite union of squares in S εr (B 2r (z)), z − 2r, z + 2r ∈ V r , and with probability at least p 1 , there are points u, v ∈ V r (z) ∩ B r (z) with the following properties.
1. We have (5.16) and the D h -geodesic from u to v is unique and is contained in Figure 10: Illustration of the statement and proof of Lemma 5.5. Building on the setting of Figure 9, we show that there is a deterministic long narrow "tube" V r (z) (light green), which is the interior of the set of ε 1 r × ε 1 r squares with corners in ε 1 rZ 2 which intersect a certain path from z − 2r to z + 2r, with the following property. With positive probability, every path in the tube from a point near z − 2r to a point near z + 2r has to get near a pair of points u, v in the tube for which D h (u, v) ≤ c 1 D h (u, v). We will eventually add a bump function to h which takes a very negative value in such a tube in order to force a geodesic between points which are far away from B 2r (z) to get near u and v. 3. Each point of O u lies at D h (·, ·; V r (z))-distance at most η D h (u, v) from u, and the same is true with v in place of u.

Let
Proof. Let α be as in Lemma 5.4 and set b 1 := 1 − α. On the event that points u ∈ ∂B αr (z) and v ∈ ∂B r (z) satisfying the conditions on Lemma 5.4 exist (which happens with probability at least p 0 /8), choose one such pair of points (u, v) in some measurable manner. Otherwise, let u = v = 0.
On the event {u = 0}, let P be the unique D h -geodesic from u to v and let H r (z) ⊂ A αr,r (z) be the half-annulus with P ⊂ H r (z) as in Lemma 5.4. We will now extend P to a path P in B 2r (z) from z − 2r to z + 2r (which will no longer be a D h -geodesic). To this end, we first let v := (3/2)(v − z) + z ∈ ∂B 3r/2 (z) and we let L − (resp. L + ) be the linear segment from z to u (resp. v to v ). We note that the Euclidean distance between L − and L + is at least b 1 r. We can choose a path π − from z − 2r to z and a path π + from v to z + 2r in B 2r (z) such that the Euclidean distances from π − ∪ π + to H r (z) and from π − ∪ L − to π + ∪ L + are each at least b 1 r. Let P be the concatenation of π − , L − , P , L + , π + .
Since |u − v| ≥ b 1 r on the event {u = 0}, Axiom V together with Lemma 2.9 imply that we can find ε 1 ∈ (0, b 1 /100) depending only on c 1 , c 2 , µ, ν such that with probability at least p 0 /9, the event of Lemma 5.4 occurs (i.e., u = 0) and also The number of subsets of S ε 1 r (B 2r (z)) is bounded above by a deterministic constant depending only on ε 1 . Consequently, we can choose p 1 ∈ (0, 1) depending only on µ, ν, D and a deterministic K r (z) ⊂ S ε 1 r (B 2r (z)) such that with probability at least p 1 , the events of Lemma 5.4 and (5.17) occur and also K r (z) = S ∈ S ε 1 r (B 2r (z)) : S ∩ P = ∅ .
Let V r (z) be the interior of the union of the squares in K r (z). Since z − 2r, z + 2r ∈ P and P is connected, it follows that V r (z) is connected and contains z − 2r and z + 2r. Henceforth assume that the events of Lemma 5.4, (5.17), and (5.18) occur. We will check the conditions in the lemma statement with V r (z) as above.
Condition 1. This is immediate from the conditions on u and v from Lemma 5.4. Condition 2. By the above definitions of π − and L − , the Euclidean ε 1 r-neighborhood of each square of S ε 1 r which intersects both B 2ε 1 r (π − ∪ L − ) and B 2ε 1 r (H r (z)) must be contained in B 10ε 1 r (u). Furthermore, using that L − is a linear segment, we get that the ε 1 -neighborhood of each such square which intersects B 2ε 1 r (π − ∪ L − ) and belongs to K r (z) (as defined in (5.18)) must be contained in O u , with O u as in the lemma statement. Since the Euclidean distance between π − ∪ L − and π + ∪ L + is at least b 1 r ≥ 100ε 1 r and P ⊂ H r (z), we see that removing O u disconnects V r (z) into at least two connected components, and the Euclidean distance between the connected component which contains z − 2r and the union of the other connected components is at least ε 1 r. A similar argument applies with v in place of u.
Condition 3. Each point of O u is contained in a square of K r (z) which lies at graph distance at most 40 from a square which contains u in the adjacency graph of squares of K r (z). The same is true with v in place of u. It therefore follows from (5.17) that condition 3 in the lemma statement is satisfied.
For z ∈ C and r > 0, let F r (z) be the event that there exists u, v ∈ V r (z) ∩ B r (z) such that the three numbered conditions in the statement of Lemma 5.5 are satisfied. In the next subsection, we will use the local independence properties of the GFF (in the form of Lemma 2.7) to argue that for a small enough ρ ∈ (0, 1) and for all r ∈ ρ −1 R 0 , it is very likely that F ρr (z) occurs for many points z ∈ B r (0). To apply the lemma, we will need the following measurability statement.
Proof. First note that the occurrence of F r (z) is unaffected by scaling each of D h and D h by the same constant factor. Therefore, Axiom III implies that F r (z) is determined by h, viewed modulo additive constant. So, we only need to show that F r (z) ∈ σ h| B 3r (z) .
We first observe that for u, v ∈ B r (z), we have D h (u, v) ≤ (c * /C * ) 2 D h (u, ∂B 2r (z)) if and only if v is contained in the D h -metric ball of radius (c * /C * ) 2 D h (u, ∂B 2r (z)) centered at v. Since this  Figure 11: Illustration of the statement and proof of Lemma 5.7. To get an event with probability close to 1, instead of just an event with uniformly positive probability, we consider a large number of disjoint balls B ρr (z) centered at a finite set of points Z ⊂ ∂B r (0) and use Lemma 2.7 to argue that with high probability, the event F ρr (z) of Lemma 5.5 occurs for a suitably "dense" set of points z ∈ Z. Then, we link up the tubes V ρr (z) for z ∈ Z (light green) via deterministic paths L k (blue). For a given choice of points x, y ∈ ∂B 2r (0) with |x − y| ≥ δr, we define U x,y r to be the union of the sets V ρr (z) for points z ∈ Z along the counterclockwise arc of ∂B r (0) from x/2 to y/2, the squares of S ε 1 ρr (B r (0)) which intersect the deterministic paths joining these sets V ρr (z), and paths of squares starting from each of x and y (light blue).
D h -metric ball is contained in B 2r (z), we infer from the locality of D h that the set of u, v ∈ B 2r (z) for which D h (u, v) ≤ (c * /C * ) 2 D h (u, ∂B 2r (z)) is determined by h| B 3r (z) . If D h (u, v) ≤ (c * /C * ) 2 D h (u, ∂B 2r (z)), then each D h -geodesic from u to v is contained in B 2r (z), so the set of D h -geodesics from u to v is the same as the set of D h (·, ·; B 2r (z))-geodesics from u to v.
Furthermore, by the definition (1.21) of c * and C * , we see that By combining these observations with the locality of the metrics D h and D h , it follows that F r (z) is determined by h| B 3r (z) .

Building a tube which contains a shortcut with high probability
In the rest of this section we no longer need to vary z, so we take z = 0. We will now prove a variant of Lemma 5.5 which holds with probability close to 1, not just with uniformly positive probability. This will be accomplished by using Lemma 2.7 to produce a large number of small balls contained in B 2r (0) for which the event of Lemma 5.5 occurs. See Figure 11 for an illustration.
1. (Existence of a shortcut) We have (5.20) and the D h -geodesic from u to v is unique and is contained in U x,y r . Proof.

(5.21)
Then the balls B 4ρr (z) for z ∈ Z are disjoint and each such ball is contained in A (1−4ρ)r,(1+4ρ)r (0). By Lemmas 5.5 and 5.6, if r ∈ ρ −1 R 0 , then each of the events F ρr (z) for z ∈ Z has probability at least p 1 and is determined by (h − h 4ρr (z))| B 3ρr (z) . Each arc I ⊂ ∂B r (0) with Euclidean length at least δr/4 satisfies #(Z ∩ I) ≥ n * . Therefore, Lemma 2.7 (applied with the whole-plane GFF h(·/(3ρr)) in place of h) implies that for each such arc I, We can choose at most 4πδ −1 arcs of ∂B r (0) with Euclidean length δr/4 in such a way that each arc of ∂B r (0) with Euclidean length at least δr/2 contains one of these arcs. By a union bound, we therefore get that with probability at least 1 − (1 − p)/4, Each arc of ∂B r (0) with length at least δr/2 contains a point z ∈ Z s.t. F ρr (z) occurs. (5.23) We will show that the statement of the lemma is satisfied with ε = ε 1 ρ and b = b 1 ρ. (5.24) Step 2: defining U x,y r . Enumerate Z = {z 1 , . . . , z N }, where N := 100n * δ −1 and z k := r exp 2πiδk 100n * . Also set z 0 := z N . We now join up the balls B 2ρr (z k ), in a manner which is illustrated in Figure 11.
For k ∈ [1, N ] Z , choose in a deterministic manner a piecewise linear path L k from z k−1 + 2ρr to z k − 2ρr which is contained in A (1−4ρ)r,(1+4ρ)r (0). We can choose the paths L k in such a way that the L k 's do not intersect any of the balls B 2ρr (z) for z ∈ Z and lie at Euclidean distance at least ρr from one another. Now consider points x, y ∈ ∂B 2r (0) with |x − y| ≥ δr. By possibly re-labeling, we can assume without loss of generality that the counterclockwise arc of ∂B 2r (0) from x to y is shorter than the clockwise arc. Let J ⊂ ∂B r (0) by the counterclockwise arc from x/2 to y/2, so that J has length at least δr/2. Let k x , k y ∈ [1, N ] Z be chosen so that J ∩ Z = {z kx , . . . , z ky }. Let L x (resp. L y ) be a smooth path from x to z ky − 2r (resp. from z ky + 2r to y) which does not intersect any of the B 2ρr (z)'s for z ∈ Z and such that L x and L y lie Euclidean distance at least ρr from each other and from each L k for k ∈ [k x + 1, k y ] Z .
Recall that for X ⊂ C, S ε 1 ρr (X) denotes the set of closed Euclidean squares of side length ε 1 ρr with corners in ε 1 ρrZ 2 which intersect X. With V ρr (z) as in the definition of F ρr (z), we define and we let U x,y r be the interior of U x,y r . Since each V r (z k ) is the interior of a finite union of squares in S ε 1 ρr (B ρr (z k )), it follows that U x,y r is the interior of a finite union of squares S ε 1 ρr (A r/2,2r (0)). Since the V r (z k )'s are connected, it is clear that U x,y r is connected and contains x, y. We also note that U x,y r is deterministic.
Step 3: checking the conditions for u and v. On the event that (5.23) holds, there is a random k ∈ [k x , k y ] Z for which F ρr (z k ) occurs. If this is the case, choose such a k and point u, v ∈ V ρr (z k ) ∩ B ρr (z k ) as in the definition of F ρr (z k ) in some measurable manner. We will show that for ε, b as in (5.24), the conditions in the lemma statement hold whenever (5.23) holds.
Condition 1. Since B 2ρr (z k ) ⊂ B 4ρr (u) and V ρr (z k ) ⊂ U x,y r , it is immediate from Condition 1 in the definition of F ρr (z k ) that this condition with b = b 1 ρ whenever (5.23) holds.
Condition 2. Assume (5.23). Let and let W k (x) and W k (y) be the interiors of W k (x) and W k (y), respectively. By (5.25), U x,y Since L x , L y , and the L k 's for k ∈ [k x + 1, k y ] Z each lie at Euclidean distance at least ρr from one another and do not intersect the interiors of the balls B ρr (z) for z ∈ Z and ε 1 < 1/100, the sets W k (x) and W k (y) lie at Euclidean distance at least ρr/2 from each other and from B ρr (z k ). We have U x,y r ∩ B 20ε 1 ρr (u) = V ρr (z k ) ∩ B 20ε 1 ρr (u), (5.27) so the definition of O u is unaffected if we replace U x,y r by V ρr (z k ). Furthermore, the connected component of U x,y r \ O u which contains x is the same as the union of W k (x) and the connected component of V ρr (z k ) \ O u which contains z k − 2ρr; and the union of the other connected components of U x,y r \ O u is the same as the union of W k (y)) and the connected components of Eqn. (5.29) Lem. 5.7 Lem. 5.9 , which is the support of the bump function f x,y r . The yellow regions are the supports of the bump functions g x r and g y r , which are used to force D h -geodesics started from points outside of B 3r (0) to enter B ηr (U x,y r ). The figure shows the relevant set for one pair of points x, y ∈ ∂B 2r (0), but all of the conditions in the event E r are required to hold simultaneously for all pairs of points x, y ∈ ∂B 2r (0) with |x − y| ≥ δr. This is important since in Section 5.6 we will take x = (3/2)x and y = (3/2)y to be the random points where the metric balls based at the starting and ending points of a given geodesic (here shown in grey) first hit ∂B 3r (0). Right: Schematic diagram of how the various quantities in the definitions of E r and G r are chosen. An arrow between two parameters indicates that one is chosen in a way which depends directly on the other. The colors indicate where the choice is made. The parameters δ and ∆ are chosen simultaneously. Most of the choices in the figure depend on p, but this is not illustrated. In the end, all of the parameters depend only on p, µ, ν (and the choice of metric).
do not contain z k + 2ρr. By condition 2 in the definition of F ρr (z k ), we find that these two sets lie at Euclidean distance at least ε 1 ρr from one another.
Condition 3. By (5.27), condition 3 in the definition of F ρr (z k ) implies that each point of O u lies at D h (·, ·; U x,y r )-distance at most η D h (u, v) from u. The same is true with v in place of u.

Definition of the event E r and the bump functions G r
The goal of this subsection is to define the event E r and the collection of smooth bump functions G r appearing in Proposition 5.1. We will also check Properties (A)and (B)from that proposition. Property (C) will be checked in Section 5.6. The definitions in this section are illustrated in Figure 12, left. Before proceeding with the details, we briefly discuss the main ideas involved. Following Section 5.1, we want to define G r to include for each x, y ∈ ∂B 3r (0) a function φ which is equal to a large positive constant on the region U x,y r of Lemma 5.7 and which is supported on the union of a small neighborhood of U x,y r and two even narrower "tubes" which approximate the segments [x, 3x/2] and [y, 3y/2] (shown in yellow in the figure). The event E r will consist of the conditions of Lemma 5.7 plus several regularity conditions discussed below.
We will eventually consider a fixed pair of points z, w ∈ C \ B 4r (0) and choose x, y ∈ ∂B 2r (0) in such a way that x := 3x/2 and y := 3y/2 are the first points of ∂B 3r (0) hit by the D h -metric balls grown from z and w, respectively. Since these points are random, it is important that the conditions in our event hold simultaneously for all possible choices of x and y. We will show in Section 5.6 that on E r , subtracting a suitable φ ∈ G r from the field makes distances in the support of φ much shorter than distances outside, so the D h−φ -geodesic has to travel through the support of φ and hence has to get close to the points u, v of Lemma 5.7.
There are several subtleties involved in this argument which are dealt with via regularity conditions in the definition of E r . For example, Lemma 5.7 requires that |x − y| ≥ δr, so we need to ensure that our random metric ball hitting points x , y are separated. This is the purpose of condition 4 in the definition of E r . Another difficulty is that it is relatively straightforward to get D h−φ -geodesics into the support of φ, but we want such geodesics to actually enter the region U x,y r where φ is equal to a large positive constant. The reason for this is that we will be comparing ratios of distances via Weyl scaling (Axiom III) and it could be that φ is much smaller on some parts of its support than it is on U x,y r . To deal with this, we will include a condition to the effect that paths which stay in a small neighborhood of ∂U x,y r without entering U x,y r are very long (condition 6).
We also need functions in G r to be supported on A r,3r (0) so we need to make the yellow tubes in Figure 12 very close to x and y without actually allowing these tubes to contain x and y (condition 8). The choice of constants involved in these conditions is somewhat delicate, so the event E r will include several parameters.
We now commence with the definitions. Fix a parameter δ ∈ (0, 1), to be chosen in a manner depending only on p in Lemma 5.9 below. Let ρ, b, ε be as in Lemma 5.7 for this choice of δ and with p = 1 − (1 − p)/2, so that ρ, b, ε depend only on δ, p, µ, ν. The definitions of E r and G r involve several additional small parameters ∆, a ∈ (0, 1) and ζ, a, θ ∈ (0, ε) and large parameters A, M, Λ 0 > 1 which we will choose in Lemma 5.9 below, in a manner depending only on p, µ, ν. See Figure 12, right for a schematic illustration of how the parameters are chosen.

Definition of G r
We first give the definition of G r in terms of the above parameters. For each x, y ∈ ∂B 2r (0) with |x − y| ≥ δr, choose in a deterministic manner depending only on U x,y r (not on the particular values of x and y) a smooth, compactly supported bump function f x,y r : C → [0, 1] which is identically equal to 1 on U x,y r and vanishes outside of B ζr (U x,y r ). Since each U x,y r is the interior of a finite union of squares in S εr (B 2r (0)), there are at most a finite, r-independent number of possibilities for U x,y r as x and y vary. From this and the scale invariance of Dirichlet energy (i.e., (f (r·), f (r·)) ∇ = (f, f ) ∇ ) it follows that we can arrange that the Dirichlet energy (f x,y r , f x,y r ) ∇ is bounded above by a constant depending only on ε, ζ.
If we subtract a large constant multiple of f x,y r from h, then LQG geodesics for the resulting field between points of U x,y r will tend to stay in U x,y r . However, we also need to get geodesics between points of C \ B 4r (0) into U x,y r . For this purpose, we will also subtract even larger constant multiples of bump functions g x r and g y r which are supported in narrow tubes which approximate the segments [x, 3x/2] and [y, 3y/2]. The supports of these bump functions are shown in yellow in Figure 12.
To define these bump functions, we first define for x ∈ ∂B 2r (0) the set where here we recall from (5.14) that S θr ([x, (3/2 − θ)x]) is the set of θr × θr squares with corners in Since the constant K f of (5.29) depends only on A, ∆, a, which have already been chosen in a manner depending only on p, µ, ν, we can find a small enough θ ∈ (0, ζ/100) depending only on p, µ, ν such that condition 8 holds with probability at least 1 − (1 − p)/100.
Recall from (5.28) that W x r is the interior of the union of a set of squares in S θr (B 3r (0)). By Axiom V and Lemma 2.9, we can find a sufficiently large M > 0 depending only θ (hence only on p, µ, ν) such that condition 9 holds with probability at least 1 − (1 − p)/100.
The definition of the set of bump functions G r above does not use the parameter Λ 0 . As discussed just after (5.32), the number of functions in G r and the Dirichlet energies of these functions are each bounded above by constants which depend only on p, µ, ν and the other parameters which we have already chosen in a manner depending only on p, µ, ν. Consequently, we can find a constant Λ 0 > 0 depending only on p, µ, ν such that condition 10 holds with probability at least 1 − (1 − p)/100.
Combining our above estimates gives the statement of the lemma.

Subtracting a bump function to move a geodesic
To prove Proposition 5.1, it remains to check Property (C) for the event E r and the collection of smooth bump functions G r defined above. To this end, fix distinct points z, w ∈ C \ B 4r (0) and let P = P z,w be the (a.s. unique) D h -geodesic from z to w. We first grow the D h -metric balls until they hit ∂B 3r (0). Let σ r (resp. σ r ) be the smallest s > 0 for which the D h -metric ball B s (z; D h ) (resp. B s (w; D h )) intersects B 3r (0). Also let x (resp. y ) be a point of ∂B 3r (0) ∩ B σr (z; D h ) (resp. B σr (w; D h )), chosen in some manner depending only on the appropriate D h -metric ball 6 , and define the points of ∂B 2r (0) x := (2/3)x and y := (2/3)y .
In the notation (5.30), we set Then φ ∈ G r , as defined in (5.31), and φ is determined by x, y and hence by h| C\B 3r (0) . Hence to prove Property (C) it remains only to prove the following.
Lemma 5.10. Let P φ be the (a.s. unique) D h−φ -geodesic from z to w. If P ∩ B 2r (0) = ∅ and E r occurs, then there are times 0 < s < t < D h−φ (z, w) such that The rest of this section is devoted to the proof of Lemma 5.10. To lighten notation, write Throughout, we assume that E r occurs and P ∩ B 2r (0) = ∅. The proof is an elementary (though somewhat technical) deterministic argument using the conditions in the definition of E r , and is divided into several lemmas.
Lemma 5.11 allows us to apply all of the conditions in the definition of E r with x = x and y = y (note that these conditions hold for all x, y ∈ ∂B 2r (0) with |x − y| ≥ δr simultaneously). We will use this fact without comment throughout the rest of the proof.
Proof of Lemma 5.11. Since P is a D h -geodesic, the D h -distance between the metric balls B σr (z; D h ) and B σr (w; D h ) is equal to the D h -distance traveled by P between the times when it hits these two metric balls. Since P enters B 2r (0), it must cross between the inner and outer boundaries of A 2r,3r (0) at least twice between hitting these two metric balls, so the D h -distance between B σr (z; D h ) and B σr (w; D h ) must be at least 2D h (∂B 2r (0), ∂B 3r (0)). Condition 4 in the definition of E r implies that if |x − y| < δr then D h (∂B 2r (0), ∂B 3r (0)) ≥ D h (x , y ; A r,4r (0)) which is at least the D h -distance between B σr (z; D h ) and B σr (w; D h ). This is a contradiction and therefore |x − y| ≥ δr.
We now prove an upper bound for D h−φ (x , y ). Since P φ is a D h−φ -geodesic, this upper bound will allow us to constrain the behavior of P φ since P φ cannot have any segment whose D h−φ -length is larger than D h−φ (x , y ) (see Lemma 5.13 below). Since W x r and W y r each intersect U , we can combine (5.39), (5.40), and (5.41) and use the triangle inequality to get (5.38).
Lemma 5.13. In the notation (5.37), P φ is contained in B 2ζr (U ∪ W). Furthermore, there is no segment of P φ of Euclidean diameter ≥ εr/100 which is contained in B 2ζr (∂U ) \ W.
Proof. Since φ is supported on B 3r (0), the definitions of σ r , σ r , B σr (z; D h ), and B σr (w; D h ) are unaffected if we replace h by h − φ. Since the segment of P φ between hitting these metric balls is the D h−φ -shortest path between these metric balls, Lemma 5.12 implies that D h−φ -length of segment of P φ between B σr (z; D h ), and B σr (w; D h ) ≤ e −ξK f (A + 4)c r e ξhr(0) .
(5.42) We will now explain how (5.42) together with the definition of E r allows us to constrain the behavior of P φ .
As in the proof of Lemma 5.11, condition 4 in the definition of E r implies that the D h -distance between B σr (z; D h ), and B σr (w; D h ) is at least 2∆c r e ξhr(0) , which is larger than e −ξK f (A+4)c r e ξhr(0) by the definition (5.29) of K f . If P φ did not enter the support B ζr (U )∪W of φ, then the D h−φ -length Since |u−v| ≥ br, we have |P φ (s)−P φ (t)| ≥ (b−40ε)r and since u, v ∈ B r (0) we have P φ (s), P φ (t) ∈ B 3r/2 (0).
It remains to check the condition (5.43). Recall that D h (u, v) ≤ (c * /C * ) 2 D h (u, ∂B 4ρr (u)) and the D h -geodesic from u to v is contained in U . Since φ ≡ K f on U and φ ≤ K f on B 4ρr (u), it follows that By (5.45) and the triangle inequality, which is bounded above by the right side of (5.43) by the definition (5.15) of η.
Proof of Lemma 5.10. Let s and t be as in Lemma 5.14. By that lemma, it remains only to check that D h−φ P φ (s), P φ (t) ≤ c 2 D h−φ P φ (s), P φ (t) .
By (5.45) and the definitions of c * and C * , By the triangle inequality, (5.46) implies that which re-arranges to give It follows from this that each D h−φ -geodesic from u to v is disjoint from W and D h−φ (u, v) ≥ e −ξK f D h (u, v). By combining this with (5.44) and condition 1 in the definition of E r , we get ≤ c 1 (1 + 2η) 1 − 2c −1 * C * η D h−φ P φ (s), P φ (t) (by (5.47)) ≤ c 2 D h−φ (P φ (s), P φ (t)) (by the definition (5.15) of η). (5.49) 6 Proof of Theorem 1.9 Assume we are in the setting of Theorem 1.9 and let h be a whole-plane GFF. Also recall the definitions of the optimal bi-Lipschitz constants c * and C * from (1.21) and the events G r (C , β) and G r (c , β) from (3.2) and (3.3). We want to show that c * = C * . To do this we will assume that c * < C * and derive a contradiction. The following proposition will be used in conjunction with Proposition 3.3 to tell us that there are many scales for which the pairs (u, v) such that D h (u, v)/D h (u, v) is close to C * are very sparse.
Proof. Assume c * < C * . Let ν * be as in Theorem 4.2 and fix parameters 0 < µ < ν ≤ ν * and c * < c 1 < c 2 < C * chosen in a manner depending only on c * and C * . The proof follows the strategy outlined in the "main idea" part of the outline in Section 1.5. Theorem 4.2 and Proposition 4.3 will allow us to show that if q > 0 is fixed, then with probability tending to 1 as ε → 0, the following is true. For every pair of points z, w ∈ (ε q Z 2 ) ∩ B r (0) with |z − w| ≥ βr, the D h -geodesic P from z to w has to hit a pair of points P (s), P (t) such that |P (s) − P (t)| ≥ const ×ε 1+ν r and D h (z, w) ≤ c 2 D h (z, w). This allows us to show that D h (z, w)/D h (z, w) is bounded above by C * minus a γ-dependent power of ε for all such pairs of points z, w. We can then use Hölder continuity to get the same statement for all pairs of points z, w ∈ B r (0) with |z − w| ≥ βr simultaneously.
Choosing ε to be an appropriate γ-dependent power of δ then gives (6.1).
Fix χ ∈ (0, ξ(Q − 2)) and χ > ξ(Q + 2), as in Lemma 2.8. By Axiom V, for each p ∈ (0, 1) we can find a bounded open set U ⊂ C which contains B 2 (0) such that P[sup u,v∈B 2r (0) D h (u, v) < D h (B 2r (0), r∂U)] ≥ p for every r > 0. On the event of the preceding sentence, every D h -geodesic between two points of B 2r (0) is contained in rU. By applying Lemma 2.8 with K = U and then sending p → 1, we get that with probability tending to 1 as ε → 0, at a rate which is uniform in r, that for any two points z, w ∈ C with |z − w| ≤ (ε q ∨ (bε 1+ν ))ρ −1 r which are either contained in Henceforth assume that this is the case.

Dimension calculations
An important remaining question concerning the LQG metric is the following.
Problem 7.1 (Hausdorff dimension of γ-LQG). Compute the exponent d γ appearing in (1.5), which is the Hausdorff dimension of C with respect to the γ-LQG metric (this is proven in [GP19b]).
Since ξ = γ/d γ and Q = 2/γ + γ/2, Problem 7.1 is equivalent to determining the relationship between these two parameters. The only case in which d γ is known is when γ = 8/3, in which case d √ 8/3 = 4. Due to existing results in the literature, d γ can equivalently be defined in a large number of other equivalent ways, e.g., the following.
1. For a large class of infinite-volume random planar maps in the γ-LQG universality class, the number of vertices in the graph distance ball of radius r centered at the root vertex is of order r dγ +or(1) [DG18, Theorem 1.6] and the graph distance traveled by a simple random walk started from the root vertex and run for n steps is of order n 1/dγ +on(1) [GM17,GH18].
(7.1) However, this prediction is known to be false at least for small values of γ due to the results of Ding-Goswami [DG16]. See [DG18,GP19a] for rigorous upper and lower bounds for d γ as well as additional discussion about various possibilities for its value. In addition to d γ , there are a number of other interesting dimensions related to γ-LQG which have not yet been computed, for example the following.
Problem 7.2 (Geodesic dimension). Compute the Euclidean Hausdorff dimension of the γ-LQG geodesic between two typical points of C.
In this paper, we gave a characterization of the γ-LQG metric in terms of its coupling with the GFF. In light of Problem 7.8, it is natural to ask if there is also a characterization solely in terms of the metric space structure, which does not require reference to the GFF. Such a characterization of the Brownian map (equivalently, the 8/3-LQG sphere) is proven in [MS15c]. A purely metric characterization of γ-LQG could potentially play an important role in a solution to Problem 7.5.
Problem 7.11 (Metric space characterization). Is there a characterization of (C, D h ) as a metric space (or of (C, µ h , D h ) as a metric measure space), without reference to the GFF and the embedding of this metric space into C?

Additional properties of the LQG metric
The construction of the 8/3-LQG metric in [MS15b,MS16b,MS16c] yields many special properties of the metric in this case which are not known (and in many cases not expected to hold) for general γ ∈ (0, 2). For example, one has d √ 8/3 = 4. Moreover, in the case when h is the GFF associated with a quantum sphere or 8/3-quantum wedge, the quantum surfaces obtained by restricting h to the complementary connected components of a 8/3-LQG metric ball are conditionally independent quantum disks given their boundary lengths. Many further properties can be obtained using the equivalence of 8/3-LQG surfaces and Brownian surfaces. However, there is nothing obviously special about γ = 8/3 from either of the definitions of the LQG metric given in this paper (the limit of LFPP or the axiomatic definition).
Problem 7.12. Can one prove that d √ 8/3 = 4, the independence properties for complementary connected components of a 8/3-LQG metric ball, or any other special property of the 8/3-LQG metric directly from the LFPP definition or the axiomatic definition?
There has been a recent proliferation of exact formulas for quantities related to the γ-LQG area and boundary length measures for general γ ∈ (0, 2), proven using ideas from conformal field theory: see, e.g., [KRV17,Rem17,RZ18]. In the special case when γ = 8/3, exact formulas for various quantities associated with the 8/3-LQG metric can be obtained using its connection to the Brownian surfaces. Exact formulas for the γ-LQG metric, if they can be found, could be very useful in attempts to solve most of the other problems listed above.
Problem 7.14 (Topology of geodesics). For a general value of γ ∈ (0, 2), what is the maximal possible number of γ-LQG geodesics joining two points in C? Is this number finite, and, if so, does it depend on γ? More generally, can one prove results about the possible topologies of the set of γ-LQG geodesics joining two points in C analogous to the results for the Brownian map in [AKM17]?
Liouville Brownian motion [GRV16,Ber15] is the natural "quantum time" parameterization of Brownian motion on an LQG surface. If we condition Liouville Brownian motion to travel a macroscopic distance (e.g., from the origin to the unit circle) in a short amount of time, then it is natural to expect that it would roughly follow a path of minimal LQG length.
Problem 7.15 (Liouville Brownian motion and LQG geodesics). Does Liouville Brownian motion conditioned to travel a macroscopic (Euclidean or quantum) distance in a short amount of time approximate an LQG geodesic?
There is a one-parameter family of infinite-volume γ-LQG surfaces with boundary called quantum wedges, which can be indexed by the weight parameter w > 0. See [DMS14] for details. In [DMS14], building on [She16a], it is shown that one can conformally together a weight-w 1 quantum wedge and a weight-w 2 quantum wedge according to the quantum length measure along their boundaries to get a weight-w 1 + w 2 quantum wedge decorated by an SLE κ (w 1 − 2; w 2 − 2) curve which corresponds to the gluing interface. In [GM16], it is shown that in the special case when γ = 8/3, this conformal welding is compatible with the 8/3-LQG metric in the following sense: the weight-(w 1 + w 2 ) quantum wedge, equipped with its 8/3-LQG metric, is the metric space quotient of the weight-w 1 and weight-w 2 quantum wedges, equipped with their 8/3-LQG metrics, under the same equivalence relation used to define the conformal welding.
The main missing ingredient needed to solve Problem 7.16 is suitable estimates for distances between points of ∂D with respect to the γ-LQG metric induced by a free-boundary GFF on D (or a variant thereof, like the quantum disk). For γ = 8/3, the needed estimates are proven in [GM16, Section 3.2] using results for the Brownian disk.

Extensions of the theory
Throughout this paper, we have neglected the critical case when γ = 2.
See [DRSV14a,DRSV14b] for a construction of the γ-LQG measure for γ = 2. One possible approach to Problem 7.17 is to try to take a limit of the γ-LQG metrics as γ increases to 2 (it is shown that the 2-LQG measure is the γ 2 limit of the γ-LQG measures, appropriately renormalized, in [APS18]). Another (likely more involved) possibility is to adapt the arguments of this paper and its predecessors [GM19c, DFG + 19, GM19a] to the critical case, corresponding to LFPP with parameter ξ = 2/d 2 . A major difficulty in the critical case is that the 2-LQG metric is not expected to be Hölder continuous w.r.t. the Euclidean metric (indeed, the optimal Hölder exponent from [DFG + 19, Theorem 1.7] converges to zero as γ → 2 − ), so more refined estimates for the continuity of the metric and for LFPP are likely to be required.
For ξ > 2/d 2 , we do not expect LFPP to converge to a continuous metric on C, but it is still possible that it has a scaling limit (in some topology) which is a metric on C with an uncountable fractal set of "singularities" whose distance to every other point is infinite. These singularities should arise from thick points of the Gaussian free field with high enough thickness [HMP10]. This metric should be related to Liouville quantum gravity with central charge c ∈ (1, 25). Note that the central charge associated with γ-LQG for γ ∈ (0, 2] is c = 25 − 6(2/γ + γ/2) 2 ∈ (−∞, 1]. We refer to [GHPR19] and the references therein for more on LQG with c ∈ (1, 25).
The γ-LQG measure is a special case of a more general theory of random measures called Gaussian multiplicative chaos (GMC) [Kah85,RV14], which studies limits of regularized versions of "e γX dz" for certain Gaussian random distributions X. Here, X is a random distribution on R n for some n ∈ N and dz denotes Lebesgue measure on R n . Problem 7.19 (More general random metrics). Is there a more general theory of random metrics associated with random Gaussian distributions analogous to GMC? In particular, can one construct metrics with similar properties to the γ-LQG metric in higher dimensions? Some of the arguments in this paper are specific to the GFF. One reason for this is that we make extensive use of the Markov property of the GFF: for an open set U ⊂ C, h| U decomposes as a zero-boundary GFF in U plus an independent random harmonic function on U . We also use the local independence properties of the GFF, as discussed in Section 2.3, and the fact that adding a smooth, compactly supported function to h affects the law of h in an absolutely continuous way.