The geodesics in Liouville quantum gravity are not Schramm-Loewner evolutions

We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLE$_\kappa$. These hypotheses are in particular satisfied by the LQG metric for $\gamma=\sqrt{8/3}$, which is isometric to a Brownian surface, constructed by the first author and Sheffield. As a consequence of our analysis, we also establish certain regularity properties of LQG geodesics which imply, among other things, that they are conformally removable.


Introduction
Suppose that D ⊆ C is a domain and h is an instance of the Gaussian free field (GFF) h on D. Fix γ ∈ (0, 2]. The γ-Liouville quantum gravity (LQG) surface described by h is the random Riemannian manifold with metric tensor (1.1) e γh(z) (dx 2 + dy 2 ) where dx 2 + dy 2 denotes the Euclidean metric tensor. This expression is ill-defined as h is a distribution and not a function, hence does not take values at points. The volume form associated with (1.1) was constructed by Duplantier-Sheffield in [7] (though measures of this type were constructed earlier by Kahane [14] under the name Gaussian multiplicative chaos; see also [11]). The construction in the case γ ∈ (0, 2) proceeds by letting for each z ∈ D and > 0 with B(z, ) ⊆ D, h (z) be the average of h on ∂B(z, ) and then taking where dz denotes Lebesgue measure on D. The construction in the case γ = 2 is similar but with the normalization factor taken to be log −1 2 [5,6]. The limiting procedure (1.2) implies that the measures µ γ h satisfy a certain change of coordinates formula. In particular, suppose that h is a GFF on D, ϕ : D → D is a conformal transformation, and If µ γ h is the γ-LQG measure associated with h, then we have that µ  (1.3). A quantum surface is an equivalence class with respect to this equivalence relation and a representative is referred to as an embedding of a quantum surface. The purpose of the present work is to study the properties of geodesics for γ-LQG surfaces and their relationship with the Schramm-Loewner evolution (SLE) [24]. At this point, the metric space structure for LQG has only been constructed for γ = 8/3 in [20,21,22,19]. In this case, the 2 JASON MILLER AND WEI QIAN resulting metric measure space is equivalent to that of a Brownian surface, the Gromov-Hausdorff scaling limit of uniformly random planar maps. The first result of this type was proved by Le Gall [16] and Miermont [18] for uniformly random quadrangulations of the sphere. The works [16,18] have since been extended to the case of uniformly random quadrangulations of the whole-plane [3], the disk [2,8], and the half-plane [1,9]. The type of Brownian surface that one obtains from the 8/3-LQG metric depends on the type of GFF h. Our results will apply in this setting, but we will work in a more general framework which should apply to any metric space structure that one could naturally associate with γ-LQG for γ ∈ (0, 2]. In particular, this work is independent of [20,21,22,19]. Recall that a metric space (X, d) is said to be geodesic if for every x, y ∈ X there exists a path in X connecting x to y with length equal to d(x, y). Recall also that (X, d) is said to be a length space if for every > 0 and x, y ∈ X there exists a path η connecting x and y with length at most d(x, y) + . We will look at a metric d h associated with a GFF instance h which satisfies the following assumption. We let B h (z, r) denote the open metric ball under d h centered at z with radius r > 0. Assumption 1.1. We assume that there exists an h-measurable metric d h so that (D, d h ) is a length space which is homeomorphic to the Euclidean metric on D and which satisfies: (i) Locality: for all z ∈ D and r > 0, B h (z, r) is a local set for h.
(ii) Scaling: there exists a constant β > 0 such that for each C ∈ R we have that d h+C (x, y) = e βC d h (x, y). (iii) Compatibility with LQG coordinate changes: if ϕ : D → D is a conformal map, h = h • ϕ + Q log |ϕ | then d h (z, w) = d h (ϕ(z), ϕ(w)) for all z, w ∈ D.
(We will review the definition of GFF local sets in Section 2.1.) The final part of Assumption 1.1 implies that d h is intrinsic to the quantum surface structure of h. Note that Assumption 1.1 is known to hold for γ = 8/3. In this article, in order to avoid dealing with boundary issues we will take D = C and work with a whole-plane GFF h. Since the whole-plane GFF is only defined modulo an additive constant, to be concrete we will often fix the additive constant by taking the average of the field on ∂D to be equal to 0. Note that (C, d h ) is a geodesic metric space, due to the Hopf-Rinow theorem and the fact that it is complete and locally compact being homeomorphic to the Euclidean whole plane. We emphasize that the geodesics of d h are the same as those of d h+C by part (ii) of Assumption 1.1, so the particular manner in which we have fixed the additive constant is not important for the purpose of analyzing the properties of geodesics. We also emphasize that the a.s. properties we will establish for geodesics in this work for the whole-plane GFF then transfer to the setting of the GFF on a general domain D ⊆ C (or to the other types of quantum surfaces considered in [4]) by absolute continuity. Finally, we remark that, if we are only interested in the whole-plane metric space (C, d h ), then we can in fact weaken the part (iii) of Assumption 1.1 so that we require the metric to be compatible with LQG coordinate changes only for translations and scalings (since these are the only properties that we will use in our proofs, in particular we will not use rotational invariance).
Our first main result is the a.s. uniqueness of geodesics connecting generic points in our domain.
Theorem 1.2. Suppose that h is a whole-plane GFF with the additive constant fixed as above and that x, y ∈ C are distinct. There is a.s. a unique d h -geodesic η connecting x and y.
We note that Theorem 1.2 was shown to hold for γ = 8/3 in [20,21,22] when x, y are taken to be quantum typical (i.e., sampled from µ h ). Taking x, y to be quantum typical corresponds to adding −γ log | · | singularities at deterministic points x, y (see, e.g., [7]). The proof of Theorem 1.2 given in the present work applies to this setting for γ = 8/3, but is also applicable in greater generality. Theorem 1.2 will be important because it allows us to refer to the geodesic connecting generic points x, y. We emphasize that Theorem 1.2 does not rule out the existence of exceptional points between which there are multiple geodesics, which are known to exist in the case γ = 8/3.
Since the GFF is conformally invariant and satisfies the spatial Markov property, one is led to wonder whether the geodesics in γ-LQG should satisfy Schramm's conformal Markov characterization of SLE [24] (see Section 2.2 for a review) and hence be given by SLE-type curves. We note that this could only be possible for κ ∈ (0, 4] as SLE κ curves with κ > 4 are self-intersecting [23] while a geodesic is necessarily simple. The main result of the present work is to show that these geodesics are in fact singular with respect to SLE κ . Recall that whole-plane SLE is the variant which describes a random curve connecting two points in the Riemann sphere, so it is the natural one to compare to LQG geodesics. Theorem 1.3. Suppose that h is a whole-plane GFF with the additive constant fixed as above and that x, y ∈ C are distinct. Let η be the a.s. unique geodesic from x to y. The law of η is singular with respect to the law of a whole-plane SLE κ curve from x to y for any value of κ > 0. The proof of Theorem 1.3 applies in other settings, as well. For example, the same technique applies to show in the case D ⊆ C is a simply connected domain that the law of a geodesic between distinct boundary points (resp. a boundary point to an interior point) is singular with respect to chordal (resp. radial) SLE.
We will prove Theorem 1.3 by analyzing the fine geometric properties of geodesics in LQG. In particular, we will show that geodesics in LQG are in a certain sense much more regular than SLE curves. As a consequence of our analysis, we will obtain the following theorem which serves to quantify this regularity (in a reparameterization invariant manner). Theorem 1.4. Suppose that h is a whole-plane GFF with the additive constant fixed as above and that x, y ∈ C are distinct. Let η be the a.s. unique geodesic from x to y. For any parameterization of η with time interval [0, T ], for each δ > 0 there a.s. exists a constant C δ > 0 so that An important concept in the theory of LQG is conformal removability. Recall that a compact set K ⊆ C is said to be conformally removable if the following is true. Suppose that U ⊇ K is an open set and ϕ : U → V is a homeomorphism which is conformal on U \ K. Then ϕ is conformal on U .
Theorem 1.5. Suppose that h is a whole-plane GFF with the additive constant fixed as above and that x, y ∈ C are distinct. Let η be the a.s. unique geodesic from x to y. Then η is a.s. conformally removable.
The conformal removability of a path in LQG is important because it implies that a conformal welding in which the path arises as the gluing interface is uniquely determined (see, e.g., [28,4,17]). In the case γ = 8/3, the conformal removability of geodesics is especially important as it is shown in [19] that metric balls in the Brownian map can be decomposed into independent slices obtained by cutting the metric ball along the geodesics from its outer boundary to its center (see Figure 1.1). Theorem 1.5 implies in the context of 8/3-LQG that the conformal structure associated with a metric ball is uniquely determined by these slices. We also note that the conformal removability of geodesics in the case γ = 8/3 was posed as [21, Problem 9.3] and Theorem 1.5 solves this problem. We will prove Theorem 1.5 by checking that a sufficient condition for conformal removability due to Jones-Smirnov [13] is necessarily satisfied for the geodesics in LQG using Theorem 1.4.
We finish by mentioning that there are many other sets of interest that one can generate using a metric from LQG. Examples include the boundaries of metric balls (see Figure 1.1) and the boundaries of the cells formed in a Poisson-Voronoi tessellation (see [10]). We expect that the techniques developed in this paper could be used to show that these sets are both not given by any form of SLE curve and also are conformally removable. This leaves one to wonder whether there is any natural set that one can generate from a metric for LQG which is an SLE.
Outline. The remainder of this article is structured as follows. We begin in Section 2 by reviewing a few of the basic facts about the GFF and SLE which will be important for this work. Next, in Section 3, we will prove the uniqueness of the d h -geodesics (Theorem 1.2). Then, in Section 4, we will analyze the regularity of the d h -geodesics, thus establish Theorem 1.3 and 1.4. Finally, in Section 5, we will prove the removability of the d h -geodesics (Theorem 1.5). In Appendix A, we will estimate the annulus-crossing probabilities for SLE curves. Theorem 1.3 (as well as Theorem 1.4) will be established by showing that the geodesics in LQG are in a certain sense much more regular than SLE curves. In particular, we will show that the probability that a geodesic has four (or more) crossings across an annulus B(z, ) \ B(z, α ) for α > 1 and > 0 decays significantly more quickly as → 0 than for SLE κ for any value of κ > 0.

Preliminaries
2.1. The Gaussian free field. We will now give a brief review of the properties of the Gaussian free field (GFF) which will be important for the present work. See [27] for a more in-depth review.
We will first remind the reader how the GFF on a bounded domain is defined before reviewing the definition of the whole-plane GFF. Suppose that D ⊆ C is a bounded domain. We let C ∞ 0 (D) be the space of infinitely differentiable functions with compact support contained in D. We define THE GEODESICS IN LQG ARE NOT SLES 5 the Dirichlet inner product by We let · ∇ be the corresponding norm. The space H 1 0 (D) is the Hilbert space completion of C ∞ 0 (D) with respect to (·, ·) ∇ . Suppose that (φ n ) is an orthonormal basis of H 1 0 (D) and that (α n ) is an i.i.d. sequence of N (0, 1) variables. Then the Gaussian free field (GFF) h on D is defined by Since the partial sums for h a.s. diverge in H 1 0 (D), one needs to take the limit in a different space (e.g., the space of distributions).
In this work, we will be mainly focused on the whole-plane GFF (see [28, Section 3.2] for a review). To define it, we replace H 1 0 (D) with the closure with respect to (·, ·) ∇ of the functions in C ∞ 1 modulo additive constants, where C ∞ 1 is the space of all smooth functions in C whose gradients are in L 2 (C). The whole-plane GFF is then defined using a series expansion as in (2.2) except the limit is taken in the space of distributions modulo additive constant. This means that if h is a whole-plane GFF and φ ∈ C ∞ 1 (C) has mean-zero (i.e., φ(z)dz = 0) then (h, φ) is defined. There are various ways of fixing the additive constant for a whole-plane GFF so that one can view it as a genuine distribution.
Note that (h, ψ − φ) is well-defined as ψ − φ has mean zero. This definition extends by linearity to any choice of ψ ∈ C ∞ 1 (C). It can also be convenient to fix the additive constant by requiring setting the average of h on some set, for example a circle (see more below), to be equal to 0.
Circle averages. The GFF is a sufficiently regular distribution that one can make sense of its averages on circles. We refer the reader to [7,Section 3] for the rigorous construction and basic properties of GFF circle averages. For z ∈ D and > 0 so that B(z, ) ⊆ D we let h (z) be the average of h on ∂B(z, ).
Markov property. Suppose that U ⊆ D is open. Then we can write h = h 1 + h 2 where h 1 (resp. h 2 ) is a GFF (resp. a harmonic function) in U and h 1 , h 2 are independent. This can be seen by noting that H 1 0 (D) can be written as an orthogonal sum consisting of H 1 0 (U ) and those functions in H 1 0 (D) which are harmonic on U . The same is also true for the whole-plane GFF except h 1 and h 2 are only defined modulo additive constant.
We emphasize that h 2 is measurable with respect to the values of h on D \ U . To make this more precise, suppose that K is a closed set and δ > 0. We then let F δ K be the σ-algebra generated by (h, φ) for φ ∈ C ∞ 0 (D) with support contained in the δ-neighborhood of K and then take Local sets. The notion of a local set of the GFF serves to generalize the Markov property to the setting in which K = D \ U can be random, in the same way that stopping times generalize the Markov property for Brownian motion to times which can be random (see [25] for a review). More precisely, we say that a (possibly random) closed set K coupled with h is local for h if it has the property that we can write h = h 1 + h 2 where, given F K , h 1 is a GFF on D \ K and h 2 is harmonic on D \ K. Moreover, h 2 is F K -measurable.
Conformal invariance. Suppose that ϕ : D → D is a conformal transformation. It is straightforward to check that the Dirichlet inner product (2.1) is conformally invariant in the sense that (f • ϕ, g • ϕ) ∇ = (f, g) ∇ for all f, g ∈ C ∞ 0 (D). As a consequence, the GFF is conformally invariant in the sense that if h is a GFF on D then h • ϕ is a GFF on D.
Perturbations by a function. Suppose that f ∈ H 1 0 (D). Then the law of h + f is the same as the law of h weighted by the Radon-Nikodym derivative exp((h, f ) ∇ − f 2 ∇ /2). Consequently, the laws of h + f and h are mutually absolutely continuous. This can be seen by writing f = ∞ n=1 β n φ n where (φ n ) is an orthonormal basis of H 1 0 (D), noting that the Radon-Nikodym derivative can be written as ∞ n=1 exp(α n β n − β 2 n /2) and weighting the law of α n by exp(α n β n − β 2 n /2) is equivalent to shifting its mean by β n .
2.2. The Schramm-Loewner evolution. The Schramm-Loewner evolution SLE was introduced by Schramm in [24] as a candidate to describe the scaling limit of discrete models from statistical mechanics. There are several different variants of SLE: chordal (connects two boundary points), radial (connects a boundary point to an interior point), and whole-plane (connects two interior points). We will begin by briefly discussing the case of chordal SLE since it is the most common variant and the one for which it is easiest to perform computations. As the different types of SLE's are locally absolutely continuous (see [26]), any distinguishing statistic that we identify for one type of SLE will also work for other types of SLEs.
Suppose that η is a simple curve in H from 0 to ∞. For each t ≥ 0, we can let H t = H \ η([0, t]) and g t be the unique conformal transformation H t → H with g t (z) − z → 0 as z → ∞. Then the family of conformal maps (g t ) satisfy the chordal Loewner equation (provided η is parameterized appropriately): Here, U : [0, ∞) → R is a continuous function which is given by the image of the tip of η at time t. That is, U t = g t (η(t)). SLE κ for κ ≥ 0 is the random fractal curve which arises by taking U t = √ κB t where B is a standard Brownian motion. (It is not immediate from the definition of SLE that it is in fact a curve, but this was proved in [23].) It is characterized by the conformal Markov property, which states the following. Let F t = σ(U s : s ≤ t) = σ(η(s) : s ≤ t) and f t = g t − U t . Then: • Given F t , we have that s → f t (η(s + t)) is equal in distribution to η.
Since this work is focused on geodesics which connect two interior points, the type of SLE that we will make a comparison with is the whole-plane SLE. Whole-plane SLE is typically defined in terms of the setting in which 0 is connected to ∞ and then for other pairs of points by applying a Möbius transformation to the Riemann sphere. Suppose that U t = √ κB t where B is a two-sided (i.e., defined on R) standard Brownian motion and we let (g t ) be the family of conformal maps which solve The whole-plane SLE κ in C from 0 to ∞ encoded by U is the random fractal curve η with the property that for each t ∈ R, g t is the unique conformal transformation from the unbounded component of C \ η([0, t]) to C \ D which fixes and has positive derivative at ∞. We will prove in Appendix A the following proposition, which is the precise property that will allow us to deduce the singularity between SLE and d h -geodesics.
Suppose that η is a whole-plane SLE κ in C from 0 to ∞. For each n ∈ N there exists α > 1 such that the following is true. There a.s. exists 0 > 0 so that for all ∈ (0, 0 ) there exists z ∈ B(0, 2) \ D such that η makes at least n crossings across the annulus B(z, ) \ B(z, α ).
We will in fact deduce Proposition 2.1 in Appendix A from the analogous fact for chordal SLE, by local absolute continuity between the different forms of SLE.
such that η makes at least n crossings across the annulus B(z, ) \ B(z, α ).

Distortion estimates for conformal maps.
Here, we recall some of the standard distortion and growth estimates for conformal maps which we will use a number of times in this article.
The following is a consequence of Koebe-1/4 theorem and the growth theorem, for example see [15,Corollary 3.23].
Lemma 2.5. Suppose that D, D ⊆ C are domains and f : D → D is a conformal transformation. Fix z ∈ D and let z = f (z). Then for all r ∈ (0, 1) and all |w − z| ≤ rdist(z, ∂D), 2.4. Binomial concentration. We will make frequent use of the following basic concentration inequality for binomial random variables. Lemma 2.6. Fix p ∈ (0, 1) and n ∈ N and let X be a binomial random variable with parameters p and n. For each r ∈ (p, 1) we have that Similarly, for each r ∈ (0, p) we have that We emphasize that for fixed r, c p,r → ∞ as p → 0 and also as p → 1.
Proof. We will prove (2.4). The proof of (2.5) follows by replacing X with n − X, p with 1 − p, and r with 1 − r. We have that Optimizing over λ > 0 implies (2.4).  and B h (y, u) are drawn in grey. We cover ∂B h (x, r) by balls of radius /2. We investigate the probability that there are two geodesics from x to y that respectively intersect See Figure 3.1 for an illustration. Fix x, y ∈ C distinct. For any r > 0, let B h (x, r) be the d h metric ball centered at x of radius r and let s : To prove the theorem, it suffices to show that for any r > 0, on the event {r < d h (x, y)}, ∂B h (x, r) ∩ ∂B h (y, s) a.s. contains a unique point. Indeed, if η is a geodesic from x to y, then we can continuously parameterize η by t ∈ [0, d h (x, y)] so that d h (η(t), x) = t, since d h is homeomorphic to the Euclidean metric. In particular, for all r ∈ [0, d h (x, y)], we have η(r) ∈ ∂B h (x, r) ∩ ∂B h (y, s). If for every r > 0, on the event {r < d h (x, y)}, ∂B h (x, r) ∩ ∂B h (y, s) a.s. contains a unique point, then for any two geodesics η and η from x to y, we a.s. have that η(r) = η(r) for all rational r ∈ [0, d h (x, y)] simultaneously. This can only be the case if we a.s. have that η = η.
From now on, fix r, ξ > 0. We will argue that on the event {r < d h (x, y)}, ∂B h (x, r) ∩ ∂B h (y, s) a.s. does not contain points which have distance more than 8ξ from each other. This will imply the desired result as we have taken r, ξ > 0 to be arbitrary. For R, , δ > 0, we define E(R, , δ) to be the event that , any geodesic from a to b has Euclidean diameter at most ξ. Since we have assumed that d h induces the Euclidean topology, it follows that the probability of (i) tends to 1 as R → ∞. For the same reason, for fixed R and ξ, as δ → 0, the probability of (iii) tends to 1. Moreover, for fixed R and δ, as → 0, the probability of (ii) tends to 1. Therefore, we can choose R, , δ in a way that < δ < ξ and the probability of E(R, , δ) is arbitrarily close to 1.
Let x 1 , . . . , x n be a collection of points on ∂B h (x, r) so that ∂B h (x, r) ⊆ ∪ n j=1 B(x j , /2). We aim to prove that, conditionally on {r < d h (x, y)} ∩ E(R, , δ), there a.s. do not exist two geodesics η and η from x to y such that η intersects B(x i , /2) and η intersects B(x j , /2), where i, j ∈ [1, n] are such that B(x i , 2ξ) ∩ B(x j , 2ξ) = ∅. This implies that any two intersection points of ∂B h (x, r) and ∂B(y, s) must have distance at most 8ξ from each other. Since the probability of E(R, , δ) can be made arbitrarily close to 1, this will complete the proof.
From now on, we further fix R, , δ and work on the event E : . We also assume that the additive constant for h is fixed so that its average on ∂B(R + 2, 1) is equal to 0 (recall that the d h -geodesics do not depend on the choice of additive constant; the choice here is made so that the circle is disjoint from B(0, R) but is otherwise arbitrary).
If u ≥ s, then obviously there do not exist two geodesics η and η from x to y such that η intersects We are going to prove that on E ∩ {u < s}, we have X i = X j a.s. This will imply that, on the event E, there a.s. do not exist two geodesics η and η from x to y such that η intersects B(x i , /2) and η intersects B(x j , /2), which will complete the proof.
Let us now work on E ∩ {u < s}. We will further condition on the sets B h (x, r) and B h (y, u) (which are local for h by Assumption 1.1). It suffices to show that under such conditioning, . We emphasize that we can choose φ as a deterministic function of B h (x, r), B h (y, u) and x i , x j , ξ. For α ∈ R, we let X α i be the infimum of d h+αφ -lengths of paths which connect a point on ∂B h (x, r) ∩ B(x i , δ) to a point on ∂B h (y, u) ∩ B(x i , δ) and which are contained in B(x i , ξ). We note that X 0 i = X i . Observe that X α i is strictly increasing and continuous in α by part (ii) of Assumption 1.1. Thus if we take A to be uniform in [0, 1] then the probability that X A i = X j is equal to 0. Since the conditional law of h + Aφ in U i given the values of h outside of U i is mutually absolutely continuous with respect to the conditional law of h in U i given its values outside of U i , it follows that the joint law of (X A i , X j ) is mutually absolutely continuous with respect to the joint law of (X i , X j ). In particular, the probability that X i = X j is also equal to 0.

Regularity
In this section, we will give the proofs of Theorems 1.3 and 1.4. The first step is carried out in Section 4.1, which is to show that (with high probability) the whole-plane GFF at an arbitrarily high fraction of geometric scales exhibits behavior (modulo additive constant) which is comparable to the GFF with zero boundary conditions. We will then use this fact in Section 4.2 to show that (with high probability): • At an arbitrarily high fraction of geometric scales (depending on a choice of parameters), the shortest path which goes around an annulus is at most a constant times the length of the shortest path which crosses an annulus (Proposition 4.6) and that • There exists a geometric scale at which the former is strictly shorter than the latter (consequence of Lemma 4.7).
The first statement is the main ingredient in the proofs of Theorems 1.3 and 1.4 since it serves to rule out a geodesic making multiple crossings across annuli. The second statement will be used to prove an upper bound for the dimension of the geodesics (Proposition 4.8) which will be used in the proof of Theorem 1.5 in Section 5.
Throughout, we let h be a whole-plane GFF. For any z ∈ C and r > 0, let F z,r be the σ-algebra generated by the values of h outside of B(z, r). By the Markov property for the GFF, we can write h as a sum of a GFF h z,r on B(z, r) with zero boundary conditions and a function h z,r which is harmonic on B(z, r) and agrees with h outside of B(z, r). Note that h z,r is measurable w.r.t. F z,r and h z,r is independent of F z,r . Let h r (z) be the average of h on ∂B(z, r).
4.1. Good scales. In this subsection, we will first define the M -good scales and show in Lemma 4.1 that they are important because on such scales the law of a whole-plane GFF and the law of a GFF with zero boundary conditions are mutually absolutely continuous with well-controlled Radon-Nikodym derivatives. Then we will prove the main result of this subsection, which is Proposition 4.3, which says that an arbitrarily large fraction of scales are M -good with arbitrarily large probability provided we choose M large enough.
Fix a constant M > 0. Fix z ∈ C and r > 0. We say that B(z, r) is M -good for h if: sup Let E M z,r be the event that B(z, r) is M -good and note that E M z,r is F z,r -measurable. Let Z z,r (·) (resp. W z,r (·)) be the Radon-Nikodym derivative of the former w.r.t. the latter (resp. latter w.r.t. the former). (Note that Z z,r (resp. W z,r ) is itself measurable w.r.t. F z,r and takes as argument h z,r | B(z,7/8) (resp. h z,r | B(z,7/8) ).) On E M z,r , for all p ∈ R, there exists a constant c(p, M ) depending only on p and M such that Note that E[Z z,r ( h z,r | B(z,7r/8) ) p | F z,r ] and E[(W z,r ( h z,r | B(z,7r/8) )) p | F z,r ] are both measurable w.r.t. F z,r .
Proof of Lemma 4.1. Note that when restricted to B(z, r), h z,r admits the Markovian decomposition h z,r = h z,r + h z,r where h z,r = h z,r − h r (z) is harmonic in B(z, r). Fix φ ∈ C ∞ 0 (B(z, 29r/32)) with φ| B(z,7r/8) ≡ 1 and let g = h z,r φ. Then h z,r +g is equal to h z,r in B(z, 7r/8). Moreover, if we take the law of h z,r and then weight it by the Radon-Nikodym derivative Z 0 z,r ( h z,r ) = exp(( h z,r , g) ∇ − g 2 ∇ /2), then the resulting field has the same law as h z,r + g. Therefore Z z,r is given by integrating Z 0 z,r over the randomness of h z,r in B(z, r) \ B(z, 7r/8) given F z,r . Conversely, if we take the law of h z,r + g and weight it by the Radon-Nikodym derivative then the resulting field has the same law as h z,r .
Note that the second equality in (4.1) holds because h z,r differs from h z,r by a function which is harmonic in B(z, r) and g is supported in B(z, r). Since h z,r + g and h z,r agree on B(z, 7r/8), we get that if we take the law of h z,r and weight it by W 0 z,r = exp(( h z,r , −g) ∇ − g 2 ∇ /2), then the restriction of the resulting field to B(z, 7r/8) has the same law as the corresponding restriction of h z,r . Therefore W z,r is given by integrating W 0 z,r over the randomness of h z,r in B(z, r) \ B(z, 7r/8) given F z,r . This proves the mutual absolute continuity. Now suppose that we are working on the event E M z,r . Then | h z,r | ≤ M in B(z, 15r/16). Recall the following basic derivative estimate for harmonic functions. There exists a constant c > 0 so that if R > 0 and u is harmonic in B(z, R) then for w ∈ B(z, R) we have that Applying this with u = h z,r , R = 15r/16, and w ∈ B(z, 29r/32) we see that h z,r 2 ∇ (with the norm computed on B(z, 29r/32)) is bounded by a constant which depends only on M . Therefore the same is true for g 2 ∇ . The second part of the lemma follows because for all p ∈ R, In particular, on E M z,r , the above quantities are bounded by a constant which depends only on p and M . The same is therefore true for Z z,r and W z,r by Jensen's inequality, which completes the proof. Now let us mention a few consequences of this lemma and its proof that we will use later on.   One main input into the proof of Proposition 4.3 is the following bound for the probability that a given ball is not M -good.
Lemma 4.4. There exist constants c 1 , c 2 > 0 such that for any z ∈ C, r > 0, and M > 0, we have Proof. By the scale and translation invariance of the whole-plane GFF, the quantity P (E M z,r ) c is independent of z and r, hence we will choose z = 0 and r = 1. We are going to bound the supremum of |h 0,1 (w) − h 0,1 (0)| when w ∈ B(0, 15/16) and show that it has a Gaussian tail.
Let p be the Poisson kernel on B(0, 31/32). Then there exists an absolute constant C > 0 so that p(w, y) ≤ C for all w ∈ B(0, 15/16) and y ∈ ∂B(0, 31/32). Letting dy denote the uniform measure on ∂B(0, 31/32), we have that for all w ∈ B(0, 15/16) We note that h 0,1 (y) − h 0,1 (0) is a Gaussian random variable with bounded mean and variance. It thus follows that by choosing a > 0 sufficiently small we have E exp a sup The result therefore follows by Markov's inequality. Proof of Proposition 4.3. By the translation and scale-invariance of the whole-plane GFF, the statement is again independent of z and r, hence we will choose z = 0 and r = 1 so that r k = 2 −k . Our strategy is to explore h in a Markovian way from outside in and to control (using Lemma 4.4) the number of scales we need to go in each time in order to find the next M -good scale.
We start by looking for the first k 0 ∈ N for which B(0, r k 0 ) is an M -good scale. Let and then to estimate the goodness of the scale B(0, r ). By applying the derivative estimate (4.2) to the harmonic function h 0,1 we see that there exists c > 0 such that if we choose = c log 2 (R) , then (4.7) is satisfied. Lemma 4.4 implies that P[R > t] ≤ c 1 e −c 2 t 2 for constants c 1 , c 2 > 0. Consequently, Now let us estimate the following quantity, which represents how good B(0, r ) is: Note that h 0,r (w) = h 0,1 (w) + h 0,r (w), where h 0,r is harmonic in B(0, r ) and agrees with a zero-boundary GFF in B(0, 1) outside of B(0, r). Therefore, combining with (4.7), we have that Note that h 0,r is independent of F 0,1 . Applying (4.6) to (4.8), we know that there exist c 1 , c 2 > 0 (depending only on C) such that P[ R > t | F 0,1 ]1 R>M ≤ c 1 e − c 2 t 2 . In particular, it implies that the conditional probability of R ≤ M is at least some p M,C > 0. We emphasize that p M,C depends only on M and C and can be made arbitrarily close to 1 if we fix C > 0 and choose M > 0 sufficiently large. From now on, we will fix C and reassign the values of c 1 , c 2 , p M , c 1 , c 2 , p M,C so that If B(0, r ) is M -good, then k 0 = . Otherwise we continue our exploration, conditionally on F 0,r and on the event {R > M } ∩ { R > M } (which is measurable w.r.t. F 0,r ). Similarly to (4.7), we define = c log 2 ( R) so that sup w∈B(0,r + ) Therefore, the goodness of B(0, r + ) has the same tail bound as R. Hence we know that the probability that B(0, r + ) is M -good (i.e., k 0 = l + l) is also at least p M,C and that otherwise we can look at the next scale B 0, r l+2 . We can thus iterate.
The above procedure implies that where the A i 's are i.i.d. random variables with P[A i ≥ t] ≤ c 1 e −c 2 t 2 and G is a geometric random variable with success probability p M . Moreover, the A i 's and G are all independent. It thus follows that k 0 has an exponential tail. Indeed, Since A 1 has a Gaussian tail, E e λA 1 is finite for any λ > 0. We also know that p M can be made arbitrarily close to 1 as M → ∞. Therefore, for all λ > 0 we can choose M big enough so that E e λk 0 < 1. (4.9) Once we find the first good scale k 0 , we can repeat the above procedure to find the next good scale k 0 + k 1 . As a first step, instead of going c log 2 R or c log 2 R further (for R, R > M ), we just need to go c log 2 M further (and then repeat the same procedure). We therefore get that k 1 is stochastically dominated by k 0 . Moreover, k 1 is independent of k 0 . Therefore, for any b ∈ (0, 1) and λ > 0, we have where the k i 's are i.i.d. and distributed like k 0 . For any a > 0, by Markov's inequality, the right hand-side of (4.10) is less than or equal to Then it completes the proof due to (4.9).

Annulus estimates.
We now proceed to establish the main estimate which will be used to prove Theorems 1.3 and 1.4. Proposition 4.6. Fix z ∈ C and r > 0. For each k, we let r k = 2 −k r. We also let L 1,k be the infimum of d h -lengths of paths contained in B(z, 7r k /8) \ B(z, r k /2) which separate z from ∞ and let L 2,k be the d h -distance from ∂B(z, 7r k /8) to ∂B(z, r k /2). Fix K ∈ N, c > 0, and let N (K, c) be the number of k ∈ {1, . . . , K} with the property that L 1,k ≤ cL 2,k . For each a 1 > 0 and b 1 ∈ (0, 1), The following lemma is the main input into the proof of Proposition 4.6.
Lemma 4.7. Fix z ∈ C and r > 0. Let L 1 be the infimum of d h -lengths of paths contained within the annulus B(z, 7r/8) \ B(z, r/2) and which separate z from ∞. Let L 2 be the d h distance from ∂B(z, 7r/8) to ∂B(z, r/2). On E M z,r , for all q > 0, there exists c 0 > 0 depending only on M such that for all c > c 0 and all z ∈ C and r > 0, we have (4.11) P[L 1 ≥ cL 2 | F z,r ] ≤ q a.s.
Since we have assumed that the d h metric is a.s. homeomorphic to the Euclidean metric, it follows that L 1 and L 2 are both a.s. positive and finite random variables. It therefore follows that (4.13) holds.
Let us now prove (4.14). Let φ be a non-negative, radially symmetric C ∞ 0 function supported in B(0, 3/4) and which is equal to 1 in B(0, 5/8). Then adding cφ to h does not affect S 1 but it multiplies S 2 by e βc where β is as in part (ii) of Assumption 1.1. Since S 1 , S 2 are a.s. positive and finite, it follows that by replacing h by h + cφ and taking c > 0 sufficiently large we will have that S 1 < S 2 with positive probability. This completes the proof as h + cφ is mutually absolutely continuous w.r.t. h.
Proof of Proposition 4.6. Fix z ∈ C and r > 0. Let E(K, b) denote the event that the fraction of k ∈ {1, . . . , K} for which B(z, r k ) is M -good is at least b. Proposition 4.3 implies that for any b ∈ (0, 1) and a > 0, there exists M > 0 sufficiently large so that We thereafter fix a, b and M so that (4.15) holds.
Let L 1,k , L 2,k be as in Lemma 4.7 for B(z, r k ). Lemma 4.7 implies that for each q > 0 there exists c > 0 so that at each M -good scale B(z, r k ), we have P[L 1,k ≥ cL 2,k | F z,r k ] ≤ q a.s. Note that both L 1,k and L 2,k are measurable w.r.t. F z,r k+1 , hence we can explore h according to the filtration (F z,r k ) k≥0 . More precisely, if we explore h from outside in, then each time we encounter a new good scale, conditionally on the past, the probability of achieving {L 1,k < cL 2,k } for that scale is uniformly bounded from below by 1 − q. For each k, let g k be the index of the kth good scale. It thus follows that the number N (K, c) of k ∈ {1, . . . , bK} that we achieve {L 1,g k < cL 2,g k } is at least equal to a binomial random variable with success probability 1 − q and bK trials. By Lemma 2.6, this proves that for any b 1 ∈ (0, b) and a > 0, if we make q > 0 sufficiently small and a sufficiently large, then we have Therefore where a 1 = a ∧ a. Since we can choose a and a to be arbitrarily large, a 1 can also be arbitrarily large. Also note that we can choose b arbitrarily close to 1 and b 1 arbitrarily close to b.
Finally, let us deduce the following upper bound for the Minkowski dimension of a geodesic using (4.12). Proof. Fix z ∈ C and r > 0 and also consider the event E(K, b). Fix a, b and M so that (4.15) holds. Let S 1,k be the infimum of d h -lengths of paths contained in B(z, 7r k /8) \ B(z, 3r k /4) which separate 0 from ∞. We also let S 2,k be the d h distance from ∂B(z, 5r k /8) to ∂B(z, r k /2). Let g k and E(K, b) be as in the proof of Proposition 4.6. Let F (K, b) be the event that S 1,g k ≥ S 2,g k for every k ∈ {1, . . . , bK} and let F (K) be the event that S 1,k ≥ S 2,k for every k ∈ {1, . . . , K}. Then we have that (4.12)).
If S 1,k < S 2,k for some k ∈ {1, . . . , K}, then it is impossible for a geodesic with endpoints outside of B(z, r) to hit B(z, 2 −K r), hence also B(z, r/2), see Figure 4.1. This implies that the upper Minkowski dimension of any d h -geodesic is at most 2 − δ < 2. The fact that its dimension is at least 1 is obvious, since the geodesic is a.s. a continuous curve. Combining these facts proves the result.  We draw the successive scales. Left: We show in red (resp. blue) the path which realizes the minimal length S 1,k (resp. S 2,k ). If for some k ∈ {1, . . . , K}, one has S 1,k < S 2,k , then any geodesic with both endpoints outside of B(z, r) cannot enter B(z, r k+1 ). Right: We show in red (resp. blue) the path which realizes the minimum length L 1,k (resp. L 2 ). If L 1,k < L 2 , then any geodesic cannot make more than four crossings across the annulus B(z, r) \ B(z, r k+1 ). In both pictures, the dashed curves represent configurations of geodesics which are impossible, since the red curves are shortcuts.

Proof of Theorems 1.3 and 1.4.
Proof of Theorem 1.3. Fix z ∈ C, > 0, ζ > 1. Let L 2 be the d h -distance from ∂B(z, ζ ) to ∂B(z, ). Fix K = log 2 1−ζ . For k ∈ [1, K], let L 1,k and L 2,k be as in Proposition 4.6 for r k = 2 −k . See Figure 4.1. Note that Consequently, the fraction ρ of k ∈ {1, . . . , K} for which is at least 1 − 1/c 1 . We will chose c 1 = 100 so that ρ ≥ 99/100. By Proposition 4.6, for any a > 0, we can choose a value of c 2 > 0 large so that the fraction of k ∈ [1, K] with We emphasize that the values of c 1 , c 2 do not depend on . Therefore by choosing > 0 sufficiently small (hence K is big), we have that L 1,k 0 < L 2 . This implies that it is not possible for a geodesic to have more than four crossings across the annulus B(z, ) \ B(z, ζ ) because in this case we have exhibited a shortcut. See the right side of Figure 4.1. Therefore, the probability for a geodesic to have more than four crossings across the annulus B(z, where the exponent a(ζ − 1) log 2 e can be made arbitrarily large, since a can be made arbitrarily big. In particular, it implies that if η is a geodesic from 0 to any point outside of B(0, 2), then by the Borel-Cantelli lemma there a.s. exists 0 > 0 so that for all ∈ (0, 0 ) and all z ∈ B(0, 2) \ D, η does not make more than four crossings across the annulus B(z, ) \ B(z, ζ ). However, this same event has probability zero for any whole-plane SLE κ curve (provided we choose ζ > 1 sufficiently close to 1 depending on κ), by Proposition 2.1. Therefore, the law of the geodesic η is singular w.r.t. the law of a whole-plane SLE curve. We have thus completed the proof.

Conformal removability
In this section, we aim to prove that for any x, y ∈ C, the geodesic η connecting x, y is a.s. conformally removable, thus proving Theorem 1.5. We will rely on a sufficient condition by Jones and Smirnov [13] to prove the removability of η, which we will now describe. Let W be a Whitney cube decomposition of C \ η. Among other properties, W is a collection of closed squares whose union is C \ η and whose interiors are pairwise disjoint. Moreover, if Q ∈ W then dist(Q, η) is within a factor 8 of the side-length |Q| of Q. Let ϕ : D → C \ η be the unique conformal transformation with ϕ(0) = ∞ and lim z→0 zϕ(z) > 0. We define the shadow s(Q) as follows (see Figure 5.1). Let I(Q) be the radial projection of ϕ −1 (Q) onto ∂D. That is, I(Q) consists of those points e iθ for θ ∈ [0, 2π) such that the line re iθ , r ∈ [0, 1], has non-empty intersection with ϕ −1 (Q). We then take s(Q) = ϕ(I(Q)).
It is shown by Jones and Smirnov in [13] that to prove that η is conformally removable, it suffices to check that This is the condition that we will check in order to prove Theorem 1.5. We depict the conformal map ϕ from D onto C \ η, where η is a geodesic from x to y shown on the right. On the right, we show one Whitney cube Q centered at w and its shadow s(Q) in red. The blue arc L is used in the proof of Lemma 5.1. The pre-images of Q, s(Q) and L under ϕ are shown on the left. Lemma 5.1. For each δ > 0 there exists a constant C δ > 0 so that the following is true. For each Q ∈ W with |Q| = 2 −n we have that Proof. Fix Q ∈ W with |Q| = 2 −n . By the definition of the Whitney cube decomposition, we have that dist(Q, η) ∈ [2 −n−3 , 2 −n+3 ]. Let w be the center of Q. See Figure 5.1 for illustration. By Lemma 2.5, for all r ∈ (0, 1) and all z such that |z − w| ≤ rdist(w, η), we have This implies that ϕ −1 (Q) is contained in a ball centered at ϕ −1 (w) with radius at most a constant times dist(ϕ −1 (w), ∂D). This implies that there exists c 0 > 0 such that diam(I(Q)) ≤ c 0 dist(ϕ −1 (w), ∂D).
Let L be the connected component of ∂B(w, 2 −n(1−δ) ) \ η which together with η separates w from ∞. The Beurling estimate implies that the probability that a Brownian motion starting from w exits C \ (η ∪ L) in L is O(2 −nδ/2 ). By the conformal invariance of Brownian motion, we therefore have that the probability that a Brownian motion starting from ϕ −1 (w) hits ϕ −1 (L) before hitting ∂D is O(2 −nδ/2 ). If ϕ −1 (L) had an endpoint in I(Q), then due to (5.2), this probability would be bounded from below. Therefore this cannot be the case, so ϕ −1 (I) must contain I(Q). That is, I contains s(Q).
Proof of Theorem 1.5. As we have mentioned above, it suffices to show that the sum (5.1) is a.s. finite.
Proposition 4.8 implies that there exists d ∈ [1, 2) and n 0 > 0 such that for all n ≥ n 0 , one can cover η with a collection of O(2 nd ) balls of radius 2 −n . We denote by C n the collection of the centers of these balls. For any Q ∈ W with |Q| = 2 −n , since dist(Q, η) ∈ [2 −n−3 , 2 −n+3 ], Q must be contained in B(z, 2 −n+4 ) for some z ∈ C n . Since all the cubes in W are disjoint, a ball B(z, 2 −n+4 ) can contain at most 2 10 cubes in W of side length 2 −n . This implies that the number of cubes in W of side length 2 −n is O(2 nd ).
On the other hand, Lemma 5.1 implies that the diameter of a shadow of a cube in W with side length 2 −n is O(2 −n(1−δ) ). Therefore the total contribution to (5.1) coming from cubes of side length 2 −n is O(2 −2n(1−δ) × 2 dn ). We can take δ ∈ (0, 1) small enough so that d − 2(1 − δ) < 0 so that the sum over n is finite. This completes the proof.

Appendix A. SLE almost surely crosses mesoscopic annuli
The purpose of this appendix is to prove Propositions 2.1 and 2.2. We will begin by proving a lower bound for the probability that chordal SLE κ makes k crossings across an annulus (Lemma A.1) and then use this lower bound to complete the proof of Propositions 2.1 and 2.2. Throughout, we will assume that we have fixed κ > 0 and that η is an SLE κ in H from 0 to ∞.
Lemma A.1. There exist constants c 2 , c 3 > 0 depending only on κ so that the following is true. For each z ∈ D with Im(z) ≥ 1/50 and > 0, the probability that η makes at least 2k crossings from ∂B(z, ) to ∂B(z, 1/100) before exiting B(0, 2) is at least c 2 c 3 k 2 .
We believe that the exact exponent in the statement of Lemma A.1 should be equal to the interior arm exponent for SLE. This was computed in [29] but in a setup which we cannot use to prove Propositions 2.1 and 2.2. We will give an elementary and direct proof of Lemma A.1.
Before we give the proof of Lemma A.1, we will first recall the form of the SDE which describes the evolution in t of π times the harmonic measure of the left side of the outer boundary of η([0, t]) and R − as seen from a fixed point in H. Let U = √ κB be the Loewner driving function for η, fix z ∈ H, and let Z t (z) = X t + iY t = g t (z) − U t and Θ t = arg Z t .
Then Θ t gives π times the harmonic measure of the left side of the outer boundary of η([0, t]) and R − as seen from z. Let Θ be given by Θ reparameterized according to log conformal radius as seen from z. Then Θ t satisfies the SDE where B is a standard Brownian motion (see, for example, [12,Section 6]).
Proof of Lemma A.1. Let ϕ be the unique conformal transformation from H to the half-infinite cylinder C = R + × [0, 2π] (with the top and bottom identified) which takes z to ∞ and 0 to 0. See Figure A.1. Since z ∈ D and Im(z) ≥ 1/50, we note that the distance between 0 and ϕ(∞) in C is bounded from below. We will consider η = ϕ(η) in place of η and we will define an event for η which implies that η makes at least k crossings from ∂B(z, ) to ∂B(z, 1/100) before exiting B(0, 2). We can choose a universal constant c 0 > 0 large enough such that the following holds simultaneously for all z ∈ D with Im(z) ≥ 1/50: ). We then define a deterministic path Γ as follows. For 0 ≤ j ≤ k, let Let Γ be the path which visits the points x 0 , . . . , x 4k in order by: • traveling from x 4j to x 4j+1 linearly to the right, • from x 4j+1 to x 4j+2 counterclockwise along an arc connecting x 4j+1 and x 4j+2 , • from x 4j+2 to x 4j+3 linearly to the left, and • from x 4j+3 to x 4j+4 clockwise along an arc connecting x 4j+3 and x 4j+4 . We choose the arcs in the definition of Γ so that it is a C 2 curve. The rest of the proof will be dedicated to proving that the following event holds with probability at least c 2 c 3 k 2 for some c 2 , c 3 > 0: η reaches distance (2c 0 k) −1 of x 4k before leaving the (2c 0 k) −1 -neighborhood of Γ. (A. 3) Note that this will complete the proof, since the event (A.3) implies that η makes at least k crossings from ∂B(z, ) to ∂B(z, 1/100) before exiting B(0, 2).
As Γ is a C 2 curve, we can parameterize it at unit speed on some time interval [0, T ] so that T k log −1 . Let 0 = t 0 < t 1 < · · · < t n = T be equally spaced times with n = c 1 k 2 log −1 where c 1 > 0 is a large constant we will adjust later. For each 1 ≤ j ≤ n, we let y j = Γ(t j ). Note that the spacing between the y j is of order c −1 1 k −1 . Let D j be the sector formed by the two infinite lines with slopes c −19/64 1 and −c −19/64 1 relative to the tangent of Γ at Γ((t j−1 + t j )/2) (see Figure A.2). Let τ j = inf{t ≥ τ j−1 : η(t) ∈ ∂D j }. Let Θ j t be the harmonic measure of the left side of the outer boundary of η([0, t]) and ϕ(R − ) as seen from y j . We inductively define events E j as follows. Let E 0 be the whole sample space. Given that E 0 , . . . , E j have been defined, we let E j+1 be the event that E j occurs, τ j+1 < ∞, and Let us first prove by induction that the following statement is true for all 1 ≤ j ≤ n: (I j ) On the event E j , η([0, τ j ]) is contained in the c −3/2 1 (2k) −1 -neighborhood of Γ. Note that (I 0 ) is obviously true. Suppose that (I j ) holds, let us prove that (I j+1 ) also holds. It suffices to prove that η([τ j , τ j+1 ]) is contained in the c −3/2 1 (2k) −1 -neighborhood of Γ. Suppose that it is not the case, so there exists t ∈ (τ j , τ j+1 ] such that the distance between η(t) and Γ is equal to c /c −1 1 ) 1/2 ), which is impossible since we are on E j+1 . This completes the induction step, hence (I j ) is true for all 1 ≤ j ≤ n.
Our next goal is to give a lower bound on the probability of E n . We will first prove the following fact for all 1 ≤ j ≤ n − 1: On the event E j , Θ Let B 1 (resp. B 2 ) be a Brownian motion started at y j (resp. y j+1 ) and stopped upon hitting η([0, τ j ]). Let T 1 (resp. T 2 ) be the first time that B 1 (resp. B 2 ) hits ∂D j . We will work on the event that B 1 (resp. B 2 ) stops in D j ∩ B(y j , k −1 c provided we choose c 1 > 0 large enough (recall that Γ is a C 2 curve, so it differs at distance x from the linear approximation corresponding to the tangent line by O(x 2 ) and this error term is at most a constant times k −1 c −19/64 1 x for x ≤ k −1 c −19/64 1 provided we choose c 1 > 0 large enough). Let a and b be points respectively on the upper and lower boundary of D j such that the distances between a, b to Γ((t j−1 + t j )/2) are k −1 c −19/16 1 . The points a, b divide ∂D j into 3 parts: one finite part that we denote by A a,b and two infinite half-lines with endpoints a and b that we denote by A a and A b . See Figure A.2.
Let f 1 (resp. f 2 ) be the conformal map from C \ D j onto H which sends y j (resp. y j+1 ) to i and ∞ to ∞. Then for i = 1, 2, the harmonic measure seen from y j+i−1 of A a,b is O(c −3/32 1 ). (This is because f 1 (resp. f 2 ) is a map of the form w → cw a for a > 1/2 and a constant c ∈ C, hence get (c Note that we have the following facts for B i for i = 1, 2: ). Conditionally on this event, the probability that B i stops on the same side of η([0, Indeed, on E j , by (I j ) we know that η(τ j ) is in the c −3/2 1 (2k) −1 -neighborhood of Γ, hence has distance at most k −1 c −77/64 1 /2 to Γ((t j−1 + t j )/2). We condition on the point B i (T i ) and let d i denote the distance between B i (T i ) and Γ((t j−1 + t j )/2). Note that Since the slope of the lines which make the two sides of ∂D j is c

). (A.5)
We can further express P[B 1 left | B 1 (T 1 ) ∈ A a,b ] as an integration w.r.t. the position of B 1 (T 1 ) on A a,b . Note that conditionally on the event that B 1 (T 1 ) hits A a,b , the point f 1 (B 1 (T 1 )) is distributed according to a measure on f 1 (A a,b ) which has Radon-Nikodym derivative 1 − O(c −3/16 1 ) w.r.t. the uniform measure on f 1 (A a,b ). (This is because the exact Radon-Nikodym derivative is a constant times 1/(1 + .) The same is true for B 2 and T 2 and f 2 (A a,b ). Note that the uniform measures on f 1 (A a,b ) and f 2 (A a,b ) are the same. This implies that )-neighborhood of 1/2 at t = τ j+1 . Let F t := σ( η| [0,t] ). It follows that for all 1 ≤ j ≤ n − 1, we have This implies that P[E n ] ≥ p n 0 . Since n = c 1 k 2 log −1 , this completes the proof. We will prove Proposition 2.2 by iteratively applying Lemma A.1 as η travels from 0 to ∂D. Let m 1 , m 2 > 0 be constants that we will adjust later. For any > 0 and j ∈ N, we define the stopping times σ j = inf{t ≥ 0 : η(t) ∈ ∂B(0, (m 1 + m 2 )j )}. Let us first prove the following lemma.
Indeed, (A.7) implies that the number of 1 ≤ j ≤ n( ) for which Im(η(σ j )) ≥ C is stochastically dominated from below by a binomial random variable with parameters p = p 0 and n( ). Thus (A.6) with q 0 = 1 − p 0 follows from Lemma 2.6. To see that (A.7) holds, fix a value of j ∈ N and let θ j = arg(η(σ j )). Let θ j (resp. θ j ) be such that [θ j , θ j ] is the set of θ ∈ [0, π] so that the imaginary part of (j + 1)e iθ is at least 2C . We then let z j be the point on ∂B(0, (m 1 + m 2 )(j + 1) ) with argument (θ j ∨ θ j ) ∧ θ j . We note that the harmonic measure as seen from z j of the part of ∂H σ j which is to the left (resp. right) of η(σ j ) is at least some constant a 0 > 0. Moreover, if Im(η(σ j+1 )) ≤ C , then the harmonic measure seen from z j of either the part of ∂H σ j+1 which is to the left or right of η(σ j+1 ) will be at most some constant a 1 > 0. We note that from the explicit form of (A.1) that there is a positive chance that Θ (with w = z j ) in the time interval [σ j , σ j+1 ] starting from a point (a 0 , 1 − a 0 ) ends in (a 1 , 1 − a 1 ). On this event, Im(η(σ j+1 )) ≥ C , which completes the proof of (A.7).
Let us then prove (A.9). For a Brownian motion started at z k and stopped upon exiting H σ j k , the probability that it hits the right hand-side of η[0, σ j k ] or R + (resp. the left-hand side of η[0, σ j k ] or R − ) is bounded below by some constant c > 0. Since we have imposed Im(φ k (z k )) = 1/10, it follows that there exists r > 0 such that |Re(φ k (z k ))| ≤ r, because otherwise the harmonic measure seen from φ k (z k ) of either R − or R + will be less than c. This completes the proof of (A.9).
Proof of Proposition 2.2. Fix α > 1. We will adjust its value later in the proof. By Lemma A.1 and Lemma A.3, the conditional probability given F σ j k that η makes n crossings across B(z k , )\B(z k , α ) before exiting B(0, (m 1 + m 2 )(j k + 1) ) is at least c 1 c 2 n 2 (α−1) for constants c 1 , c 2 > 0. Since this is true for all k, by combining with Lemma A.2 we see that the probability that η fails to make n such crossings for all k with σ j k before η first hits ∂D is at most (1 − c 1 c 2 n 2 (α−1) ) 1/(q 0 ) . This tends to 0 as → 0 provided we take α > 1 sufficiently close to 1, which completes the proof.
Proof of Proposition 2.1. This follows from Proposition 2.2 and the local absolute continuity between whole-plane and chordal SLE κ [26].