A multifractal boundary spectrum for SLE$_\kappa(\rho)$

We study SLE$_\kappa(\rho)$ curves, with $\kappa$ and $\rho$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed"angle"and determine the almost sure Hausdorff dimension of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $g_t$, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.


Introduction
The Schramm-Loewner evolution (SLE) is a one parameter family of random fractal curves, that was introduced by Oded Schramm as a conformally invariant candidate for the scaling limit of twodimensional discrete models in statistical mechanics. Consider the half-plane Loewner differential equation where the driving function, W t , is continuous and real-valued. The chordal Schramm-Loewner evolution with parameter κ > 0 (SLE κ ) is the curve with corresponding conformal maps given by the Loewner equation with W t = √ κB t . SLE κ exhibits interesting geometric behaviour. If 0 < κ ≤ 4, the curves are simple and do not intersect the real line, if κ > 4 they have non-traversing self-intersections and collide with the real line and if κ ≥ 8, the curves are space-filling. For κ > 0, the almost sure Hausdorff dimension is d κ = min{2, 1 + κ 8 }, see e.g. [4]. For κ > 4, the intersection of the curves with the real line are random fractals of almost sure Hausdorff dimension min{2 − 8/κ, 1}, see [2] and [21]. In [1], Alberts, Binder and Viklund studied and computed the almost sure Hausdorff dimension spectrum of random sets of points, where the SLE κ curve, for κ > 4, hit the real line at a prescribed "angle".
If we again, consider the Loewner equation, but instead let W t be the solution to the following system of SDEs where x R is a point on the real line, called the force point, and ρ is an appropriately chosen associated weight, then the Loewner chain is generated by a random curve, called an SLE κ (ρ) curve. This two parameter family of random fractal curves is a natural generalization of SLE κ , in which one keeps track of the force point as well as the curve. The weight ρ determines a repulsion (if ρ > 0) or attraction (if ρ < 0) of the curve, from the boundary. For ρ = 0, it is the ordinary SLE κ . For κ ∈ (0, 8), ρ ∈ ((−2) ∨ ( κ 2 − 4), κ 2 − 2) and x R = 0 + , the Hausdorff dimension of the intersection of SLE κ (ρ) with the real line is almost surely 1 − (ρ + 2)(ρ + 4 − κ/2)/κ (see [18] and [24]). What we are interested in studying in this article, is the dimension spectrum studied by Alberts, Binder and Viklund in [1], but for SLE κ (ρ). In [3], the authors apply the main result of this paper to describe the boundary hitting behaviour of the loops of the two-valued sets of the Gaussian free field.
Let (g t ) be the SLE κ (ρ) Loewner chain, let g t denote the spatial derivative of g t and for β ∈ R, define V β = x > 0 : lim that it, V β is the set of points x in R + , at which g τs decays like e −βs . This can be viewed as a generalized hitting "angle" of the real line (see the discussion in the introduction in [1]). We shall view the decay of g τs (x) as a decay of a certain harmonic measure. Indeed, let r t be the rightmost point of η([0, t]) ∩ R and let H t be the unbounded connected component of H \ η([0, t]). Then the harmonic measure from infinity of (r t , x] in H t is defined as Assume that we have c −1 e −βs ≤ g τs (x) ≤ ce −βs for some β. By the Koebe 1/4 theorem, we then have c −1 e −αs ≤ ω ∞ ((r τs , x], H τs ) ≤ ce −αs , α = β + 1.
for some constant c. As we see in (2), however, we allow a subexponential error, as up to constant asymptotics are too restrictive to require. For fixed κ > 0, ρ ∈ R, let and write β − = inf{β : d(β) > 0} and β + = sup{β : d(β) > 0}. The main theorem of the paper is the following.
To prove Theorem 1.1, it will be more convenient to consider the sets V * β = x > 0 : lim s→∞ 1 s log g τs (x) = −β(1 + ρ/2), τ s = τ s (x) < ∞ ∀s > 0 , and then use that V β = V * β/(1+ρ/2) . We now give an overview of the paper. In Section 2, we introduce the preliminary material needed in the rest of the paper, such as SLE κ (ρ) processes, the Gaussian free field and the imaginary geometry coupling. In order to make the paper more self-contained, the section on imaginary geometry is more extensive than necessary. Furthermore, we use the Girsanov theorem to weigh the measure with the local martingale given by the product of a time-change of g t (x) ζ (where ζ is a parameter in one-to-one correspondence with β) and a compensator. With this new measure we can compute the asymptotics of g τs (x) ζ which we use in Section 3 to find a one-point estimate. It turns out to be strong enough to give the upper bound on the dimension of V β , so this can actually be achieved immediately after Corollary 3.2. The rest of Section 3 is dedicated to studying the mass concentration of the weighted measures, which we need for the correlation estimate. Section 4 is dedicated to the proof of the twopoint estimate needed to prove Theorem 1.1 for κ ∈ (0, 4]. This is done by employing the coupling between SLE and the Gaussian free field. We finish the paper in Section 5 by first establishing the upper bound on the dimension of V β using the one-point estimate of Section 3 and then constructing Frostman measures and using the two-point estimate to show that the s-dimensional energy is finite for every s < d(β), and hence that the Hausdorff dimension can not be smaller than d(β).
We believe Theorem 1.1 to be true for all κ and β ∈ [β − (κ), β + (κ)], and our upper bound is actually valid for all parameters. Given the spectrum for κ ∈ (0, 4] it seems natural to try and prove the result for κ > 4 using SLE duality, i.e., that the outer boundary of the SLE κ (ρ) curves are variants of SLE 16/κ curves, similar to what is done in [18]. This is not as straightforward here, however, as what we are interested in is not the dimension of the intersection of the curve with the real line, but the set of points where the curve intersects the boundary with the prescribed behaviour of the derivatives of the conformal maps (or equivalently, the decay of ω ∞ ) as the curve approaches the boundary. How to do this is not clear at the moment.
However, using the method of [1] to get a two-point estimate, one can deduce that in the case κ > 4, the theorem holds for β ∈ [β − , β 0 ]. This is done by considering three events which exhaust the possible geometries of the curve approaching the two points (this is possible as for boundary interactions, the geometries are rather simple) and then separately estimating each of them. The correlation estimate that we have is actually more closely related to the one in [18].
An almost sure multifractal spectrum of SLE curves was first derived in [9], where the reverse flow of SLE was used to study the behaviour of the conformal maps close to the tip of the curve. Another result in this direction is [8], where the imaginary geometry techniques, developed and demonstrated in the articles [14], [15], [16] and [17], were used to find an almost sure bulk multifractal spectrum. We also mention [12], where Lawler proved the existence of the Minkowski content of an SLE curve intersected with the real line, which is related to what was done in [1] and what we do here. Lastly, we mention [5], where the authors computed average multifractal spectra of SLE.
As for notation, we write , and the constants do not depend on f , g or x. We say that φ is a subpower function if lim x→∞ φ(x)x −ε = 0 for every ε > 0. In the same way, we say that ψ is a subexponential function if for every ε > 0, lim x→∞ ψ(x)e −εx = 0. In what follows, implicit constants, subpower functions and subexponential functions may change between the lines, without a change of notation.

Preliminaries
We begin by introducing some preliminaries on complex analysis, SLE κ (ρ) processes, the Gaussian free field and imaginary geometry.

Measuring distances and sizes
Let D be a simply connected domain, z ∈ D, and let ϕ : D → D be a conformal map of D onto D such that ϕ(z) = 0. We define the conformal radius of D with respect to z as crad D (z) = 1 |ϕ (z)| .

It behaves well under conformal transformations
(that is, it is conformally covariant) and by the Schwarz lemma and the Koebe 1/4 theorem, one easily sees that We say that A is a compact H-hull if A = H ∩ A and H \ A is simply connected and we let Q be the set of compact H-hulls. For any domain H \ A, A ∈ Q, we define the harmonic measure from infinity as where E ⊂ ∂(H \ A), where ω denotes the usual harmonic measure. By Proposition 3.36 of [10], there exists, for every A ∈ Q, a unique conformal map g A : H \ A → H which satisfies the hydrodynamic normalization, i.e., The function g A is called the mapping-out function of A. By (6) and the conformal invariance of ω, we have that where |g A (E)| is the total length of the set g A (E) ⊂ R.

SLE κ (ρ) processes
In this section we will introduce SLE κ and SLE κ (ρ) processes. As stated in the introduction, a chordal SLE κ Loewner chain is the collection of random conformal maps (g t ) t≥0 , given by solving (1) with where B t is a standard Brownian motion with B 0 = 0 and filtration F t , satisfying (6). We define f t to be the centered conformal map For fixed z ∈ H, the solution to (1) exists until time T z = inf{t ≥ 0 : f t (z) = 0}. The domain of g t is H t = H \ K t where K t = {z : T z ≤ t} ∈ Q is the SLE hull at time t and g t is the unique conformal map from H t onto H such that lim z→∞ |g t (z) − z| = 0. Rohde and Schramm proved that the family of SLE κ hulls is almost surely generated by a curve η : [0, ∞] → H, i.e., H t is the unbounded component of H \ η([0, t]) (see [19]). We call η the SLE κ process or SLE κ curve and say that K t is the filling of η([0, t]). Now we will define the SLE κ (ρ) process. Let x L = (x l,L , . . . , x 1,L ), x R = (x 1,R , . . . , x r,R ), where x l,L < . . . < x 1,L ≤ 0 ≤ x 1,R < . . . < x r,R . Also, let ρ L = (ρ 1,L , . . . , ρ l,L ), ρ R = (ρ 1,R , . . . , ρ r,R ), ρ j,q ∈ R, q ∈ {L, R}. We call ρ j,q the weight of x j,q . Let W t be the solution to the system of SDEs where N L = l and N R = r. An SLE κ (ρ L ; ρ R ) Loewner chain with force points (x L ; x R ) is the family of conformal maps (g t ) t≥0 obtained by solving (1) with W t being the solution to (7). The SLE κ (ρ L ; ρ R ) hulls, (K t ), are defined analogously and they are almost surely generated by a continuous curve, η, the SLE κ (ρ L ; ρ R ) process or SLE κ (ρ L ; ρ R ) curve (see Theorem 1.3 in [14]). SLE κ (ρ L ; ρ R ) is a generalization of the SLE κ (SLE κ = SLE κ (0; 0)), where one also keeps track of the force points and their assigned weights either attract (ρ j,q < 0) or repel (ρ j,q > 0) the curve. If η is an SLE κ (ρ) curve, we write η ∼ SLE κ (ρ). Exactly how the weights of the force points affect the curve η is explained in Lemma 2.1.
The solution to the system of SDEs (7) exists up until the continuation threshold is hit, that is, the first time t such that either as is explained in Section 2.2 of [14]. Moreover, for every t > 0 before the continuation threshold, P(W t = V j,q t ) = 0 for j ∈ N and q ∈ {L, R}. Geometrically, hitting the continuation threshold means the curve η swallowing force points on either side such that the sum of their weights is less than −2, that is, η hits an interval (x m+1,L , x m,L ) (or (x n,R , x n+1,R )) such that m j=1 ρ j,L ≤ −2 (resp. n j=1 ρ j,R ≤ −2). Now, we write ρ 0,L = ρ 0,R = 0, x 0,L = 0 − , x l+1,L = −∞, x 0,R = 0 + and x r+1,R = +∞, and let for q ∈ {L, R} and j ∈ N The following lemma describes the interaction η with the real line. It is written down in [18], and just as they did, we refer to Remark 5.3 and Theorem 1.3 of [14] and Lemma 15 of [6] for the proof.
, then η can hit (x k,R , x k+1,R ), but then can not be continued afterwards, , then η can hit and bounce off of (x k,R , x k+1,R ) and η ∩ (x k,R , x k+1,R ) has empty interior.
The same holds if we replace R by L and consider (x k+1,L , x k,L ).
Note that in (ii) in the above lemma, the curve has swallowed force points with a total weight at least as negative as −2, and hence it cannot be continued. In (iii) and (iv), the total weight of the force points swallowed is greater than −2, and hence the curve can be continued.
Lemma 2.2. Fix κ > 0 and let (x n L ) and (x n R ) be sequences of vectors of numbers x n l,L < · · · < x n 1,L < 0 < x n 1,R < · · · < x n r,R , converging to vectors (x L ) and (x R ) such that x 1,L = 0 − and x 1,R = 0 + . For each n, denote by (W n , (V n,L ), (V n,R )) the driving processes of an SLE κ (ρ L , ρ R ) process with force points (x n L ; x n R ). Then (W n , (V n,L ), (V n,R )) converges weakly in law, with respect to the local uniform topology, to the driving process (W, It turns out that if we, using the Girsanov theorem, reweight an SLE κ process by a certain martingale (how this is done is explained briefly below), then we obtain an SLE κ (ρ) process at least until the first time that W t = V j,q t for some (j, q). Let x 1,L < 0 < x 1,R and define Then M t is a local martingale and an SLE κ process weighted by M t has the law of an SLE κ (ρ L ; ρ R ) process with force points (x L ; x R ) (see [20]). So far, we have only defined chordal SLE κ (ρ) processes in H, but we can define it in any Jordan domain, by a conformal coordinate change. More precisely, an SLE κ (ρ L ; ρ R ) in a Jordan domain D, from z 0 to z ∞ , with force points (x L , x R ) is the image of an SLE κ (ρ L ; ρ R ) in H from 0 to ∞ under a conformal map ϕ such that ϕ(H) = D, ϕ(0) = z 0 , ϕ(∞) = z ∞ and such that the force points of the SLE κ (ρ L ; ρ R ) in H are mapped to (x L ; x R ). We say that the constructed The configuration of the SLE κ (ρ L ; ρ R ) process we defined in the beginning of the section is (H, 0, x L , x R , ∞).

The case of one force point
Let a = 2/κ. We will parametrize the SLE κ (ρ) so that hcap(K t ) = at, i.e., as the solution to with where B t is a one-dimensional standard Brownian motion with B 0 = 0 and filtration F t . We say that the conformal maps (g t ) t≥0 are driven by (W t , V t ) t≥0 . The solution to (11) (i.e. dW t , dV t ) exists for all t ≥ 0, if κ > 0 and ρ > −2, so henceforth, we assume this. If ρ ≥ κ 2 − 2, then η does not hit the real line, almost surely, and hence we are interested in the case ρ < κ 2 − 2. If κ > 4 and ρ ∈ (−2, κ 2 − 4], then by Lemma 2.1, η ∩ (x R , ∞) is almost surely an interval, and thus we will consider ρ ∈ ((−2) ∨ ( κ 2 − 4), κ 2 − 2). Fix κ > 0 and ρ > −2 and let (K t , t ≥ 0) be the hulls of an SLE κ (ρ) process with force point x R . The SLE κ (ρ) satisfies two important properties. The first is the following scaling rule: for any m > 0, (m −1 K m 2 t , t ≥ 0) has the same law as the hulls of an SLE κ (ρ) process with force point x R /m. If x R = 0 + , then it is scaling invariant. The second is the Domain Markov Property: for any finite stopping time, τ , the curve defined as (η(t) = f τ (η(t + τ )), t ≥ 0) is an SLE κ (ρ) curve with force point Note that f t follows the SDE Taking the spatial derivative of (12) results in an ODE that, upon solving, yields While g t is defined in H t , it (and hence f t and g t ) extends continuously to the real line and is realvalued there. For x ∈ R + , g t (x) ∈ [0, 1] and is decreasing in t. Due to symmetry, it is enough to consider x, x R ∈ R + . By applying Itô's formula to log f t (z) and exponentiating, we see that The same procedure, applied to v t (z) : Observe that, considered as functions on R + , f t , g t and v t are increasing in x. We will mostly work with these functions on or close to R + .

Local martingales and weighted measures
Fix and let, as above, a = 2/κ. For each such pair (κ, ρ), we will define a one-parameter family of local martingales which will play a major role in our analysis. Let ζ be the variable with which we will parametrize the martingales and let Note that with our choice of ζ, µ > µ c and β > 0. The reason for letting ζ > − µ 2 c 2a will be clear by the end of the next subsection. The parameters are related as follows: For each x > 0, is a local martingale on 0 ≤ t ≤ T x , such that Note that under the measure weighted by M ζ t , the SLE κ (ρ) process becomes an SLE κ (ρ, −µκ) process with force points (x R , x). We shall write the local martingale in a different way, which is very convenient for our analysis. Define the random processes and We will often not write out the dependence on x, as it will cause no confusion. Since we V t ≥ W t , we see that Q t ∈ [0, 1] for every t ≥ 0. Further, we note that if x R = 0 + , then Q is the ration between the harmonic measure from infinity of two sets, more precisely, if η R denotes the right side of the curve η and r t is the rightmost point of η([0, t]) ∩ R, then .
With these processes, we can write This will be very convenient, as we will relate the process δ t to the conformal radius at the point x, and thus, it will be comparable to dist(x, η[0, t]). With this in mind, we will then make a random time change so that the time-changed process decays deterministically, and it will give us good control over the decay of the distance between the curve and a certain point. It does not, however, make sense to talk about the conformal radius of a boundary point, so we will begin by sorting this out. We let H t = H t ∪ {z : z ∈ H t } ∪ {x ∈ R + : t < T x } be the union of the reflected domain and the points on R + that have not been swallowed at time t. Then, it makes sense to talk about crad Ht (x) for 0 < x ∈ H t , and a calculation shows that if x R = 0 + , then for t < T x , This implies that While this only holds for x R = 0 + , similar bounds can be acquired for other x R , as the following lemma shows. See also [22]. Lemma 2.3. Let κ > 0, ρ > −2, and let η t be an SLE κ (ρ) curve with force point x R ≥ 0. Let (g t ) t≥0 be the conformal maps, driven by (W t , V t ) t≥0 , and let T x be the swallowing time of x. Let x > x R and 0 < t < T x , then Proof. Denote by (K t ) t≥0 the compact hulls corresponding to the SLE κ (ρ) process. Extend the maps (g t ) t≥0 by Schwarz reflection to C \ (K t ∪ {z : z ∈ K t } ∪ R − ), let r t be the rightmost point of K t ∩ R and let O t = g t (r t ). Then, by the Koebe 1/4 theorem, that is, and thus the upper bound is done.
for t < T u . Then, Using the Loewner equation, we see that Thus, for every u ∈ (0, x R ] and t < T u , Fixing t and letting u → r t , we get which implies that and thus With this lemma in mind, we will proceed and make a random time change. The process δ t satisfies the (stochastic) ODE so we define the processt(s) as the solution to the equation Then,δ s := δt (s) satisfies the ODE dδ s = −aδ s ds, i.e., This time change is called the radial parameterization. Note that this time-change is depending on x. We letg s = gt (s) etc. denote the time-changed processes. They are all adapted to the filtratioñ F s = Ft (s) . In this parametrization, andQ s follows the SDE whereB s is a Brownian motion with respect to the filtrationF s . Combining (22) and (25), we see thatM For each x > 0 and ζ > − µ 2 c 2a , we define the new probability measure, which we denote by P * = P * x,ζ , as for every A ∈F s . We denote the expectation with respect to P * by E * . LetB * s denote a Brownian motion with respect to P * . By the Girsanov theorem, we have that under the measure P * ,Q s follows the SDE Under P * ,Q s is positive recurrent and has the invariant density see Corollary A.2. Throughout, we will denote byX s , a process that follows the same SDE asQ s , but started from the invariant density. For s ≥ 0 and y > 0, short calculations show that In Section 3 we will use that P * (t(s) < ∞) = 1 for every s (which is shown in Appendix A), and the following lemma.
Lemma 2.4. Fix x and let τ s andt(s) be as above. Theñ Thus, τ s ≤t s a + 1 a (log x + log 4) , and the proof is done.
In what follows, x and x R will be kept constant (time is the parameter which will change), and hence we will instead treat the inequality as The only place where we have to be careful with this is in the proof of Lemma 4.6, but we will discuss that there.

The Gaussian free field
We will now introduce and discuss the Gaussian free field (GFF). Let D ⊂ C be a Jordan domain and let C ∞ c (D) be the set of compactly supported smooth functions on D. The Dirichlet inner product on D is defined as This does not converge in any space of functions, however, it converges almost surely in a space of distributions. The GFF is conformally If we denote the standard L 2 (D) inner product by (·, ·) and U ⊆ D is open, then for f ∈ C ∞ c (U ), g ∈ C ∞ c (D), we obtain by integration by parts that is the set of functions in H(D) which are harmonic in U and have finite Dirichlet energy. Hence, we may write and H ⊥ (U ), respectively. Note that h U is a GFF on U and that h U ⊥ is a random distribution which agrees with h on D \ U and can be viewed as a harmonic function on U and that h U and h U ⊥ are independent. Hence, the law of h restricted to U , given the values of h restricted to ∂U , is that of a GFF on U plus the harmonic extension of the values of h on ∂U . This is the so-called Markov property of the GFF. With this in mind, one can make sense of GFF with non-zero boundary conditions: let f : ∂D → R, let F be the harmonic extension of f to D and let h be a zero boundary GFF on D, then the law of the of the GFF with boundary condition f is given by the law of h + F .
The GFF also exhibits certain absolute continuity properties, the key (for us) of which we state now. (This is the content of Proposition 3.4 (ii) in [14], where the reader can find a proof.) Proposition 2.5. Let D 1 , D 2 be simply connected domains, such that D 1 ∩ D 2 = ∅ and for j = 1, 2, let h j and F j be a zero-boundary GFF and a harmonic function on D j , respectively. If U ⊆ D 1 ∩ D 2 is a bounded simply connected domain such that D 1 ∩ U = D 2 ∩ U and F 1 − F 2 tends to zero when approaching ∂D j ∩ U for j = 1, 2, then the laws of (h 1 + F 1 )| U and (h 2 + F 2 )| U are mutually absolutely continuous.
In other words, if h 1 and h 2 are GFFs on D 1 and D 2 , whose boundary conditions agree in some set E ⊂ ∂D 1 ∩ ∂D 2 , then the laws of h 1 and h 2 restricted to any simply connected bounded subdomain U of D 1 and D 2 , such that U ∩ ∂D 1 ∩ ∂D 2 ⊂ E, are mutually absolutely continuous.
The result holds for unbounded domains U as well, but we shall only need the bounded case.

Imaginary geometry
In this section, we describe the coupling of SLE with the GFF. As stated in the introduction, this section will be slightly longer than necessary, in order to make this paper more self-contained.
Suppose for now that h is a smooth, real-valued function on a Jordan domain D and fix constants χ > 0 and θ ∈ [0, 2π). A flow line of the complex vector field e i(h/χ+θ) , with initial point z, is a solution to the ordinary differential equation If η is a flow line of e ih/χ and ψ : This follows by the chain rule and the fact that a reparametrization of a flow line is a flow line. Hence, the following definition makes sense: we say that an imaginary surface is an equivalence class of pairs (D, h) under the equivalence relation We say that ψ is a conformal coordinate change of the imaginary surface. The idea is that if h is a GFF, then we are interested in the flow lines of h and we want to see that these are SLE κ (ρ) curves. However, while (33) makes sense if h is a GFF, the ODE (32) does not, as h is then a random distribution and not a continuous function. Thus, the approach to defining the flow lines of the GFF will be a little less "direct". Instead, the following characterization will be used: let h be a smooth function and η a smooth curve in H, starting in the vertical direction, so that as t → 0 the winding number is ≈ π/2. Furthermore, let (f t ) t≥0 be the centered Loewner chain of η. Then, for any t > 0, we have two parametrizations of η| [0,t] : Theorem 2.6. Fix κ > 0 and a vector of weights (ρ L ; ρ R ). Let (K t ) and (f t ) be the hulls and centered Loewner chain, respectively, of an SLE κ (ρ L ; ρ R ) process in H from 0 to ∞ with force points (x L ; x R ) and let h be a zero-boundary GFF in H. Furthermore, let and define If τ is any stopping time for the SLE κ (ρ) process which almost surely occurs before the continuation threshold, then the conditional law of (h + Φ 0 )| H\Kτ given K τ is equal to the law of h • f τ + Φ τ . In this coupling, η ∼ SLE κ (ρ) is almost surely determined by h, that is, η is a deterministic function of h (see Theorem 1.2 of [14]). When κ ∈ (0, 4), a flow line of the GFF h + Φ 0 on H is an SLE κ (ρ) curve, η, coupled with h + Φ 0 as in Theorem 2.6. This definition can be extended to other simply connected domains than H, using the conformal coordinate change described in (33), see Remark 2.7. If we add θχ to the boundary values, i.e., replace h + Φ 0 by h + Φ 0 + θχ then the resulting flow line is called a flow line of angle θ, and we denote it by η θ .
Note that if κ ∈ (0, 4) then χ > 0 and if κ > 4 then χ < 0. If we let κ ∈ (0, 4) and write κ = 16/κ, then κ > 4 and χ(κ) = −χ(κ ). From Theorem 2.6, it is clear that the conditional law of h + Φ 0 given an SLE κ or SLE κ curve is transformed in the same way under a conformal map, up to a sign change, which motivates the following definition. A counterflow line of the GFF h + Φ 0 is an SLE κ (ρ) curve coupled with −(h+Φ 0 ) as in Theorem 2.6. Note that the sign of the GFF is changed so that it matches the sign of χ(κ ) and that in the notation of the theorem, the λ is replaced by λ = π √ κ = λ − π 2 χ. In the figures, we often write a ∼ , where a is some real number. This is to be interpreted as a + χ times the winding of the curve, see Figure 3. This makes perfect sense for piecewise smooth curves, but for fractal curves the winding is not defined pointwise. However, the harmonic extension of the winding of the curve makes sense, as we can map conformally to H with piecewise constant boundary conditions. The term −χ arg f τ (z) in Theorem 2.6 is interpreted as the harmonic extension of χ times the winding of the curve η.
Remark 2.7. Let D be a simply connected domain, with x, y ∈ ∂D distinct and let ψ : D → H be a conformal transformation with ψ(x) = 0 and ψ(y) = ∞. Let x L (x R ) consist of l (r) marked prime Figure 3: Each time the curve makes a turn to the right, the boundary values decrease by χ times the angle and each time it makes a turn to the left the angle increases by χ times the angle, for example, a quarter turn to the right (left) decreases (increases) the boundary values by π 2 χ. We illustrate a + χ · winding as a ∼ , and hence, the three pictures of this figure give the same information. ends in the clockwise (counterclockwise) segment of ∂D, which are in clockwise (counterclockwise) order. The orientation of ∂D is as defined by ψ. Write x 0,L = x 0,R = x and x l+1,L = x r+1,R = y and let ρ L and ρ R be vectors of weights corresponding to the points in x L and x R respectively. Let h be a GFF on D with boundary values given by The same statement holds for counterflow lines with κ ∈ (4, ∞) if we replace λ with λ in the boundary values (but keep χ = χ(κ)).
We write the following statements for flow lines in H, but they hold true for other simply connected domains as well.
Let h be a GFF in H with piecewise constant boundary values. It turns out that (Theorem 1.4, [14]) if η is a counterflow line of h in H, from ∞ to 0, then the range of η is almost surely equal to the points that can be reached by the flow lines in H, from 0 to ∞, with angles in the interval − π 2 , π 2 . Also, it almost surely holds that the left boundary of η is equal to the trace of the flow line of angle − π 2 and the right boundary is equal to the trace of the flow line of angle π 2 (seen from the viewpoint of travelling along η , from the flow lines' point of view, it is the other way). Here, we talk about counterflow lines corresponding to the parameter κ and flow lines corresponding to κ, so that they can be coupled with the same GFF.
Again, let h be a GFF in H with piecewise constant boundary values. For each x ∈ R and θ ∈ R, we denote by η x θ the flow line of h from x to ∞ with angle θ. Fix x 1 , x 2 ∈ R such that x 1 ≥ x 2 , then the following holds (see Figure 5 for illustrations).
(i) If θ 1 < θ 2 , then η x1 θ1 almost surely stays to the right of η x2 θ2 . If θ 2 − θ 1 < πκ 4−κ , then the paths might hit and bounce off of each other, otherwise they almost surely never collide away from the starting point.
2 and η π 2 will hit and merge with the respective sides of η , as they will be the outer boundary of η . Note that the outer boundary conditions agree.
The above flow line interactions are the content of Theorem 1.5 of [14]. We shall make use of property (ii), as it is instrumental in our two-point estimate.

Level lines
The coupling is valid for κ = 4 as well. We then interpret the resulting SLE κ (ρ) curve as the level line of the GFF. Note that χ(4) = 0, that is, there is no extra winding term, and hence the boundary values of the level line are constant along the curve, −λ on the left and λ on the right. As in the case of flow and counterflow lines, level lines can be defined in other domains and with different starting and ending points via conformal maps. For level lines, the terminology is a bit different: we say that η is a level line of height u ∈ R if it is a level line of the GFF h + u. The same interactions as for flow lines hold for level lines. We let η x u denote the level line of height u starting from x. Let x 1 ≥ x 2 , then (i) If u 1 < u 2 , then η x1 u1 almost surely stays to the right of η x2 u2 . (ii) If u 1 = u 2 , then η x1 u1 and η x2 u2 can intersect and if they do, they merge and never separate. For more on the level lines of a GFF with piecewise constant boundary data, see [23].

Deterministic curves and Radon-Nikodym derivatives
We now recall two lemmas about flow and counterflow line behaviour and two lemmas on absolute continuity, all from [18], which we will need in Section 4. Note that while they are proven for κ = 4, (c) If θ2 < θ1 < θ2 + π, then η x 1 θ 1 and η x 2 θ 2 cross when intersecting for the first time, and never cross back. the case κ = 4 follows by the same argument as κ < 4, when the SLE 4 (ρ) curves are coupled as level lines.
Next, we shall describe the Radon-Nikodym derivatives between SLE κ (ρ) processes in different domains, see also [6] and [18]. The results that we need are Lemma 2.10 and Lemma 2.11, which we will use in Section 4. Let be a configuration, that is, a Jordan domain D with boundary points z 0 , x L x R and z ∞ , and let U be an open neighborhood of z 0 . Denote the law of an SLE κ (ρ L ; ρ R ) process with configuration c, stopped the first time τ it exits U , by µ U c . Let H D be the Poisson excursion kernel of D, that is, if ϕ : D → H is conformal, then Furthermore, let ρ ∞ = κ − 6 − j,q ρ j,q and Moreover, let where x τ j,q = x j,q if x j,q is not swallowed by η at time τ and x τ j,q the leftmost (resp. rightmost) point of K τ ∩ ∂D on the clockwise (resp. counterclockwise) arc of ∂D if q = L (resp. q = R). Moreover, let µ loop be the Brownian loop measure, a σ-finite measure on unrooted loops (see [13]), and write

One-point estimates
In this section we will find first moment estimates, which will be of importance, as they will give us the means to get good two-point estimates as well as give us the upper bound for the dimension of V * β . Recall thatg s = gt (s) is the Loewner chain under the radial time change, see Section 2.2.2. 1−a+µ)s )). Using this, together with the fact that P * (t(s) < ∞) = 1 for every s, we have Using Proposition 3.1 and Lemma 2.4, recalling that we can choose a C * = C * (x, x R ) such that (31) holds, we get the following corollary.
Proof. By Lemma 2.4 and that the map t → g t is decreasing, we have By the previous proposition, we have where the constants in O depend on x and x R (since C * does). Thus, the proof is done.
At this point, we already have what is needed for the upper bound on the dimension of V * β .

Mass concentration
In this subsection, we will see that the mass of the weighted measure P * is concentrated on an event where the behaviour ofg s (x), for fixed x, is nice. On this event, we will show thatg s (x) satisfies a number of inequalities which will be helpful in proving the two-point estimate of the next section. The ideas here are similar to those of Section 7 of [11]. We define the processL s , by recalling (26), as As stated in Section 2.2.2 (and shown in the appendix),Q s has an invariant distribution, pQ, under P * . Therefore, by the ergodicity ofQ s (Corollary A.2) and a computation, holds P * -almost surely, that is, the time average converges P * -almost surely to the space average. We shall prove that, roughly speaking, as s → ∞,L s ≈ β(1 + ρ/2)s, with an error of order √ s. To prove this, we need to prove the next lemma first.
There is a positive constant c < ∞ such that for p > 0 sufficiently small, and t ≥ 1, The proof idea is as follows. Note that we can writeM ζ s as M ζ s = e −aζLs e aµ(1+ρ/2)sQµ s .
Proof. We begin by the first inequality. We have that where ε is small enough forÑ t to be well-defined. Then, and exponentiating, we get Consider the case δ < 0, i.e., ≥ 1 for sufficiently small ε, and thus Consider the case δ > 0. We will split the expectation into the casesQ t ≤ y andQ t > y for some y ∈ (0, 1]. First, for sufficiently small ε. For the other part, note that sinceL t ≥ 0, for some constant, c ,where the last equality follows by Corollary A.2 and (30). If we let then we see that both the "Q t ≤ y"-part and the "Q t > y"-part are bounded by positive constants. Thus, we are done.
With the previous lemma at hand, we can now prove the following.
Proposition 3.4. There exists a constant, c, such that if we fix t > 0 and letĨ u t be the event that for all 0 ≤ s ≤ t, then, for every ε > 0 there exists a u < ∞ such that Proof. There is a constant, c, such that for any k ∈ N, Thus, by splitting into subintervals of length 1, Chebyshev's inequality and Lemma 3.3 (with p > 0 accordingly) We shall denote both the event and the indicator function of the event asĨ u t , and we will more often than not drop the u in the notation and writeĨ t . Straightforward calculations, using thatg s (x) = e −Ls , show that on the event of the above proposition, we have ψ 0 (s) −1 e −aβ(1+ρ/2)s ≤g s (x) ≤ ψ 0 (s)e −aβ(1+ρ/2)s where ψ 0 is the subexponential function ψ 0 (s) = e au √ s log(2+s)+c . Next, we want to convert these facts into the corresponding for g τt (x). We let C * = C * (x, x R ) denote the constant as remarked after Lemma 2.4, that is, the constant such that, for t > 0,t((t/a − C * ) ∨ 0) ≤ τ t (x) ≤t(t/a + C * ). What we will do now, is to define an F τt -measurable version ofĨ u t (the indicator of the event of Proposition 3.4) and the natural way is to define this as the conditional expectation with respect to this filtration. Fix u > 0 and write In the next proposition, we will see that this indeed works the same way for g τt (x) asĨ u t does forg t (x). We will omit the superscript and write I t = I u t .
Lemma 3.5. Let u > 0 and I t = I u t be as above. Then there is a subexponential function ψ such that where the implicit constants depend on x and x R .
Proof. Fix u > 0 and write t + = t/a + C * and t − = t/a − C * (where C * is as described above). Since t → g t (x) is decreasing, we haveg Hence, by (36) In the same way, and the lemma is proven. denote the event where we replace c byc > c, the same estimates hold with the subexponential function ψ 1 (s) = e au √ s log(2+s)+c in place of ψ 0 . Hence, the correct asymptotic behaviour on g is preserved onĨ u,c t . Furthermore,Ĩ u t ⊂Ĩ u,c t .
Remark 3.7. We could actually add the condition to the eventĨ u t while still retaining that, uniformly in t, P * (Ĩ u t ) ≥ 1 − ε for large enough u. This would then give the bounds g τt (x)ψ(t − s) −1 e β(1+ρ/2)(t−s) I t ≤ g τs (x)I t ≤ g τt (x)ψ(t − s)e β(1+ρ/2)(t−s) I t for s ≤ t. The key to adding this condition toĨ u t is proving that 4 Two-point estimate

Outline
In this section, we use the imaginary geometry techniques to prove a two-point estimate that we need for the lower bound on the dimension of V * β (and hence V β ). We follow the ideas of Section 3.2 of [18] and we will keep the notation similar. Note that we will write the proof for flow lines, i.e., κ < 4, but the merging property and every lemma that we will need, hold for the level lines of the GFF as well, so the method also gives the two-point estimate in the case κ = 4. The main idea is to use the merging of the flow lines and the approximate independence of GFF in disjoint regions to "move the problem between scales" and separate the points when at the right scale.
We let h be a GFF in H with boundary conditions such that the flow line η from 0 is an SLE κ (ρ) process from 0 to ∞. We define a sequence of random variables E n (x), for x ∈ R and n ∈ N, such that if E n (x) > 0 for every n ∈ N, then x ∈ V * β and we say that x is a perfect point. The idea for the construction of the random variables is as follows. Consider the event A 1 0 , that η hits the ball B(x, ε 1 ), ε 1 = e −α1 and let E 0 (x) = 1 A 1 0 (x) I u,Λ α1 , where I u,Λ α1 is the random variable of (37) but with a larger constant Λ. That is, if E 0 (x) > 0, then η gets within distance ε 1 of x and the derivative g τt (x) decays approximately as e −β(1+ρ/2)t until η hits B(x, ε 1 ).
We proceed inductively. Assume that E k (x) is defined and that ε j = e − j l=1 α l , α l > 0. Let η x k+1 be the flow line started from the point x k+1 = x − ε k+1 /4. Let A 1 k+1 (x) be the event that η x k+1 hits B(x, ε k+2 ), plus some regularity conditions. Furthermore, let I u,Λ,k+1 k+1 denote the random variable corresponding to (37), but for η x k+1 until hitting B(x, ε k+2 ). Next, given that A 1 k (x) and A 1 k+1 (x) occur, let A 2 k+1 (x) be the event that η x k hits η x k+1 plus some regularity conditions. We then set In short, we let a sequence of flow lines, on smaller and smaller scales, approach the point x, such that each flow line has the correct geometric behaviour as they approach x. Moreover, each flow line hits and merges with the next. In this way, the SLE κ (ρ) process η inherits its geometric behaviour from each of the flow lines. This is very convenient to derive the two-point estimate, that is, we want to prove that the correlation of E n (x) and E n (y) is small when |x − y| is large. The key property that we use is that the flow lines started within the balls B(x, |x − y|/(2 + δ 0 )) and B(y, |x − y|/(2 + δ 0 )) be approximately when δ 0 > 0 (in the sense that the Radon-Nikodym derivative between the measures with and without the other set of flow lines present is bounded above and below by a constant). Moreover, the flow lines outside of those balls will also be approximately independent, in the same sense. See Lemma 4.4. Moreover, the probability of two subsequent flow lines merging is proportional to 1, see Lemma 4.5.
Having a certain decay rate of the derivatives of the conformal maps is equivalent to having a certain decay rate of the harmonic measure from infinity of some set on the real line. This will be essential to us, as it is the tool with which we show that the perfect points actually belong to V * β . Moreover, it is important that α j → ∞, but not too quickly. If α j would not tend to ∞, then the perfect points would just be points where lim s→∞ In the next subsection, there will be parameters which at first may look redundant, but in fact play an important role in the regularity conditions. We conclude this subsection by listing them and writing short comments about their use.
• δ ∈ (0, 1 2 ): Chosen to be very small and makes sure that the curve η x k does not hit B(x, 1 M ε k ) or B(x, ε k+1 ) too close to the real line. Important, as it makes sure that the probability of η x k and η x k+1 merging does not decrease in k.
• M > 0: Crucial in the proof that the perfect points belong to V * β . It makes sure that the probability of exiting in the interval between the rightmost point on R of η x k+1 (stopped upon hitting B(x, ε k+2 )) and x for a Brownian motion started in B(x, ε k+1 ) depends mostly on η x k+1 and not on η x k . Chosen large, so that the process Q k will be close in law to its P * -invariant distribution when η x k+1 reaches B(x, 1 M ε k ).
• Λ > 0: Chosen large so that the eventĨ u,Λ t for η x k contains the eventĨ u t for the image of η x k under some map F .
• u > 0: Chosen large enough, so that the eventĨ u t has sufficiently large P * -probability.

Perfect points and the two-point estimate
Throughout this section we fix κ ∈ (0, 4) and ρ ∈ (−2, κ 2 − 2) and let h be a GFF in H with boundary values −λ on R − and λ(1 + ρ) on R + , so that the flow line η from 0 to ∞ is an SLE κ (ρ) curve with force point located at 0 + (so the configuration is (H, 0, 0 + , ∞)). Note that the interval for ρ is chosen so that η can hit R + . We denote the flow line from x by η x and note that for x > 0, η x is an SLE κ (2 + ρ, −2 − ρ; ρ) with configuration (H, x, (0, x − ), x + , ∞). We fix δ ∈ (0, 1 2 ), M > 0 large and an increasing sequence, α j → ∞, write α k = k j=1 α j and let ε k = e −α k . The constants δ and M will be chosen later. As for α j , we define it as α j = α 0 + log j, where α 0 = log N for some large integer N . For x ≥ 1 and k ∈ N, we write (when x = 0, we omit the superscript) and and note that σ(B(x, ε k )) = τ α k (x). Furthermore, let σ x k,M = σ x k (B(x, 1 M ε k )).We let η x k ,R denote the right side of the flow line η x k and let r k t = max{y ∈ η x k ([0, t]) ∩ R} and define Q k t by .
Recall that by (21), Q t = Q 0 t is the diffusion (20). For k ≥ 0, letĨ u,Λ,k t =Ĩ u,Λ,k t (x) denote the event (as well as the indicator of the event) of Proposition 3.4, with constant Λ (see Remark 3.6) but for the flow line η x k , and as previously. The constant Λ will be chosen in Lemma 4.6. Note that the eventĨ u,Λ,k t is a condition on the geometry of the curve which does not change when we rescale (it can be expressed in terms of Q k , which is invariant under scaling of the SLE κ (ρ) process). We let A 1 1 2 ε k ) and it does not hit B(x, 1 M ε k ) or B(x, ε k+1 ) "too far down" (the latter being due to the condition on Q k t ). Now, we set ) for t > σ x k−1 but before merging with η x k , and let E 2 k (x) = 1 A 2 k (x) be the indicator of that event. Next, we let E k (x) = E 1 k (x)E 2 k (x) and write and E n (x) = E −1,n (x).
and the derivatives of the Loewner chain for η x k−1 behave as we want. Furthermore, given that : Condition (ii) on A 2 k (x) ensures that we will not have the case in the above figure, instead there will be some sector which the flow lines will not enter.
Why this is the right setting and these conditions are the correct ones to look for might not be clear at first sight. This, we prove in the next lemma.
First, we note that by the Koebe 1/4 theorem, for each integer k ∈ N. Hence, it is enough to see that the decay rate of ω ∞ is the correct one. Let K n denote the closure of the complement of the unbounded connected component of H \ (η([0, τ αn+1 ]) ∪ η xn ([0, σ x n ])). Clearly, on the event E n (x) > 0, since K τα n+1 ⊂ K n and (r n σ x n , x] ⊂ (r τα n+1 , x]. In view of ω as the hitting probability, it is easy to see that where the implicit constant is independent of n. Indeed, a Brownian motion must hit the line segment L n = [x, η xn (σ x n )] before hitting either of the two intervals (r τα n+1 , r n σ x n ] and (r n σ x n , x]. However, from any point z ∈ L n , dist(z, (r τα n+1 , r n σ x n ]) > ε n+1 , dist(z, (r n σ x n , x]) ≤ ε n+1 and x − r n σ x n > ε n+1 . Hence, the conditional probability of the Brownian motion exiting in (r n σ x n , x], given that it will exit in (r τα n+1 , x] is greater than somep > 0. That the constant is independent of n follows from scale invariance. See Figure 10 for the illustration of the last inequality. Thus, we have proven that on the event E n (x) > 0, Finally, we shall prove that ω ∞ ((r n σ x n , x], H \ K n ) has the correct decay rate, that is, that We start with the upper bound. By the Markov property for Brownian motion, (the probability increases as we are removing obstacles, allowing more Brownian paths). Next, we shall see that where C is independent of j. Indeed, for z ∈ ∂B x, 1 2 ε j−1 , we have that where the implicit constant is independent of both z and j. Moreover, noting that on the event we have that We now turn to the lower bound. We begin by noting that ifτ k = inf{t > 0 : where the implicit constant depends only on δ, M and Λ. In fact, the probability of the Brownian motion hitting (r σ x k , x] and some arc That is, let υ denote the exit time of H \ η([0,τ 0 ]) for the Brownian motion, then where, again, the implicit constants depend only on δ, M and Λ. By the same reasoning, if S k = { 1 2 ε k e iθ : θ ∈ [θ 1 (δ), θ 2 (δ)], θ 1 (δ) > 0}, then for every z ∈ S k−1 , where υ k denotes the exit time from H \ K k for the Brownian motion, and the implicit constant does not depend on k. Thus, by the Markov property, (41) and (42), Hence, Therefore, by (39), and consequently by (38) lim n→∞ 1 α n log g τα n (x) = −β(1 + ρ/2).
Thus, if E n (x) > 0 for every n ∈ N, then x ∈ V * β . Figure 10: A Brownian motion started from the dash-dotted line will hit the green interval with probability proportional to that of hitting the union of the green and the blue intervals.
The two-point estimate which we will acquire here is the following.
The main ingredients in the proof are divided into three lemmas; the first of which establishes "approximate independence" between flow line interactions in different regions; the second states that merging of these flow lines happens with high enough probability and the third is a one-point estimate.
The constants in may depend on κ and ρ.

Lemma 4.5.
For each x ≥ 1 and m, n ∈ N such that m ≤ n, it holds that where the constants can depend on κ, ρ and δ.
Proof. The upper bound follows from the first part of Lemma 4.4 and hence it remains to show that For the remainder of the proof, assume that E m (x), E m,n (x) > 0. Let K 1 and K 2 be the closure of the complement of the unbounded connected component of H \ (η([0, τ αm+1 ]) ∪ η xm ([0, σ x m ])) and H\(η xm+1 ([0,τ ])∪η xn ([0, σ x n ])), respectively, whereτ = σ xm+1 (B(x, ε n+1 )). Let K 1,L and K 1,R denote the boundaries of K 1 to left and right of η(τ αm+1 ). Clearly, Arguing as in the proof of Lemma 4.1 (i.e., a Brownian motion must first hit the arc of ∂B(x, ε m+1 ) with endpoints η(τ αm+1 ) and x + ε m+1 , from there, the probabilities of hitting the different sets are proportional), we see that and that and hence we can apply Lemma 2.9 to conclude the result.
Lemma 4.6. For each δ ∈ (0, 1 2 ), sufficiently small, there exist a constant c(δ) > 0 and a subexponential function ψ such that the for each x ≥ 1, Proof. By Lemma 4.4, so we need to show that there exist a constantc(δ) and a subexponential function ψ such that However, (44) follows from the very same argument as Lemma 4.5, so we will now concern ourselves with (43). .
As stated above, η x k is an SLE κ (2+ρ, −2−ρ; ρ) curve with configuration (H, x k , (0, x − k ), x + k , ∞) and by Lemma 2.11, the Radon-Nikodym derivative between the law of η x k and an SLE κ (−2−ρ; ρ) curve with configuration (H, x k , x − k , x + k , ∞), both stopped upon exiting B(x, 1 2 ε k ) is bounded above and below by constants and hence we can (and will) instead consider the latter. Also, we can translate and rescale the process so that we consider an SLE κ (−2 − ρ; ρ) curve,η, started from 0 and the point that we want the curve to get close to being 1. Then, since dist(x k , x) = 1 4 ε k , the event {σ x k < σ x k (H \ B(x, 1 2 ε k ))} turns into the event {η hits B(1, e −α k+1 ) before leaving B(1, 2)} and the event {Q k σ x k,M , Q k σ x k ∈ [δ, 1−δ]} remains roughly the same (it contains in the same event, but with a different δ). We denote by (g t ) the Loewner chain corresponding toη and weigh the probability measure P with the local martingale (recall (8)) and denote the resulting measure by P * . Note that it is under P * that we can choose u such that has probability arbitrarily close to 1 (in the case with no force point to the left). Let for some subexponential function ψ. Furthermore, Q σ k 1 and by Lemma 2.3 δ σ k e −α k+1 , that is, δ −µ(1+ρ/2) σ k e α k+1 µ(1+ρ/2) . Moreover, we see that g σ k (1) − V L σ k 1, since it is the harmonic measure from infinity of the left side ofη, so that it is upper bounded by ω ∞ (B(1, 2), H), which is finite, and lower bounded by ω ∞ ([0, 1/2], H) = 1/2.
we have that ). where F (z) =fσ 1 (z) fσ 1 (1) , , γ(1)) <ε}, and (f t ) and (K t ) are the centered Loewner chain and hulls respectively. Then, the curve η(t) = F (η(σ 1 + t)) has the law of a time-changed SLE κ (−2 − ρ; ρ, −µκ) curve with configuration (H, 0, x L , (x R , 1), ∞), where x L < −2 and x R ∈ [0 + , 1). Note that we may choose γ andε so that 1 − x R >δ for someδ > 0 (to get a bound on the constant C * , chosen as remarked after Lemma 2.4). By Lemma 2.8, the above happens with positive probability, say p 0 . By Lemma 2.11, the Radon-Nikodym derivative between the law of η and the law of a correspondingly time-changed SLE κ (ρ, −µκ) curve with force points (x R , 1), is bounded above and below by some constants. Thus we may consider such an SLE κ (ρ, −µκ) process. Note that, if (ĝ t ) and (g t ) are the Loewner chains ofη and η, respectively, then Thus, we can choose Λ to be sufficiently large, so that is the event of Proposition 3.4 for η. We have lower bounded the probability of {σ 1 ≤σ 2 } and next, we prove that {η hits 1 before ∂B(1, 2)} ∩ I u α k+1 a +C * k (x) (η) ∩ {Q-event} has positive probability, which completes the proof of the Lemma. (Note that we do not need η to hit 1 before exiting B(1, 2), but rather that it hits some small set, separating 1 from ∞, before exiting B (1, 2), which is of course weaker).
We begin by lower bounding P * (η hits 1 before ∂B (1, 2)). We now make a conformal coordinate change with the Möbius transformation [20]). Furthermore, 1 is mapped to ∞ and ϕ(B(1, 2)) = B(−1, 1 2 ) and thus the event of hitting 1 before exiting B(1, 2) turns into hitting the event that ϕ(η) does not hit B(−1, 1 2 ). We have that κ ≤ 4 and ρ L > κ 2 − 2, so the probability of ϕ(η) avoiding B(−1, 1 2 ) is positive. By choosing δ to be sufficiently small and M to be large enough, we can (by Corollary A.2) guarantee that the P * -probability of the event regarding Q is as close to 1 as we want. By then choosing u > 0 sufficiently large, we have that the P * -probability ofĨ u,k k a +C * k (x) is arbitrarily close to 1, and hence that P * (A 1 k ∩Ĩ u,k k a +C * k (x) ) 1, and hence (43) is proven, which gives the result.
Remark 4.7. In the above proof, we actually proved an upper bound as well: Proof of Proposition 4.2. It holds that where we used Lemma 4.4 in the second inequality, Lemma 4.5 in the fourth and Lemma 4.6 and (45) in the fifth.

Dimension spectrum
In this section, we will compute the almost sure Hausdorff dimension of the random sets (4). We will, however, compute the almost sure dimension of the sets and note that this is sufficient, due to the monotonicity of t → g t . The theorem we will prove in this section is the following.
Proof. Let x > 0 and write S x = V * β ∩ (0, x). Since x R = 0 + , the SLE κ (ρ) process is scaling invariant and hence, the law of S x is identical to the law of xS 1 . However, since dim H xS 1 = dim H S 1 (due to the invariance of the Hausdorff dimension under linear scaling), we see that the law of dim H S x is not depending on x. The sets S x are decreasing as x → 0 + , and hence dim H S x has an almost sure limit (as x → 0 + ) which is measurable with respect to F 0 + , since S x is measurable with respect to F Tx . By Blumenthal's 0-1 law, the limit must be constant and the same for every x > 0.
In the following two sections, we prove the upper and lower bounds on the dimension, Theorem 5.3 and Theorem 5.6, and together they imply Theorem 1.1.
Then every interval has length e −n /2. We denote by x j,n the midpoint of the interval J j,n and write J n := {J j,n }. By distortion estimates, we have that there is some constant c > 0, such that if then, We also have that since the curve must hit the ball of radius e −(n−2) , centered at x j,n before it hits the ball of radius e −n , centered at any point x in J j,n , as the former ball contains the latter for any x ∈ J j,n . Combining this with the fact that t → g t (x) is decreasing for every fixed x, (48) and writing c = c e 2β , shows that (47) implies that g τn−2(xj,n) (x j,n ) ≥ ce −β(1+ρ/2)(n−2) .
Proof. By (49), we get using Chebyshev's inequality in the fourth row and Corollary 3.2 in the last.
Next, we construct the cover for V β ∩ [1,2]. Let x ∈ V β ∩ [1, 2] and n > min(0, − log x) be such that g τn (x) ≤ e −β(1+ρ/2)n . By Lemma 2.4, there is a constant C * , such that where s n = n a + C * (here, C * can be chosen so that the above holds for every x ∈ [1, 2]). By the distortion principle, there is a smallest nonnegative integer k such that, if J is the unique interval in J n+k such that x ∈ J, then for every z ∈ J, where the second subscript denotes the point which the time-changet(s) is made with respect to (if no second subscript is written out, the time-change corresponds to the point in which we evaluate the function). Let x J denote the midpoint of J. Theng sn,x (x J ) e −β(1+ρ/2)n . Since dist(x, x J ) ≤ e −(n+k) /4, we have for geometric reasons and by Lemma 2.4, that Therefore, there is a constant c 2 such thatg s n (x J ) ≤ c 2 e −β(1+ρ/2)n . The constants above can be chosen to be universal. Let us recap what we have done above; we concluded that there are universal constants k, c 1 , c 2 such that every x ∈ V β is contained in contained in an interval J in J n+k and g n a +c1 (x J ) ≤ c 2 e −β(1+ρ/2)n , where x J is the midpoint of J. Therefore, choosing universal constants C 1 and C 2 , we have that if where I + n = j ∈ {0, 1, ..., 2e n − 1} :g n a +C1 (x j,n ) ≤ C 2 e −β(1+ρ/2)n , then for every m. We let N n (β) denote the number of intervals J j,n that make up J n,+ (β), i.e., N n (β) = 2e n −1 j=0 1 g n a +C1 (x j,n ) ≤ C 2 e −β(1+ρ/2)n .
Proof. Applying Chebyshev's inequality and Proposition 3.1 in the same way as in the previous lemma gives the result.
With these, we will now prove Theorem 5.3.
Proof of Theorem 5.3. We begin with (i). Lemma 5.4 implies, for s > d * (β) and n sufficiently large, that , is bounded by a constant times the t-dimensional lower Minkowski content of ∪ n≥m J n,− (β), which implies that The same argument shows that V β = ∅ almost surely for β > β * + . What is left, is to use (i) and (iii) to prove that dim . By then letting ε → 0+ and the fact that d * (β) is increasing on [β * − , β * 0 ] gives the upper bound for chosen β. In the same way, letting β ∈ (β * 0 , β * + ] and ε > 0 be such . Again, letting ε → 0+ and noting that d * (β) is decreasing on [β * 0 , β * + ] gives the desired upper bound.

Lower bound
We shall prove the lower bound using Frostman's lemma, that is, we let E s (ν) be the s-dimensional energy of the measure ν, i.e., E s (ν) = dν(x)dν(y) |x − y| s .
We then construct a Frostman measure on V * β and show that it has finite s-dimensional energy for every s < d * = d * (β), which implies that the s-dimensional Hausdorff measure is 0, and thus that the Hausdorff dimension must be greater than or equal to s. Just like in the previous section, we do this for V * β intersected with the interval [1,2], but again it will be clear that this can be done for any closed interval to the right of 0. In the following, we will construct a family of Frostman measures and show that it gives the correct lower bound on the dimension of V * β . Theorem 5.6. Let κ ∈ (0, 4], ρ ∈ (−2, κ 2 − 2) and x R = 0 + . Then, for every ς > 0, Proof. Fix κ ∈ (0, 4), ρ ∈ (−2, κ 2 −2) and δ ∈ (0, 1 2 ) small and u, Λ, M > 0 large enough for Proposition 4.2 to hold. We fix n ∈ N, divide [1,2] into ε −1 n intervals of length ε n , and let x j,n = 1+(j − 1 2 )ε n be the midpoint of the jth of these intervals. Let D n = {x j,n : j = 1, ..., ε −1 n }, let C n = {x ∈ D n : E n (x) = 1} and let J n (x) = [x − εn 2 , x + εn 2 ]. Then, We define the measure ν n by for Borel sets A ⊂ [1,2]. We want to take a subsequential limit of the sequence of measures (ν n ), which we will prove converges to the Frostman measure on V * β . To see that this limit exists, we need that the event on which we want to take the subsequential limit has positive probability and that the support of the limit is contained in V * β . That the support of the limit is contained in V * β is obvious by construction, so we turn to proving that the event has positive probability. , and we will bound the diagonal and the off-diagonal parts separately below. For the diagonal terms, we have (since E n (x) 2 ≤ E n (x)) for large enough α 0 and n, since ψ is a subpower function and o n (1) tends to 0 as n → ∞. For the off-diagonal terms, we have, again for large enough α 0 and n, by Proposition 4.2, j =k Ψ δ |x j,n − x k,n | −1+o |x j,n −x k,n | (1) |x j,n − x k,n | d * −1+o |x j,n −x k,n | (1)

1.
The implicit constants do not depend on n, and hence we have that for every ς > 0, the limiting measure ν satisfies E d * −ς (ν) < ∞ on an event of positive probability. By Lemma 5.2, the Hausdorff dimension is almost surely constant, so a lower bound with positive probability is an almost sure lower bound. Thus, the proof is done. Now Theorem 5.1 follows from Theorem 5.3 and Theorem 5.6 and thus, as remarked, Theorem 1.1 is proven. Hence p n (t, x) = P ( δ + 2 −1, δ − 2 −1) n (x) exp − n 2 (n + δ + 2 + δ − 2 − 1)t solves (53) for t > 0 and −1 < x < 1. Thus, the candidate for the transition density of Y is where f 2 δ+,δ− = f, f δ+,δ− . First, we need that (54) is absolutely convergent for t > 0. This holds, since Next, we shall check that It is sufficient to show this for polynomials. Let q(x) be a polynomial, then a n = q, P Next, lettingq (t, x) = ∞ n=0 a n p n (t, x), we note thatq(t, x) solves (53) and thatq(0, x) = q(x). We fix t 0 > 0 and let M t =q(t 0 − t, Y t ) for 0 < t ≤ t 0 . Then M t is bounded and by Itô's formula, (where denotes the spatial derivative) and hence M t is a bounded martingale. Furthermore, we have that lim t→t0 M t = q(Y t0 ), so by the optional stopping theorem, E y0 [q(Y t0 )] = M 0 =q(t 0 , y 0 ), that is, q(y)p Y (t 0 , y 0 , y)dy.
Thus, p Y (t, x, y) is the transition density of Y and sending t to infinity, we see that Y has a unique stationary distribution with density given by that is, the n = 0 term in (54), we therefore see that there is a constant C, such that Thus, p Y (t, x, y) → p Y (y) uniformly in x, y ∈ [−1, 1] as t → ∞ and furthermore, if we denote by Y the process following the SDE (52), started from the invariant density (56), then We have thus proven the following.
Lemma A.1. The transition density, p Y (t, x, y) for Y is given by (54). Furthermore, Y has a unique invariant density, p Y (y), given by (56), (58) holds and Y is ergodic.
As a corollary, we have the following (noting that δ++δ− 4 = 1 − a + µ and letting E * denote the expectation under P * , as in Sections 2 and 3).