Intersections of SLE Paths: the double and cut point dimension of SLE

We compute the almost-sure Hausdorff dimension of the double points of chordal SLEκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SLE}_\kappa $$\end{document} for κ>4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa > 4$$\end{document}, confirming a prediction of Duplantier–Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLEκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SLE}_\kappa $$\end{document} for κ>4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa > 4$$\end{document} as well as analogous dimensions for the radial and whole-plane SLEκ(ρ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SLE}_\kappa (\rho )$$\end{document} processes for κ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa > 0$$\end{document}. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field eih/χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{ih/\chi }$$\end{document}, where h is a Gaussian free field and χ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi > 0$$\end{document}, of different angles with each other and with the domain boundary.


Overview
The Schramm-Loewner evolution SLE κ (κ > 0) is the canonical model for a conformally invariant probability measure on non-crossing, continuous paths in a proper simply connected domain D in C. SLE κ was introduced by Oded Schramm  [32] as the candidate for the scaling limit of loop-erased random walk and for the interfaces in critical percolation. Since its introduction, SLE has been proved to describe the limiting interfaces in many different models from statistical mechanics [4][5][6]11,21,23,38,38,39]. The purpose of this article is to study self-intersections of SLE paths as well as the intersection of multiple SLE paths when coupled together using the Gaussian free field (GFF). Our main results are Theorems 1.1-1.6 which give the dimension of the self-intersection and cut points of chordal, radial, and wholeplane SLE κ and SLE κ (ρ) processes as well as the dimension of the intersection of such paths with the domain boundary. Theorems 1.1-1.4 are actually derived from Theorem 1.5 which gives the dimension of the intersection of two SLE κ (ρ) processes coupled together as flow lines of a GFF [8,10,12,[25][26][27][28][29]34,36,40] with different angles.
Our next result gives the dimension of the self-intersection points of the radial and whole-plane SLE κ (ρ) processes for κ ∈ (0, 4). Unlike chordal SLE κ and SLE κ (ρ) processes, such processes can intersect themselves depending on the value of ρ > −2. The maximum number of times that such a process can hit any given point for κ > 0 is given by [29,Proposition 3.31]: . (1.3) In particular, J κ,ρ ↑ +∞ as ρ ↓ −2 and J κ,ρ ↓ 1 as ρ ↑ κ 2 − 2. Recall that −2 is the lower threshold for an SLE κ (ρ) process to be defined. For radial or whole-plane SLE κ (ρ), the interval of ρ values in which such a process is self-intersecting is given by (−2, κ 2 − 2) (see, e.g., [29, Section 2.1]) (For chordal SLE κ (ρ), this is the interval of ρ values in which such a process is boundary intersecting). For ρ ≥ κ 2 − 2, such processes are almost surely simple.
Note that J κ,ρ + 1 is the value of j that makes the right side of (1.4) equal to zero. Similarly, J κ,ρ is the value of j that makes the right side of (1.5) equal to zero. Inserting j = 1 into (1.4) we recover the dimension formula for the range of an SLE κ process [3] (though we do not give an alternative proof of this result). We next state the corresponding result for whole-plane and radial SLE κ (ρ) processes with κ > 4. Such a process has two types of self-intersection points. Those which arise when the path wraps around its target point and intersects itself in either its left or right boundary (which are defined by lifting the path to the universal cover of the domain minus the target point of the path) and those which occur between the left and right boundaries. It is explained in [29,Section 4.2] that these two self-intersection sets are almost surely disjoint and the dimension of the latter is almost surely given by the corresponding dimension for chordal SLE κ (Theorem 1.1). In fact, the set which consists of the multiple intersection points of the path where the path hits itself without wrapping around its target point and are also contained in its left and right boundaries is almost surely countable. The following gives the dimension of the former: Theorem 1. 4 Suppose that η is a radial SLE κ (ρ) process in D for κ > 4 and ρ ∈ κ 2 − 4, κ 2 − 2). Assume that η starts from 1 and has a single boundary force point of weight ρ located at 1 − (immediately to the left of 1 on ∂D). For each j ∈ N, let I j denote the set of points that η hits exactly j times and which are also contained in its left and right boundaries. For each 2 ≤ j ≤ J κ ,ρ where J κ ,ρ is given by (1.3), we have that almost surely. For j > J κ ,ρ , almost surely I j = ∅. These results similarly hold if η is a whole-plane SLE κ (ρ) process. Similarly, let L j (resp. R j ) be the set of points on ∂D which η hits exactly j times while traveling in the clockwise (resp. counterclockwise) direction. Then almost surely on {L j = ∅}. (1.7) and almost surely on {R j = ∅}. (1.8) The reason that we restrict to the case that ρ > κ 2 − 4 is that for ρ ≤ κ 2 − 4 such processes almost surely fill their own outer boundary. That is, for any time t, the outer boundary of the range of the path drawn up to time t is almost surely contained in η ( [t, ∞]) and processes of this type fall outside of the framework described in [29].
The proofs of Theorems 1.1 and 1.2 are based on using various forms of SLE duality which arises in the interpretation of the SLE κ and SLE κ (ρ) processes for κ ∈ (0, 4) as flow lines of the vector field e ih/χ where h is a GFF and χ = 2 √ κ − √ κ 2 [7,8,26,28,29]. We will refer to these paths simply as "GFF flow lines". The flow line with angle θ ∈ R is the flow line with angle 0 of the field h + θχ. And the difference between the angles of two flow lines is called the angle gap (An overview of this theory is provided in Sect. 2.2). The duality statement which is relevant for the cut-set (see Fig. 5) is that the left (resp. right) boundary of an SLE κ process is given by an SLE κ flow line of a GFF with angle π 2 (resp. − π 2 ). Thus the cut set dimension is given by the dimension of the intersection of two flow lines with an angle gap of θ cut = π. (1.9) Another form of duality which describes the boundary of an SLE κ process before and after hitting a given boundary point and also arises in the GFF framework allows us to relate the double point dimension to the dimension of the intersection of GFF flow lines with an angle gap of [28] (1.10) We will explain this in more detail in Sect. 5. The set of points which a whole-plane or radial SLE κ (ρ) process for κ ∈ (0, 4) and ρ ∈ (−2, κ 2 − 2) hits j times (in the interior of the domain) is locally absolutely continuous with respect to the intersection of two flow lines with an angle gap of see [29,Proposition 3.32]. The angle gap which gives the dimension of the selfintersection set contained in the interior of the domain for κ > 4 and ρ ∈ ( κ 2 − 4, κ 2 − 2) is given by

Theorem 1.5 Suppose that h is a GFF on H with piecewise constant boundary data.
Fix κ ∈ (0, 4), angles and let For i = 1, 2, let η θ i be the flow line of h starting from 0 with angle θ i . We have that almost surely on the event {η θ 1 ∩ η θ 2 ∩ H = ∅}.
Theorem 1.5 gives the dimension of the intersection of two flow lines in the bulk. The following result gives the dimension of the intersection of one path with the boundary.
. Let η be an SLE κ (ρ) process with a single force point located at 0 + . Almost surely, (1.13) (Recall that κ 2 − 4 is the threshold at which such processes become boundary filling and −2 is the threshold for these processes to be defined). In the case that ρ = θ π (2 − κ 2 ) − 2 for θ > 0 and κ ∈ (0, 4), we say that η intersects ∂H with an angle gap of θ . This comes from the interpretation of such an SLE κ (ρ) process as a GFF flow line explained in Sect. 2.2. See, in particular, Fig. 4. By [29,Proposition 3.33], applying Theorem 1.6 with an angle gap of θ j+1 where θ j is as in (1.11) gives (1.5) of Theorem 1.3. Similarly, by [29,Proposition 4.11], applying Theorem 1.6 with an angle gap of gives (1.7) and with an angle gap of gives (1.8). Theorem 1.6 is proved first by computing the boundary intersection dimension for κ ∈ (0, 4) and then using SLE duality to extend to the case that κ > 4. We obtain as a corollary (when ρ = 0) the following which was first proved in [1]. 4,8) and let η be an SLE κ process in H from 0 to ∞. Then, almost surely One of the main inputs in the proof of Theorems 1.5 and 1.6 is the following theorem, which gives the exponent for the probability that an SLE κ (ρ) process gets very close to a given boundary point.
Let η be an SLE κ (ρ 1,R , ρ 2,R ) process with force points (0 + , 1). Let and ρ 2,R = 0, Theorem 1.8 gives the exponent for the probability that an SLE κ (ρ) process gets close to a fixed point on the boundary. Theorem 1.8 is proved (in somewhat more generality) in Sect. 3.1 and it implies the upper bound of the Hausdorff dimension in Theorem 1.6. We remark that, all results in Sect. 3.1 are independent of Imaginary Geometry. In other words, the upper bound of the dimension in Theorem 1.6 is derived without knowledge of Imaginary Geometry. Whereas, the lower bound, derived in Sect. 3.2, uses the interaction behavior of flow lines. An alternative proof to the lower bound of Theorem 1.6 for κ ∈ (8/3, 4] is given in [44] using the relationship between the SLE κ (ρ) processes for these κ values and the Brownian loop soups.

Outline
The remainder of this article is structured as follows. In Sect. 2, we will review the definition and important properties of the SLE κ and SLE κ (ρ) processes. We will also describe the coupling between SLE and the Gaussian free field. Next, in Sect. 3, we will compute the Hausdorff dimension of SLE κ (ρ) intersected with the boundary. We will extend this to compute the dimension of the intersection of two GFF flow lines in Sect. 4. Note that, Sect. 3 proves Theorem 1.6 and Sect. 4 proves Theorem 1.5, and they imply Theorems 1.2-1.4. Finally, in Sect. 5 we will explain the SLE duality describing the boundary of SLE κ process before and after hitting a given boundary point and complete the proof of Theorem 1.1 from the SLE duality and Theorem 1.5.

Preliminaries
We will give an overview of the SLE κ and SLE κ (ρ) processes in Sect. 2.1. Next, in Sect. 2.2, we will give an overview of the SLE/GFF coupling and then use the coupling to establish several useful lemmas regarding the behavior of the SLE κ and SLE κ (ρ) processes. In Sect. 2.3, we will compute the Radon-Nikodym derivative associated with a change of domains and perturbation of force points for an SLE κ (ρ) process. Finally, in Sect. 2.4 we will record some useful estimates for conformal maps. Throughout, we will make use of the following notation. Suppose that f, g are functions. We will write f g if there exists a constant C ≥ 1 such that We will write f g if there exists a constant C > 0 such that f (x) ≤ Cg(x) and f g if g f .

SLE κ and SLE κ (ρ) processes
We will now give a very brief introduction to SLE. More detailed introductions can be found in many excellent surveys of the subject, e.g., [16,43]. Chordal SLE κ in H from 0 to ∞ is defined by the random family of conformal maps (g t ) obtained by solving the Loewner ODE Then g t is the unique conformal map from H t := H\K t to H satisfying lim |z|→∞ |g t (z) − z| = 0.
Rohde and Schramm showed that there almost surely exists a curve η (the so-called SLE trace) such that for each t ≥ 0 the domain H t of g t is the unbounded connected component of H\η([0, t]), in which case the (necessarily simply connected and closed) set K t is called the "filling" of η([0, t]) [31]. An SLE κ connecting boundary points x and y of an arbitrary simply connected Jordan domain can be constructed as the image of an SLE κ on H under a conformal transformation ϕ : H → D sending 0 to x and ∞ to y (The choice of ϕ does not affect the law of this image path, since the law of SLE κ on H is scale invariant). For κ ∈ [0, 4], SLE κ is simple and, for κ > 4, SLE κ is self-intersecting [31]. The dimension of the path is 1 + κ 8 for κ ∈ [0, 8] and 2 for κ > 8 [3].
An SLE κ (ρ L ; ρ R ) process is a generalization of SLE κ in which one keeps track of additional marked points which are called force points. These processes were first introduced in [20,Section 8.3]. Fix x L = (x ,L < · · · < x 1,L ≤ 0) and x R = (0 ≤ x 1,R < · · · < x r,R ). We associate with each x i,q for q ∈ {L , R} a weight ρ i,q ∈ R. An SLE κ (ρ L ; ρ R ) process with force points (x L ; x R ) is the measure on continuously growing compact hulls K t generated by the Loewner chain with W t replaced by the solution to the system of SDEs: It is explained in [26,Section 2] that for all κ > 0, there is a unique solution to (2.2) up until the continuation threshold is hit -the first time t for which either The almost sure continuity of the SLE κ (ρ) processes is proved in [26,Theorem1.3]. Let with the convention that ρ 0,L = ρ 0,R = 0, x 0,L = 0 − , x +1,L = −∞, x 0,R = 0 + , and x r +1,R = +∞. The value of ρ k,R determines how the process interacts with the interval (x k,R , x k+1,R ) (and likewise when R is replaced with L). In particular: In this article, it will also be important for us to consider radial SLE κ and SLE κ (ρ) processes. These are typically defined using the radial Loewner equation. On the unit disk D, this is described by the ODE where W t is a continuous function which takes values in ∂D. For w ∈ ∂D, radial SLE κ starting from w is the growth process associated with (2.4) where W t = we i √ κ B t and B is a standard Brownian motion. For w, v ∈ ∂D, radial SLE κ (ρ) with starting configuration (w, v) is the growth process associated with the solution of (2.4) where the driving function solves the SDE , the force point. The continuity of the radial SLE κ (ρ) processes for ρ > −2 can be extracted from the continuity of chordal SLE κ (ρ) processes given in [26,Theorem1.3]; this is explained in [29, Section 2.1]. The value of ρ for a radial SLE κ (ρ) process has the same interpretation as in the setting of chordal SLE κ (ρ) explained in Lemma 2.1. That is, the processes are boundary filling for ρ ∈ (−2, κ 2 −4] (for κ > 4), boundary hitting but not filling for ρ ∈ ((−2) ∨ ( κ 2 − 4), κ 2 − 2), and boundary avoiding for ρ ≥ κ 2 − 2. In particular, by the conformal Markov property for radial SLE κ (ρ), such processes are self-intersecting for ρ ∈ (−2, κ 2 − 2) and fill their own outer boundary for ρ ∈ (−2, κ

Martingales
From the form of (2.2) and the Girsanov theorem, it follows that the law of an SLE κ (ρ) process can be constructed by reweighting the law of an ordinary SLE κ process by a certain local martingale, at least until the first time τ that W hits one of the force points V i,q [42]. It is shown in [41,Theorem 6,Remark 7] that this local martingale can be expressed in the following more convenient form. Suppose x 1,L < 0 < x 1,R and define Then M t is a local martingale and the law of a standard SLE κ process weighted by M (up to time τ , as above) is equal to that of an SLE κ (ρ L ; ρ R ) process with force points (x L ; x R ). We remark that there is an analogous martingale in the setting of radial SLE κ (ρ) processes [41,Equation 9], a special case of which we will describe and make use of in Sect. 4. One application of this that will be important for us is as follows. Suppose that η is an SLE κ (ρ L ; ρ R ) process with only two force points x L < 0 < x R . If we weight the law of η by the local martingale then the law of the resulting process is that of an SLE κ ( ρ L ; ρ R ) process where ρ L = κ − 4 − ρ L . If ρ L < κ 2 − 2 so that ρ L > κ 2 − 2, Lemma 2.1 implies that the reweighted process almost surely does not hit (−∞, x L ).

SLE and the GFF
We are now going to give a brief overview of the coupling between SLE and the GFF. We refer the reader to [26,Sections 1,2] as well as [27, Section 2] for a more detailed overview. Throughout, we fix κ ∈ (0, 4) and κ = 16/κ > 4.
Suppose that D ⊆ C is a given domain. The Sobolev space H 1 0 (D) is the Hilbert space closure of C ∞ 0 (D) with respect to the Dirichlet inner product The zero-boundary Gaussian free field (GFF) h on D is given by where (α n ) is a sequence of i.i.d. N (0, 1) random variables and ( f n ) is an orthonormal basis for H 1 0 (D). The sum (2.9) does not converge in H 1 0 (D) (or any space of functions) but rather in an appropriate space of distributions. The GFF h with boundary data f is given by taking the sum of the zero-boundary GFF on D and the function F in D which is harmonic and is equal to f on ∂ D. See [35] for a detailed introduction. Let In this coupling, η is almost surely determined by h [8,26,40]. For κ ∈ (0, 4), η has the interpretation as being the flow line of the (formal) vector field e i(h+φ 0 )/χ [36] starting from 0; we will refer to η simply as a flow line of h + φ 0 . See Fig. 1 for an illustration of the boundary data. The notation x ∼ is used to indicate that the boundary data for the field is given by x + χ · winding where "winding" refers to the winding of the path or domain boundary. For curves or domain boundaries which are not smooth, Suppose that h is a GFF on H whose boundary data is as indicated above. Then the flow line η of h starting from 0 is an SLE κ (ρ 2,L , ρ 1,L ; ρ 1,R , ρ 2,R ) process (κ ∈ (0, 4)) from 0 to ∞ with force points located at x 2,L < x 1,L < 0 < x 1,R < x 2,R . The conditional law of h given η (or η up to a stopping time) is that of a GFF off of η with the boundary data as illustrated on η; the notation x ∼ is shorthand for x + χ · winding and is explained in detail in [26,Figures 1.9,1.10]. The boundary data for the coupling of SLE κ (ρ) with many force points arises as the obvious generalization of the above Suppose that h is a GFF on H whose boundary data is as indicated above. Then the counterflow line η of h starting from 0 is an SLE κ (ρ 2,L , ρ 1,L ; ρ 1,R , ρ 2,R ) process (κ > 4) from 0 to ∞ with force points located at x 2,L < x 1,L < 0 < x 1,R < x 2,R . The conditional law of h given η (or η up to a stopping time) is that of a GFF off of η with the indicated boundary data; the notation x ∼ is shorthand for x + χ · winding and is explained in detail in [26,Figures 1.9,1.10]. The boundary data for the coupling of SLE κ (ρ ) with many force points arises as the obvious generalization of the above it is not possible to make sense of the winding along the curve or domain boundary. However, the harmonic extension of the winding does make sense. This notation as well as this point are explained in detail in [26,Figures 1.9,1.10]. When κ = 4, η has the interpretation of being the level line of h + φ 0 [40]. Finally, when κ > 4, η has the interpretation of being a "tree of flow lines" which travel in the opposite direction of η [26,29]. For this reason, η is referred to as a counterflow line of h + φ 0 in this case.
If h were a smooth function, η a flow line of the vector field e ih/χ , and ϕ a conformal map, then ϕ(η) is a flow line of e i h/χ where (2.11) see [26, Figure 1.6]. The same is true when h is a GFF and this formula determines the boundary data for coupling the GFF with an SLE κ (ρ L ; ρ R ) process on a domain other than H. See also [26, Figure 1.9]. SLE κ flow lines and SLE κ , κ = 16/κ ∈ (4, ∞), counterflow lines can be coupled with the same GFF. In order for both paths to transform in the correct way under the application of a conformal map, one thinks of the flow lines as being coupled with h as described above and the counterflow lines as being coupled with −h. This is because χ(κ ) = −χ(κ); see the discussion after the statement of [26,Theorem 1.1]. This is why the signs of the boundary data in Fig. 2 are reversed in comparison to that in Fig. 1. The theory of how the flow lines, level lines, and counterflow lines of the GFF interact with each other and the domain boundary is developed in [26,29]. See, in particular, [26,Theorem 1.5]. The important facts for this article are as follows. Suppose that h is a GFF on H with piecewise constant boundary data. For each θ ∈ R and x ∈ ∂H, let η x θ be the flow line of h starting at x with angle θ (i.e., the flow line of h + θχ starting at x). If θ 1 < θ 2 and x 1 ≥ x 2 then η x 1 θ 1 almost surely stays to the right of η x 2 θ 2 . If θ 1 = θ 2 , then η x 1 θ 1 may intersect η x 2 θ 2 and, upon intersecting, the two flow lines merge and never separate thereafter. See Fig. 3. Finally, if θ 2 + π > θ 1 > θ 2 , then η x 1 θ 1 may intersect η x 2 θ 2 and, upon intersecting, crosses and possibly subsequently bounces Suppose that h is a GFF on H with piecewise constant boundary data and x 1 , Fix angles θ 1 , θ 2 and, for i = 1, 2, let η then η almost surely stays to the left of (but may bounce off of) η θ upon intersecting after which the paths never separate given η . Let ϕ be a conformal map which takes the unbounded connected component of H\η ) is the flow line of h 2 starting from ϕ(x 1 ) with angle θ 1 and we can read off its conditional law from the boundary data of h 2 as in Fig. 1 off of η x 2 θ 2 but never crosses back. It is possible to compute the conditional law of one flow line given the realization of several others; see Fig. 4. For simplicity, we use η θ to indicate η x θ when x = 0. If η is a counterflow line coupled with the GFF, then its outer boundary is described in terms of a pair of flow lines starting from the terminal point of η [7,8,26,29]; see Fig. 5.
We are now going to use the SLE/GFF coupling to collect several useful lemmas regarding the behavior of SLE κ (ρ) processes.

Lemma 2.2
Fix κ > 0. Suppose that (x n,L ) (resp. (x n,R )) is a sequence of negative (resp. positive) real numbers converging to x L ≤ 0 − (resp. x R ≥ 0 + ) as n → ∞. For each n, suppose that (W n , V n,L , V n,R ) is the driving triple for an SLE κ (ρ L ; ρ R ) process in H with force points located at (x n,L ≤ 0 ≤ x n,R ). Then (W n,L , V n,L , V n,R ) converges weakly in law with respect to the local uniform topology to the driving triple The same holds if [−1, 1] 2 is replaced by a proper, simply-connected domain and the boundary data of the GFF is transformed according to (2.11). Finally, if ρ L , ρ R ≥ κ 2 − 4, then conditional law of η given η L and η R is independently that of an SLE κ ( κ 2 − 4; κ 2 − 4) in each of the bubbles of [−1, 1] 2 \(η L ∪ η R ) which lie to the right of η L and to the left of η R Then Proof See Fig. 6 for an illustration. We will use the terminology "flow line", but the proof holds for κ > 0. By running η for a very small amount of time and using that = 0 for all t > 0 before the continuation threshold is reached [26, Section 2] and then conformally mapping back, we may assume without loss of generality that ρ 1, , which contains 0. Suppose κ ∈ (0, 4) and let h be a GFF on H whose boundary data has been chosen so that its flow line η from 0 is an SLE κ (ρ L ; ρ R ) process as in the statement of the lemma. Pick a point x 0 ∈ ∂U with |γ (T ) − x 0 | ≤ . Let h be a GFF on U whose boundary conditions are chosen so that its flow line η starting from 0 is an SLE κ process from 0 to x 0 . Let Then the laws of h| U δ and h| U δ are mutually absolutely continuous [26,Proposition 3.2]. Thus the result follows since we can pick δ > 0 sufficiently small so that P[ X > δ] > 0. This proves the result for κ ∈ (0, 4). For κ > 4, one chooses the boundary data for h so that the counterflow line is an SLE κ ( κ 2 − 2; κ 2 − 2) process (recall Lemma 2.1).
Proof We know that this event has positive probability for each fixed choice of x L , x R as above by Lemma 2.3. Therefore the result follows from Lemma 2.2 and the results of [16,Section 4.7].
Proof See Fig. 7 for an illustration. We will use the terminology "flow line", but the proof holds for κ > 0. Arguing as in the proof of Lemma 2.3, we may assume without loss of generality that ρ 1,L = ρ 1,R = 0. Let U be a Jordan domain which contains γ and is contained in A( ). Assume, moreover, that ∂U ∩ [x 2,L , x 2,R ] is an interval which contains 0 and ∂U ∩ [x k,R , x k+1,R ] is also an interval, say [y L , y R ]. Suppose κ ∈ (0, 4). Let h be a GFF on H whose boundary data has been chosen so that its flow line η from 0 is an SLE κ (ρ L ; ρ R ) process as in the statement of the lemma. Let h be a GFF on U whose boundary conditions are chosen so that its flow line η starting from 0 and targeted at y R is an SLE κ (ρ) process with a single force point located at y L with ρ as in the statement of the lemma. Let σ 1 be the first time that η hits [y L , y R ]. Since almost surely. Since η almost surely hits [y L , y R ], the assertion follows using the same absolute continuity argument for GFFs as in the proof of Lemma 2.3. As in the proof of Lemma 2.3, one proves the result for κ > 4 by taking the boundary conditions for h on U so that the counterflow line starting from 0 is an SLE κ ( κ and define stopping times Then we have that Proof See Fig. 8. Lemma 2.5 implies that this event has probability strictly smaller than 1 for each fixed choice of x L , x R as above. Therefore the result follows from Lemma 2.2.

Radon-Nikodym derivative
Following [7, Lemma 13], we will now describe the Radon-Nikodym derivative between SLE κ (ρ) processes arising from a change of domains and the locations and weights of the force points.
and which takes the force points of η to those of η.
c denote the law of an SLE κ (ρ L ; ρ R ) process in c stopped at the first time τ that it exits U and define μ U c analogously. Let where H D is the Poisson excursion kernel of the domain D. We also let where K τ is the compact hull associated with η([0, τ ]) and μ loop the Brownian loop measure on unrooted loops in C (see [22] for more on the Brownian loop measure).
The following result is proved in [7,Lemma 13] in the case that U is at a positive distance from the marked points of c, c other than z 0 . We are now going to use the SLE/GFF coupling described in the previous section to extend the result to the case that U is at a positive distance from the marked points of c, c which are different.

Lemma 2.7
Assume that we have the setup described just above. Suppose that U is at a positive distance from those marked points of c, c which differ. The probability measures μ U c and μ U c are mutually absolutely continuous and Proof We are first going to prove the result in the case that is the flow line of h (resp. h) starting from z 0 . By our hypotheses, the boundary data of h and h agree with each other in the boundary segments which are also contained in ∂U . Consequently, the laws of h| U and h| U are mutually absolutely continuous [26,Proposition 3.2]. Since η (resp. η) is almost surely determined by h (resp. h) [26,Theorem 1.2], it follows that μ U c and μ U c are mutually absolutely continuous. Thus, to complete the proof, we just need to identify . By [7,Lemma13], we know that f (η) is equal to the right side of (2.14) for paths η which intersect the boundary only in the counterclockwise segment of ∂ D from x 1,L to x 1,R (and this only happens for κ > 4). Therefore, to complete the proof, we need to show that the same equality holds for paths η which intersect the other parts of the domain boundary. Note that the right hand side of (2.14) is a continuous function of η with respect to the uniform topology on paths. Therefore, to complete the proof, it suffices to show that the Radon-Nikodym derivative f (η) is also continuous with respect to the same topology. Indeed, then the result follows since both functions are continuous and agree with each other on a dense set of paths. We are going to prove that this is the case using that η, η are coupled with h, h, respectively.
Let ν U c (resp. ν U c ) denote the joint law of (η, h| U ) (resp. ( η, h| U )). As explained above, ν U c and ν U c are mutually absolutely continuous. Moreover, the Radon- is continuous in η with respect to the uniform topology on continuous paths.
Rearranging, we see that (the right side does not depend on the choice of · since the left side does not depend on ·). This implies the desired result in the case that x 1,L = z 0 = x 1,R since the latter factor on the right side is continuous in η, as we remarked above. The result follows in the case that one or both of x 1,L , x 1,R agrees with z 0 since the laws converge as one or both of x 1,L , x 1,R converge to z 0 (Lemma 2.2).
where X is one of H, H τ , D, D τ and u, v are two marked points on the boundary of X . We will complete the proof by considering several cases depending on the location of the marked points.
Case 1 At least one marked point is outside of U ζ . This is the case handled in the proof of [7, Lemma 14].
Case 2 Both marked points u, v are contained in U and u = v. It is enough to bound from above and below the ratios: We can bound A as follows. Let ϕ : D → H be the unique conformal transformation with ϕ(x) = x, ϕ(y) = y, and ϕ (x) = 1. Then A = |ϕ (y)| which, by [20,Proposition 4.1], is equal to the mass of those Brownian excursions in H connecting x and y which avoid H\ D. We will write q(H, x, y, H\ D) for this quantity. Since this is given by a probability, we have that |ϕ (y)| ≤ 1 and it follows that |ϕ (y)| is bounded from below by q(H, x, y, U ζ ) > 0. This lower bound is a positive continuous function in x, y ∈ ∂U ∩ ∂H hence yields a uniform lower bound. Consequently, A is bounded from both above and below.
Similarly, B is equal to the mass q(H\K τ , x τ , y τ , H\ D) of those Brownian excursions in H\K τ which connect x τ and y τ and avoid H\ D. As before, this quantity is bounded from above by 1. We will now establish the lower bound. Let g be the conformal map from H\K τ onto H which sends the triple (x τ , y τ , ∞) to (0, 1, ∞). Note that g can be extended to C\(K τ ∪K τ ) by Schwarz reflection wherē K τ = {z ∈ C :z ∈ K τ }. We will view g as such an extension. Then it is clear that Note that q(H, 0, 1, H\g(U ζ )) is a continuous functional on compact hulls K inside U equipped with the Hausdorff metric. Indeed, suppose that (K n ) is a sequence of compact hulls inside U converging towards K in the Hausdorff metric and, for each n, let g n be the corresponding conformal map. Then g n converges to g uniformly away from K ∪K . In particular, g n (U ζ ) converges to g(U ζ ) in Hausdorff metric. Let φ n (resp. φ) be the conformal map from H\g n (U ζ ) (resp. H\g(U ζ )) onto H which fixes 0, 1 and has derivative 1 at 1. Then φ n (0) converges to φ (0). Thus q(H, 0, 1, H\g n (U ζ )) = φ n (0) converges to q(H, 0, 1, H\g(U ζ )) = φ (0) which explains the continuity of q(H, 0, 1, H\g(U ζ )) in K . Since the set of compact hulls inside U endowed with Hausdorff metric is compact, there exists q 0 > 0 depending only on U and ζ such that Case 3 A single marked point u contained in U . The ratios which involve terms of the form H X (u, u) are interpretted using limits hence are uniformly bounded by the argument of Case 2.

Estimates for conformal maps
For a proper simply connected domain D and w ∈ D, let CR(w; D) denote the conformal radius of D with respect to w, i.e., CR(w; ⊇ D} denote the out-radius of D with respect to w. By the Schwarz lemma and the Koebe one-quarter theorem, where the right-hand inequality above holds for |ζ | ≤ 1/2. Finally, we state the Beurling estimate [16,Theorem 3.76] which we will frequently use in conjunction with the conformal invariance of Brownian motion.

Theorem 2.9 (Beurling estimate) Suppose that B is a Brownian motion in C and
There exists a constant c < ∞ such that if γ : [0, 1] → C is a curve with γ (0) = 0 and |γ (1)| = 1, z ∈ D, and P z is the law of B when started at z, then

The intersection of SLE κ (ρ) with the boundary 3.1 The upper bound
The main result of this section is the following theorem, which in turn implies Theorem 1.8.
We have that The o(1) in the exponent of (3.3) tends to 0 as → 0 and depends only on κ, δ, x R , and the weights ρ 1,R , ρ 2,R . The o(1), however, is uniform in r ≥ 2. Taking Thus Theorem 3.1 leads to the upper bound of Theorem 1.6. We begin with the following lemma which contains the same statement as Theorem 3.1 except is restricted to the case that δ ∈ (0, 1) and, in particular, is not applicable for δ = 0.

Lemma 3.2
Assume that we have the same setup and notation as in Theorem 3.1.
Then for each δ ∈ (0, 1) and r ≥ 2 fixed, we have that where the constants in depend only on κ, δ, x R , and the weights ρ 1,R , ρ 2,R .
Proof For η, the SLE κ (ρ 1,R , ρ 2,R ) process with force points (x R , 1), let (g t ) be the associated Loewner evolution and let V R t denote the evolution of x R . From (2.6) we know that is a local martingale and the law of η reweighted by M is that of an SLE κ ( It is clear that this Brownian motion has uniformly positive (though δ-dependent) probability to exit H\K through each of the left side of K , the right side of K , the interval [x R , 1], and the interval (1, ∞). Consequently, by the conformal invariance of Brownian motion, These facts imply that M τ −α on E δ,r where the constants in depend only on κ, δ, x R , and the weights ρ 1,R , ρ 2,R . Thus where P is the law of η weighted by the martingale M. As we remarked earlier, P is the law of an SLE κ (ρ 1,R , ρ 2,R ) with force points (x R , 1). We now perform a coordinate change using the Möbius transformation ϕ(z) = z/(1 − z). Then the law of the image of a path distributed according to P under ϕ is equal to that of an SLE κ (2 + ρ 1,R + ρ 2,R ; ρ 1,R ) process in H from 0 to ∞ with force points (− ; x R /(1 − x R )) (see Fig. 9). Note that 2 + ρ 1,R + ρ 2,R ≥ κ 2 − 2 by the hypotheses of the lemma. Let η be an SLE κ (2 + ρ 1,R + ρ 2,R ; ρ 1,R ) process in H from 0 to ∞ with force points (− ; x R /(1 − x R )). In particular, by Lemma 2.1, η almost surely does not hit (−∞, − ). Under the coordinate change, the event E δ,r becomes {σ 1, < ξ ,r , Im(η (σ 1, )) ≥ δ} where σ 1, is the first time that η hits ∂ B(− , 1), ξ ,r is the first time that η hits ∂ B(− r 2 /(r 2 − 1), r/(r 2 − 1)). By Lemma 2.4, the probability of the event {σ 1, < ξ ,r , Im(η (σ 1, )) ≥ δ} is bounded from below by a positive constant depending only on κ, δ, ρ 1,R , and ρ 2,R . Thus P [E δ,r ] 1 which implies P[E δ,r ] α and the constants in depend only on κ, δ, x R , and the weights ρ 1,R , ρ 2,R .
where the constants in depend only on κ, δ, r, x L , x R , and the weights ρ L , ρ 1,R , ρ 2,R .
Proof Let (g t ) be the Loewner evolution associated with η and let V L t , V R t denote the evolution of x L , x R , respectively, under g t . From (2.6) we know that is a local martingale which yields that the law of η reweighted by M is that of an SLE κ (ρ L ; ρ 1,R , ρ 2,R ) process where ρ 2,R = −2ρ 1,R − ρ 2,R − 8 + κ. Note that, by similar analysis in Lemma 3.4, the term g τ (1) − V L τ is bounded both from below and above by positive finite constants depending only on r on the event E δ,r . The rest of the analysis in the proof of Lemma 3.2 applies similarly in this setting.
Throughout the rest of this subsection, we let: Let ζ be the first time that η exits D ∩ ϑT. Then Finally, if η exits D ∩ ϑT through the right side of ∂D ∩ ϑT, then Proof For z ∈ C, we let P z denote the law of a Brownian motion B in C started at z.
Suppose C > 0; we will set its value later in the proof. For each 1 ≤ k ≤ 1 Cϑ , we let L k = {z ∈ H : Re(z) = −kCϑ} and ζ k = inf{t ≥ 0 : η(t) ∈ L k }. Fig. 10 Illustration of the justification of (3.14) in the proof of Lemma 3.5 On E ϑ , we have that ζ 1 < ζ 2 < · · · < σ 1 < T L 0 . For each k, let F k = {ζ k < T L ϑ } and let F k be the σ -algebra generated by η| [0,ζ k ] . To complete the proof, we will show that is the constant from Lemma 2.6. To see this, we just need to show that g ζ k (η| [ζ k ,ζ k+1 ] ) satisfies the hypotheses of Lemma 2.6 and that with (3.14) Let B be a Brownian motion starting from z ϑ k = η(ζ k ) − ϑ and let H k+1 = {z ∈ H : Re(z) ≥ −(k + 1)Cϑ} be the subset of H which is to the right of L k+1 (see Fig. 10). The probability that B exits H k+1 \η([0, ζ k ]) through the right side of η([0, ζ k ]) (blue) is 1, through (−(k + 1)Cϑ, −kCϑ) (green) is 1, and through L k+1 (orange) is 1/C (since this probability is less than the probability that the Brownian motion exits {z ∈ C : −(k + 1)Cϑ < Re(z) < −kCϑ} through L k+1 which is less than 1/C). Let By the conformal invariance of Brownian motion, we have that Indeed, the probability of a Brownian motion started from z ϑ k to exit H k+1 := through L k+1 is bounded from below by a positive universal constant times the probability that a Brownian motion starting from z ϑ k exits This latter probability is bounded from below by a positive universal constant times y ϑ k / d. Thus 1/C y ϑ k / d, as desired.
The conformal invariance of Brownian motion and the estimates above also imply that sin(arg( z ϑ k )) 1, hence | z ϑ k | | y ϑ k |. Combining this with (3.15) implies that Thus, by the triangle inequality, We are first going to perform a change of coordinates. Let ϕ : H → H be the Möbius transformation z → ϕ(z) := z/ (1 − z). Fix x R ∈ [0 + , 1) and let η be an SLE κ (ρ 1,R , ρ 2,R ) process with force points located at ( x R , 1) as in Theorem 3.1. Then the law of η = ϕ( η) is that of an SLE κ (ρ L ; ρ R ) process with force points Let σ 1 be the first time that η hits ∂D and let (1) . Note that the exponent α comes from the sum of the exponent of |V L t − V R t | and the exponent of |W t − V L t | in the left martingale M L from (2.7) with these weights. For Fix β ∈ (0, 1) and set ϑ = β . For u > 0, we have the bound We claim that exists constants C 1 > 0 and γ > 0 depending only on ρ L , ρ R , and κ such that Therefore the sign of the exponent of |V L t − V R t | in the definition of M L t is the same as the sign of ρ R . If ρ R ≥ 0, then the exponent has a positive sign. In this case, M L t ≥ |W t − V L t | α so that we can take γ = α. Now suppose that ρ R < 0. By (3.8) of Lemma 3.4 we know that there exists a constant C 2 > 0 such that Thus, in this case, there exists a constant C 3 > 0 such that This proves the claimed bound in (3.18).
Set u = ϑ γ /C 1 . To bound the second term on the right side of (3.17), we first note by (3.18) that (3.20) By Lemma 3.5, we know that (3.21) We will now bound the first term on the right side of (3.17). Since τ L 0 , τ L u are stopping times for the martingale M L and Recall that (see for example [24,Section 4]) the β-Hausdorff measure of a set A ⊆ R is defined as Proof of Theorem 1.6 for κ ∈ (0, 4), upper bound Fix κ ∈ (0, 4), ρ ∈ (−2, κ 2 − 2). Let η be an SLE κ (ρ) process with a single force point located at 0 + . Let α ∈ (0, 1) be as in (3.4). Fix 0 < x < y. We are going to prove the result by showing that (3.23) For each k ∈ Z and n ∈ N we let I k,n = [k2 −n , (k + 1)2 −n ] and let z k,n be the center of I k,n . Let I n be the set of k such that I k,n ⊆ [x/2, 2y] and let E k,n be the event that η gets within distance 2 1−n of z k,n . Therefore there exists n 0 = n 0 (x, y) such that for every n ≥ n 0 we have that {I k,n : k ∈ I n , E k,n occurs} is a cover of η ∩ [x, y]. Fix ζ > 0. Theorem 3.1 implies that there exists a constant C 1 > 0 (independent of n) and n 1 = n 1 (ζ ) such that P[E k,n ] ≤ C 1 2 −(α−ζ )n for each n ≥ n 1 and k ∈ I n .
We will eventually take limits as a → ∞ and δ → 0 + . For U ⊆ H, we let (3.24) We will omit the superscript in (3.24) if x = 0. For k ∈ N and x ∈ [1, ∞), we let 1 hits B(x, k ) and does so for the first time above the horizontal line through iδ k . Given that E 1 k (x) has occurred, E 2 k (x) is the event that η x k−1 merges with η x k before the path leaves the annulus B(x, 1 2 k−1 )\B(x, k+1 ). Also indicated is the boundary data for h along ∂H as well as along the paths η x k−1 and η x k before exiting the annulus B(x, 1 2 k−1 )\B(x, k+1 ) (see Fig. 11). Finally, we let The following is the main input into the proof of the lower bound.

Proposition 3.6
For each δ ∈ (0, 1), there exists a constant c(δ) > 0 such that for all x, y ∈ [1, 2] and m ∈ N such that 1 2 m+1 ≤ |x − y| < 1 2 m we have The main steps in the proof of Proposition 3.6 are contained in the following three lemmas.

Lemma 3.7
For each x ≥ 1 and m, n ∈ N with m ≤ n, we have that
Given η x m+1 | [0,τ ] , we similarly have that the Radon-Nikodym derivative between the conditional law of η x n stopped upon exiting the connected component of B(x, 1 2 n )\η x m+1 ([0, τ ]) with x n on its boundary with respect to the law in which we additionally condition on H on E m (x) is bounded from above and below by C 1 and C −1 1 , respectively, possibly by increasing the value of C 1 > 1 (see Fig. 13). Moreover, conditional on both of the paths η x m+1 | [0,σ x m+1 (B(x, n+1 ))] and η x n | [0,σ x n ] as well as the event that they have merged before exiting U , the joint law of η Fig. 13). This proves (3.26).
The second part of the lemma is proved similarly.

Lemma 3.8
For each x ≥ 1 and m, n ∈ N with m ≤ n we have that
Proof The upper bound follows from (3.26) of Lemma 3.7. To complete the proof of the lemma, it suffices to show that Throughout, we assume that we are working on E m (x)∩ E m,n (x). To see this, we let H (resp. K ) be the closure of the complement of the unbounded connected component of and let z m be the point which lies at distance δ m+1 from ω along the line segment connecting ω to x (see Fig. 13). Note that the probability that a Brownian motion starting from z m exits H\(H ∪ K ) in the left (resp. right) side of H is 1 (though this probability decays as δ ↓ 0) and likewise for the left side of K . Let ϕ : H\(H ∪ K ) → H be the conformal map which takes z m to i and ω to 0. Let x L (resp. x R ) be the image of the leftmost (resp. rightmost) point of H ∩ R under ϕ. The conformal invariance of Brownian motion implies that there exists > 0 depending only on δ such that |x q | ≥ for q ∈ {L , R}. Let y L (resp. y) be the image of the leftmost point of K ∩ R (resp. η x m+1 (σ x m+1 )) under ϕ. By shrinking > 0 if necessary (but still depending only on δ), it is likewise true that y − y L ≥ and y L ≤ −1 . Consequently, it follows from Lemma 2.5 that η x m | [σ x m ,∞) has a positive chance (depending only on δ, κ, and ρ) of hitting (hence merging into) the left side of η

Lemma 3.9
For each δ ∈ (0, 1) there exists a constant c(δ) > 0 such that the following is true. For each x ≥ 1, we have that Proof By (3.26) of Lemma 3.7, we know that Therefore we just have to show that there exists a constant c(δ) > 0 such that Note that (3.30) follows from Lemma 2.5 using the same argument as in the proof of Lemma 3.8. We know that η x k is an SLE κ (2 + ρ, −2 − ρ; ρ) process within the configuration c = (H, x k , (0, x − k ), (x + k ), ∞). Consequently, (3.29) follows by combining Corollary 3.3 and Lemma 2.8. The latter is used to get that the Radon-Nikodym derivative between the law of an SLE κ (2 + ρ, −2 − ρ; ρ) process with configuration (H, x k , (0, x − k ), (x + k ), ∞) and the law of an SLE κ (−2 − ρ; ρ) process with configuration (H, x k , (x − k ), (x + k ), ∞), where each path is stopped upon exiting B(x, k 2 ), is bounded both from below and above by universal positive and finite constants.
Proof of Proposition 3. 6 We have that,

P[E m (x)]P[E m+1,n (x)]P[E m+1,n (y)] (Lemma 3.7)
(Lemmas 3.8 and 3.9) Proof of Theorem 1. 6 We are first going to give the lower bound for κ ∈ (0, 4) and then explain how to extract the dimension result for κ > 4 from the result for κ ∈ (0, 4). For each β ∈ R and Borel measure μ, let be the β-energy of μ. To prove the lower bound, we will show that, for each ζ > 0, there exists a nonzero Borel measure supported on η∩ [1,2] that has finite (1−α−2ζ )energy. Fix n ∈ N. We divide [1,2] into −1 n intervals of equal length n and let z j,n = ( j − It is easy to see that Let μ n be the measure on [1,2] defined by 1 provided we choose n and a large enough. Set β = 1 − α − 2ζ . We also have that Consequently, the sequence (μ n ) has a subsequence (μ n k ) that converges weakly to some nonzero measure μ. It is clear that μ is supported on C and has finite (1−α −2ζ )energy. From [24,Theorem 4.27], we know that Since η is conformally invariant, by 0-1 law (see [3]), we have that for any ζ > 0. This proves the lower bound for κ ∈ (0, 4). It is left to prove the result for κ > 4. Fix ρ ∈ ( κ 2 − 4, κ 2 − 2). Consider a GFF h on [−1, 1] 2 with the boundary values as depicted in Fig. 5 with ρ R = ρ and ρ L = 0, and let η be the counterflow line of h from i to −i. Then η is an SLE κ (ρ ) process with a single force point located at (i) + , i.e., immediately to the right of i. Consequently, it follows that the dimension of η ∩ S is given by

The intersection of flow lines
In this section, we will prove Theorem 1.5. We begin in Sect. 4.1 by proving an estimate for the derivative of the Loewner map associated with an SLE κ (ρ) process when it gets close to a given point. Next, in Sect. 4.2 we will prove the one point estimate which we will use in Sect. 4.3 to prove the upper bound. Finally in Sect. 4.4 we will complete the proof by establishing the lower bound.

Derivative estimate
Recall from Sect. 2.4 that for a point w in a simply connected domain U , CR(w; U ) denotes the conformal radius of U as viewed from w. Fix κ ∈ (0, 4), let η be an ordinary SLE κ process in H from 0 to ∞ and, for each t, let H t denote the unbounded connected component of H\η([0, t]). We use the notation of [13, Section 6.1]. We let For z ∈ H, we let We note that ϒ t = 1 2 CR(z; H t ) dist(z, ∂H t ). For each r ∈ R, we also let (4.2) (In the notation of [13], a = 2/κ). Then we have that [13, Proposition 6.1]: is a local martingale. This martingale also appears in [41,Theorem 6], though it is expressed there in a slightly different form (The martingale in (2.6) is of the same type, though there we have not included the interior force points). For each > 0 and R > 0, we let τ = inf{t ≥ 0 : ϒ t = 1 2 } = inf{t ≥ 0 : CR(z; H t ) = } and σ R = inf{t ≥ 0 : |η(t)| = R}. (4.4) , and z ∈ H such that arg(z) ∈ (δ, π − δ). Let P be the law of η weighted by M. We have that, and where the constants depend only on δ, κ, and r . We also have that where constants depend only on δ, κ, and r . Finally, we have that uniformly over > 0.
Proof Note that (4.5) and (4.6) are proved in [13, Equation (6.9)], so we will not repeat the arguments here. Following [13], we define the radial parametrization (i.e., by log conformal radius) u(t) by and write η(t) = η(u(t)) and t = u(t) . Then t satisfies the SDE (see [13,Section 6.3]) where W is a P -Brownian motion. The process almost surely does not hit {0, π} (see [16,Lemma 1.27]) and the density with respect to Lebesgue measure on [0, π] for the stationary distribution for (4.9) is given by where c > 0 is a normalizing constant (see [16,Lemma 1.28]). Moreover, as t → ∞, the law of t converges to the stationary distribution with respect to the total variation norm. We can use this to extract (4.7) as follows. Fix 0 < T < ∞. We first note that by the Girsanov theorem the law of | [0,T ] stopped upon leaving ( δ 2 , π − δ 2 ) is mutually absolutely continuous with respect to that of B| [0,T ] where B is a Brownian motion starting from 0 , also stopped upon leaving ( δ 2 , π − δ 2 ). Fix 0 ≤ t ≤ T . Then a Brownian motion starting from 0 ∈ [δ, π − δ] has a uniformly positive chance of staying in ( δ 2 , π − δ 2 ) during the time interval [0, t] and then being in (δ, π − δ) at time t. Therefore it is easy to see that (4.7) holds for all 0 ≤ t ≤ T .
The lower bound, however, that comes from this estimate decays as T increases. We are now going to explain how we make our choice of T as well as get a uniform lower bound for t ≥ T . We suppose that 1 , 2 are solutions of (4.9) where 1 0 = δ and 2 0 = π − δ. We assume further that the Brownian motions driving , 1 , and 2 are independent of each other until the time that any two of the processes meet, after which we take the Brownian motions for the pair to be the same. This gives us a coupling ( 1 , , 2 ) such that 1 t ≤ t ≤ 2 t for all t ≥ 0 almost surely. Note that after 1 first hits 2 , all three processes stay together and never separate. Let q δ > 0 be the mass that the stationary distribution puts on (δ, π − δ). We then take T > 0 sufficiently large so that: 1. For all t ≥ T , the total variation distance between the law of 1 t and the stationary distribution is at most q δ 2 .
We are now going to use Lemma 4.1 to estimate the moments of g t (z) at times when η is close to z. We will actually prove this for general SLE κ (ρ) processes which is why we truncate on various events in the estimates proved below. Lemma 4.2 Fix r < 1 2 − 4 κ and δ ∈ (0, π 2 ). There exists R 0 = R 0 (r ) > 0 such that for all R ≥ R 0 the following holds. Suppose η ∼ SLE κ (ρ) in H from 0 to ∞ where the force points lie outside of 2RD. Fix z ∈ D ∩ H with arg(z) ∈ (δ, π − δ). For each > 0 and R > 0 we let τ and σ R be as in (4.4). Then where the constants depend only on δ, κ, and the weights ρ of the force points. Fix a constant C > 1 and suppose that ζ is a stopping time for η such that τ C ≤ ζ ≤ τ /C . Let Then we have that where the constants depend only on C, δ, κ, and the weights ρ of the force points.
Proof It suffices to prove the result for an ordinary SLE κ process since it is clear from the form of (2.6) that the Radon-Nikodym derivative between the law of an SLE κ and an SLE κ (ρ) process whose force points lie outside of 2RD stopped at time σ R is bounded from above and below by finite and positive constants which depend only on the total (absolute) weight of the force points and κ.
We are now going to prove the upper bound of (4.10) and the lower bound of (4.12) with τ = ζ . We have that, This proves the upper bound of (4.10). For the lower bound, we compute To bound P [E δ ,R ], we have From (4.7), we know that P [ τ ∈ (δ, π − δ)] is bounded from below uniformly in > 0. From (4.8), we know that P [σ R < τ ] converges to zero as R → ∞ uniformly over > 0. These show that P [E δ ,R ] is bounded from below which proves the lower bound for (4.12). The upper bound in the case that we replace τ with ζ is proved similarly. For the lower bound, it is not difficult to see that

The same likewise holds if h is a GFF on H with piecewise constant boundary conditions which change values a finite number of times and in the interval
takes the form in (4.14). In this case, the constants also depend on h| R ∞ .
Thus, by (4.10) of Lemma 4.2, we have that This gives the upper bound for (4.15).
The final claim of the lemma follows from (2.6) to compare the case with extra force points to the case without considered above.
In order for Lemma 4.3 to be useful, we need that as η 1 gets progressively closer to a given point z, it is unlikely that 1 / ∈ (δ, π − δ) for some δ > 0. This is the purpose of the following estimate.
For each n ∈ N, we let D n be the set of squares with side length 2 −n which are contained in H and with corners in 2 −n Z 2 . For each Q ∈ D n , let z(Q) be the center of Q and let Q n (Q) = B(z(Q), 2 1−n ). For each z ∈ H, let Q n (z) be the element of D n which contains z and let Q n (z) = Q n (Q n (z)). See Fig. 15 for an illustration. 15 Shown in the illustration are Q n (z) and Q n (z) for a given point z ∈ H Lemma 4.5 Suppose that η is an SLE κ process in H from 0 to ∞ with κ ∈ (0, 4). For each z ∈ H, let z be the process from (4.1) (with respect to z) and let ζ z,n = inf{t ≥ 0 : η(t) ∈ ∂ Q n (z)}. Let S δ n be the set of points z ∈ H such that E δ z,n = {ζ z,n < ∞, z ζ z,n / ∈ (δ, π − δ)} occurs and let S δ = ∪ ∞ n=1 ∩ ∞ m=n S δ m . There exists δ 0 > 0 such that for every δ ∈ (0, δ 0 ) we have that S δ = ∅ almost surely.
Proof Fix z ∈ H and let n z = − log 2 Im(z). Note that Q n (z) ⊆ B(z, 2 2−n ) so that Q n (z) ⊆ H provided n ≥ n z + 2. By Lemma 4.4, we have that P[∩ r m=n E δ z,m ] ≤ p(δ) r −n for all n z + 2 ≤ n ≤ r (4.20) (where p(δ) is as in the statement of Lemma 4.4). Suppose that Q ∈ D m and suppose that n ∈ N with n ≤ m. Then the function Q → R given by w → w ζ w,n is positive and harmonic. Consequently, it follows from the Harnack inequality [16,Proposition 2.26] that there exists a universal constant K ≥ 1 (independent of m, n) such that the following is true. If E δ w,m occurs for any w ∈ Q, then E K δ z(Q),m occurs. Thus letting E δ Q,m = ∪ w∈Q E δ w,m we have that Combining this with Lemma 4.4 implies that P[∩ r m=n E δ Q,m ] ≤ ( p(K δ)) r −n for any n z(Q) + 2 ≤ n ≤ r. Fix ω ∈ (0, 1) and let n = − log 2 ω. For each r ≥ n + 2, let V ω,δ r be the collection of squares Q in D r with Q ⊆ {z ∈ H : |z| < 1 ω , Im(z) ≥ ω} and for which ∩ r m=n E δ Q,m occurs. Then (4.22) implies that there exists a constant C > 0 such that Take δ 0 > 0 so that δ ∈ (0, δ 0 ) implies that 4 p(K δ) < 1. Then for δ ∈ (0, δ 0 ), the summation on the right side of (4.23) is finite. This implies that for every ω ∈ (0, 1), V ω,δ r = ∅ for all but finitely many r almost surely. This, in turn, implies the desired result since ω ∈ (0, 1) was arbitrary and V ω,δ r increases as ω decreases.

The upper bound
Now that we have established Lemmas 4.3 and 4.5, we can prove the upper bound in Theorem 1.5. 16 Suppose that h is a GFF on H with the illustrated boundary data. Let η 1 (resp. η 2 ) be the flow line of h starting from 0 with angle 0 (resp. θ ∈ (π − 2λ/χ, 0)). Shown is an illustration of the construction of the event that a given point, say z ∈ H, is a "perfect point" for the intersection of η 1 and η 2 . Each of the green flow lines has angle θ -the same as that of η 2 -and start at points along η 1 which get progressively closer to z. The reason that we introduce the auxiliary green flow lines is that this is what gives us the approximate independence necessary for the two point estimate, see e.g., Fig. 20 (color figure online) We are now going to show that, for every n ∈ N, W ,δ n = ∪ m≥n U ,δ m is a cover of I ,δ . To see this, we fix z ∈ I ,δ and let (Q k ) be a sequence of squares in ∪ m≥n D m such that z ∈ Q k for every k and |Q k | → 0 as k → ∞. Let z k = z(Q k ). Since for all k large enough, there exists K 0 = K 0 (z) such that for all k ≥ K 0 , we have that 1,z k ζ 1 z k , ∈ (δ, π − δ). Since z ∈ Q k , we have that η 1 hits Q k . If there exists a subsequence (k j ) such that, for every j, η 1 hits ∂ B(z k j , δ) after hitting ∂ B(z k j , ) and before hitting Q k j , we get a contradiction that z ∈ I ,δ . Therefore there exists K 1 = K 1 (z) such that for every k ≥ K 1 , we have that, after hitting ∂ B(z k , ), η 1 hits Q k before hitting ∂ B(z k , δ). Combing this with Lemma 4.5 implies that there exists a sequence (k j ) such that Q k j ∈ W ,δ n for all j, which proves our claim. By running η 1 until time ζ 1 z, and then conformally mapping back, Lemma 4.3 implies for Q ∈ D m with Q ⊆ B(0, δ −1 ) and Im(z(Q)) ≥ δ that P[Q ∈ U ,δ m ] ≤ 2 −m(A+o(1)) provided m is large enough and > 0 is small enough relative to δ > 0. (The purpose of choosing > 0 smaller than δ > 0 is so that the force points of η 1 are mapped far away from η 1 (ζ 1 z, ) relative to the distance of z.) Consequently, it follows that there exists C = C( , δ) > 0 such that for each ξ > 0, we have

The lower bound
We are now going to prove the lower bound for Theorem 1.5. As in the proof of Theorem 1.6, we will accomplish this by introducing a special class of points, socalled "perfect points", which are contained in the intersection of two flow lines whose correlation structure is easy to control (see Fig. 16). Fix β > β 2 > β > 1; we will eventually send β → ∞ but we will take β fixed and large.

Definition of the events
We are going to define the perfect points as follows. Suppose that γ 1 is a path in H starting from 0 and γ 2 is a path starting from x 2 ∈ [0, e β ]. Let ζ 1 be the first time that γ 1 hits ∂ B(i, e − β ) and suppose that γ 2 is a path starting from γ 1 ( ζ 1 ). Fix u ∈ R\[0, x 2 ]. We let E β, β u (γ 1 , γ 2 , γ 2 ) be the event that the following hold (see Fig. 17 for an illustration): contains i also contains u on its boundary. The purpose of Part (i) above is that, by drawing a path up until hitting ∂ B(i, e −β ) and then conformally mapping back, the resulting configuration of paths satisfies the hypotheses of Lemma 4.3.
We claim P[X ] d −1 . Indeed, X 1 ∩ X 2 ⊆ X where X 1 is the event that the Brownian motion exits ∂ B(0, d) before hitting ∂H at a point with argument in [ π 4 , 3π 4 ] and X 2 is the event that it hits ϕ(∂ B(i, e −α(β+ β) )) after hitting ∂ B(0, d) before hitting ∂H. It is easy to see that P[

Flow line estimates
Fix θ ∈ (π − 2λ/χ, 0); recall that this is the range of angles so that a GFF flow line with angle θ can hit and bounce off of a GFF flow line wiith angle 0 on its right side. We will now use the events introduced in Sect. 4.4.1 to define the perfect points. Suppose that h 1 is a GFF on H with the following boundary data: suppose x 1,1 = x 1,2 = 0 and u 1 ∈ R\{0}. If u 1 < x 1,1 = x 1,2 = 0, the boundary data is Fig. 14.) Let ζ 1 , ζ 2 be the times that η 1 , η 2 hit ∂ B(z, ), respectively, and let ϕ be the unique conformal map that uniformizes the unbounded connected component of H\(η 1 ([0, ζ 1 ]) ∪ η 2 ([0, ζ 2 ])) with z sent to i and ∞ fixed. For the lower bound of Theorem 1.5, we will also need to estimate the probability of the event H δ (z) that G δ (z) occurs (as described in Fig. 14), that the diameter of η 2 ([0, ζ 2 ]) is not too large, and that the images of η i (ζ i ) for i = 1, 2 under ϕ are not too far from i as illustrated on the right If u 1 > x 1,1 = x 1,2 = 0, then the boundary data is These two possibilities correspond to the boundary data that arises when one takes a GFF with boundary conditions as in Figs. 14 and 18 and then applies a change of coordinates which takes a given point z ∈ H to i. In either case, we let η 1,1 (resp. η 1,2 ) be the flow line of h 1 starting from x 1,1 (resp. x 1,2 ) of angle 0 (resp. θ ). We also let ζ 1,1 be the first time that η 1 hits ∂ B(i, e − β ) and let η 1,2 be the flow line of h 1 starting from (the right side of) η 1,1 ( ζ 1,1 ) with angle θ .
On the event that η k+1,1 hits ∂ B(i, e − β−β ), say for the first time at time ζ k+1,1 , we let ϕ k+1 be the conformal transformation which uniformizes the connected component of H\η k+1,1 ([0, ζ k+1,1 ]) with u k+1 on its boundary fixing i and with ϕ k+1 (η k+1,1 (ζ k+1,1 )) = 0. We then define the event E k+1 in terms of the paths η k+1,1 , η k+1,2 , and η k+1,2 analogously to E 1 as well as stopping times ζ k+1,2 , ζ k+1,2 . For each n ≥ m we let E m,n = ∩ n k=m+1 E k and E n = E 0,n . (4.24) Remark 4.8 (i) Note that E m,n for n > m ≥ 1 can occur even if only a subset of (or none of) E 1 , . . . , E m occur. (ii) The conformal maps ϕ j are measurable with respect to η 1,1 . Note that each of the paths η k,2 is given by the conformal image of a flow line which starts at a point in the range of η 1,1 . The starting points of these flow lines are likewise measurable with respect to η 1,1 . These facts will be important when we establish the two point estimate for the lower bound of Theorem 1.5 at the end of this subsection.
We will now work towards proving the one point estimate for the perfect point i.

Proposition 4.9
There exists β 0 > 1 such that for all β > β 2 > β ≥ β 0 we have where A is the constant from (4.13) and the constants in the of (4.25) depend only on u 1 , κ, and θ .
In the statement of Proposition 4.9, we write o β (1) to indicate a quantity which converges to 0 as β → ∞ and O β (1) for a term which is bounded by some constant which depends only on β. In particular, for β fixed, O β (1)o β (1) → 0 as β → ∞. The first step in the proof of Proposition 4.9 is Lemma 4.10. The second step, which allows one to iterate the estimate in (4.26), is Lemma 4.12 and is stated and proved below.

Lemma 4.10
There exists β 0 > 1 such that for all β > β 2 > β ≥ β 0 we have where A is the constant from (4.13) and the constants in the of (4.26) depend only on u 1 , κ, and θ .
Proof By Lemma 2.3, we know that η 1,1 has a positive chance of being uniformly close to [0, i] before hitting ∂ B(i, e −β ). Let τ be the first time that η 1,1 hits ∂ B(i, e −β ) and let g be the conformal transformation from the connected component of H\η 1,1 ([0, τ ]) containing i which fixes i and sends η 1,1 (τ ) to 0. By choosing β 0 sufficiently large, it is clear that g(η 1,1 ) and g(η 1,2 ) satisfy the hypotheses of (4.16) of Lemma 4.3. From this, we deduce that the probability that η 1,1 and η 1,2 both hit ∂ B(i, 2e − β ) before leaving B(i, e 2β ) and such that the harmonic measure of the left (resp. right) side of each of the paths stopped at this time as viewed from i is bounded from below by some universal constant is equal to e − β(1+O β (1)o β (1))A . The rest of the lemma follows from repeated applications of Lemmas 2.3 and 2.5.
For each z ∈ H, we let ψ z be the unique conformal transformation H → H taking z to i and fixing 0. For each k ∈ N, we let η z k,i for i = 1, 2 and η z k,2 be the paths after applying the conformal map ψ z and we let ζ z k,i , ζ z k,i be the corresponding stopping times. We define E m,n (z) = E m,n (η z 1,1 , η z 1,2 , η z 1,2 ) and E n (z) = E 0,n (z). (4.27) In other words, E m,n (z) and E n (z) are the events corresponding to E m,n and E n defined in (4.24) but with respect to the flow lines of the GFF h 1 • ψ −1 z − χ arg(ψ −1 z ) starting from 0. Let ϕ k,z be the corresponding conformal maps. We let ϕ j,k z = ϕ j+1,z • · · · • ϕ k,z for each 0 ≤ j ≤ k and ϕ k z = ϕ 0,k z . (4.28) For each n ∈ N and z ∈ H, we also let V n (z) = B(z, 2 −8n−4 Im(z)e −n(β+ β) ) and V n (z) = B(z, 2 8n+4 Im(z)e −n(β+ β) ).

Lemma 4.11
There exists β 0 > 1 such that for all β > β 2 > β ≥ β 0 , the following is true. For each m, n ∈ N with m ≥ n + 2, on E m (z) we have both . Proof We will only give the proof of (i); the proof of (ii) is analogous. We are first going to give the proof in the case that z = i. Fix m, n ∈ N with m ≥ n + 2. Throughout, we shall assume that we are working on E m . It follows from [16,Corollary 3.25] that if r ∈ (0, 1 2 ) then Iterating (4.29) implies that
Proof By applying ψ z , we may assume without loss of generality that z = i. Recall the definition of the GFF h m+1 as well as the paths η k,i for i = 1, 2 and η k,2 from just before Remark 4.8. By the definition of E m and the conformal invariance of Brownian motion, we know that there exists a constant c 1 > 0 such that the boundary data for h m+1 in (−c 1 , 0) (resp. (0, c 1 )) is given by −λ (resp. λ). The same is likewise true for h 1 . Moreover, by Lemma 4.7, it follows that the auxiliary paths coupled with h m+1 are far away from i provided β 0 is large enough. Consequently, by Lemma 2.8, the laws of η m+1,1 (given E m ) and η 1,1 stopped upon exiting the c 1 2 neighborhood of the line segment from 0 to i are mutually absolutely continuous with Radon-Nikodym derivative which is bounded from above and below by universal positive and finite constants which depend only on κ and θ .
Let K be the compact hull associated with these paths and let g be the conformal transformation H\K → H with g(z) ∼ z as z → ∞. Conditionally on all of these paths and the event that they are contained in B(i, 2e − β ), the probability that η m+1,2 (1) (as in the proof of Lemma 4.3; the extra force points only change the probability by a positive and finite factor by Lemma 2.8.) Given that η m+1,2 has hit ∂ B(i, 10e − β ), the conditional probability that it then merges with η m+1,2 before the latter has hit ∂ B(i, 1 2 e − β ) or ∂ B(i, 2e − β ) is positive by Lemma 2.5. The same is true with η 1,2 in place of η m+1,2 , which completes the proof.
Proof We are going to extract (4.33) from (4.32) of Lemma 4.12. As before, by applying ψ z , we may assume without loss of generality that z = i. Fix k ≥ m. By Proposition 4.9, it suffices to prove in place of (4.33). By Lemma 4.11, we know that the paths involved in E m,n are disjoint from those involved in E k (w) due to the choice of m. By Lemma 4.11, we know that the paths involved in E m,n are contained in V m−1 (i), and that the paths involved in E k (w) are outside of V m−2 (i), due to the choice of m. Thus, the paths involved in E m+1,n are disjoint from those involved in E k (w), and by conformally mapping back (see Fig. 20) and applying Lemma 2.8 as in the proof of Lemma 4.12, it is therefore not hard to see that Combining this with (4.32) completes the proof. . 20 Illustration of the setup for the two point estimate (Lemmas 4.12, 4.14) in the case that η 1 gets close first to w and then to z. Conformally map back η 1,1 drawn up until the path hits the neighborhood of z. Then all of the auxiliary paths are outside of a large ball which is far from i = ϕ(z), so we can apply the one point estimate for perfect points (Lemma 4.10) for this region as before. We can also apply the one point estimate for the paths near z. Finally, to complete the proof, we apply the one point estimate a final time for the paths up to when they hit a neighborhood containing both z and w Lemma 4.14 For every > 0 and δ ∈ (0, π 2 ) there exists β 0 > 1 such that for all β > β 2 > β ≥ β 0 there exist constants C > 0 and n 0 ∈ N such that the following is true. Fix z, w ∈ D ∩ H distinct with arg(z), arg(w) ∈ (δ, π − δ). Let m be the smallest Proof Suppose that z, w ∈ H are as in the statement of the lemma. Let P w be the event that η 1 hits V m (w) before hitting V m (zs) and lsset P z be the event in which the roles of z and w are swapped. We have that (4.35) We are going to bound the first summand; the second is bounded analogously. We have, We can now complete the proof of Theorem 1.5.
Proof of Theorem 1. 5 We suppose that h is a GFF on H with boundary conditions h| (−∞,0] ≡ −λ and h| (0,∞) ≡ λ − θχ and let η 1 (resp. η 2 ) be the flow line of h starting from 0 with angle 0 (resp. θ ∈ (π − 2λ/χ, 0)). We have already established the upper bound for dim H (η 1 ∩ η 2 ∩ H) in Proposition 4.6. We will now establish the lower bound. Once we have proved this, we get the corresponding dimension when h has general piecewise constant boundary data as described in the theorem statement by absolute continuity for GFFs. The proof is completed in the same manner as the proof of Theorem 1.6. Indeed, we let n = 2 8n+4 e −(β+ β)n . We divide [−1, 1] × [1, 2] into 2 −2 n squares of equal side length n and let z n j be the center of the jth such square for j = 1, . . . , 2 −2 n . Let C n be the set of centers z of these squares for which E n (z) occurs. Let S n (z) be the square with center z and length n . Finally, we let C = k≥1 n≥k z∈C n S n (z).
It is easy to see that The argument of the proof of Theorem 1.6 combined with Lemma 4.14 implies, for each ξ > 0, that P[dim H (η 1 ∩η 2 ) ≥ 2− A−ξ ] > 0. To finish the proof, we only need to explain the 0-1 argument: that for each d ∈ [0, 2], P[dim H (η 1 ∩η 2 ∩H) = d] ∈ {0, 1}. For r > 0, let D r = dim H (η 1 ∩ η 2 ∩ B(0, r ) ∩ H). It is clear that 0 < r 1 < r 2 implies D r 1 ≤ D r 2 . By the scale invariance of the setup, we have that D r 1 has the same law as D r 2 . Thus D r 1 = D r 2 almost surely for all 0 < r 1 < r 2 . In particular, P[D ∞ = D r ] = 1 for all r > 0. Thus the events {D ∞ = d} and {D r = d} are the same up to a set of probability zero. The latter is measurable with respect to the GFF restricted to B(0, r ). Letting r ↓ 0, we see that this implies that the event {D ∞ = d} is trivial, which completes the proof.
That is, dim H (P ∩ (z)) is almost surely equal to the Hausdorff dimension of the intersection of two GFF flow lines with an angle gap of θ double (recall (1.10)) as given in Theorem 1.5.
Proof See Fig. 22 for an illustration of the argument. We shall assume throughout for simplicity that z ∈ ∂ L T. A similar argument gives the same result for z ∈ ∂ U T. Suppose that h is a GFF on H with the boundary data as indicated in the left side of Fig. 22. Let η 1 0 be the flow line of h from 0 with angle π 2 . Given η 1 0 , let η 2 0 be the flow line of h with angle π 2 from ∞ in the component L of H\η 1 0 which is to the left of η 1 0 . Note that η 1 0 is an SLE κ ( κ 2 − 2; − κ 2 ) process in H from 0 to ∞. Moreover, the conditional law of η 2 0 given η 1 0 is an SLE κ (κ − 4; − κ 2 ) process in L from ∞ to 0; see [28,Lemma 3.3]. (The κ − 4 force point lies between η 1 0 and η 2 0 .) By the main result of [27], the time-reversal η 2 0 of η 2 0 is an SLE κ (− κ 2 ; κ − 4) process in L from 0 to ∞. As explained in Fig. 23, it consequently follows from Theorem 1.5 that dim H (η 1 0 ∩ η 2 0 ) = 2 − (12 − κ )(4 + κ ) 8κ almost surely (5.1) since this is the almost sure dimension of η 1 0 ∩ η 2 0 (using the notation of Fig. 23). Thus to complete the proof, we just have to argue that dim H (P ∩ (z)) is also given by this value. 0 is an SLE κ ( κ 2 − 2; − κ 2 ) in H from 0 to ∞. Moreover, the conditional law of η 2 0 given η 1 0 is that of an SLE κ (κ − 4; − κ 2 ) in the component of H\η 1 0 which is to the left of η 1 0 from ∞ to 0 (the κ − 4 force point lies between the paths). Shown is the boundary data for the conditional law of h given (η 1 0 , η 2 0 ) in the component U 1 of H\(η 1 0 ∪ η 2 0 ) which contains 1 on its boundary. Let ϕ : U 1 → H be the conformal transformation with ϕ(1) = z and which takes leftmost (resp. rightmost) point of ∂U 1 ∩ ∂H to −∞ (resp. +∞). Then h • ϕ −1 − χ arg(ϕ −1 ) has the boundary data shown on the right side. Let (η 1 1 , η 2 1 ) be a pair of paths defined in the same way as (η 1 0 , η 2 0 ) except starting from 1. Then the image of the region in U 1 between η 1 1 and η 2 1 under ϕ has the same law as P(z) described in Fig. 21 (see also [28,  Suppose that h is a GFF on H with zero boundary conditions as illustrated. Let η 1 0 (resp. η 2 0 ) be the flow line of h starting from 0 with angle − 1 2 θ double (resp. 1 2 θ double ); recall (1.10). Then η 1 0 is an SLE κ ( κ 2 −2; − κ 2 ) process in H from 0 to ∞ (Fig. 1) and the conditional law of η 2 0 given η 1 0 in the connected component of H\ η 1 0 which is to the left of η 1 0 is an SLE κ (− κ 2 ; κ −4) process from 0 to ∞ (Fig. 4). Similarly, η 2 0 is an SLE κ (− κ 2 ; κ 2 − 2) process in H from 0 to ∞ (Fig. 1) and the conditional law of η 1 0 given η 2 0 is an SLE κ (κ − 4; − κ 2 ) process from 0 to ∞ in the component of H\ η 2 0 which is to the right of η 2 0 (Fig. 4). In particular, by the main result of [27], the joint law of the ranges of η 1 0 and η 2 0 is equal to the joint law of the ranges of η 1 0 and η 2 0 from the left side of Fig. 22. Consequently, we can use Theorem 1.5 to compute the almost sure dimension of the intersection of the latter Let U 1 be the component of H\(η 1 0 ∪ η 2 0 ) which contains 1 on its boundary. Let ϕ : U 1 → T be the conformal transformation which takes 1 to z and the leftmost (resp. rightmost) point of ∂U 1 ∩ R to −∞ (resp. +∞). Let (η 1 1 , η 2 1 ) be a pair of paths constructed in exactly the same manner as (η 1 0 , η 2 0 ) except starting from 1 rather than 0.
Proof For each ω ∈ D, let t f (ω) and t (ω) be the first and last time that η hits ω.
We now have all of the ingredients to complete the proof of Theorem 1.1 for κ ∈ (4, 8).
Proof of Theorem 1.1 For κ ≥ 8 Fix κ ≥ 8 and let κ = 16 κ ∈ (0, 2]. Let η be an SLE κ process in H from 0 to ∞ and let D be the set of double points of η . Then η is space-filling [31]. For each point z ∈ H, let t (z) be the first time that η hits z and let γ (z) be the outer boundary of η([0, t (z)]). It follows from [29, Theorems 1.1, 1.13] and [3] that the dimension of γ (z) is equal to 1+ κ 8 = 1+ 2 κ . Given γ (z), η ([t (z), ∞)) is an SLE κ process in the remaining domain, and thus almost surely hits every point on γ (z) except the point z. This implies that every point on γ (z) except for z is contained in D. This gives the lower bound for dim H (D).
Let (z k ) k∈N be a countable dense set in H. For the upper bound, we will show that every element of D is in fact on γ (z k ) for some k. Note that (t (z k )) k∈N is a dense set of times in [0, ∞) because η is continuous. For each ω ∈ D, let t f (ω) and t (ω) be the first and last times, respectively, that η hits ω. For each δ > 0, D δ = {ω ∈ D : t (ω) − t f (ω) ≥ δ}. Then D = ∪ δ>0 D δ . Since the sets D δ are increasing as δ > 0 decreases, it suffices to show that D δ ⊆ ∪ k γ (z k ) for each δ > 0. Fix δ > 0 and ω ∈ D δ . Since (t (z k )) k∈N is dense, we have that k = min{ j ≥ 1 : t (ω) > t (z j ) > t f (ω)} < ∞.

Remark 5.3
We note that SLE κ for κ ∈ (4, 8) does not have triple points and, when κ ≥ 8, the set of triple points is countable. Indeed, to see this we note that if z is a triple point of an SLE κ process η then there exists rational times t 1 < t 2 such that z is a single-point of and contained in the outer boundary of η | [0,t 1 ] and a double point of and contained in the outer boundary of η | [0,t 2 ] . For each pair t 1 < t 2 there are precisely two points which satisfy these properties. The claim follows for κ ∈ (4, 8) since SLE κ for κ ∈ (4, 8) almost surely does not hit any given boundary point distinct from its starting point. The claim likewise follows for κ ≥ 8 because this describes a surjection from Q + × Q + , Q + = (0, ∞) ∩ Q, to the set of triple points.