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 SLE_kappa for kappa>4, 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 SLE_kappa for kappa>4 as well as analogous dimensions for the radial and whole-plane SLE_kappa(rho) processes for kappa>0. 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 e^{ih/chi}, where h is a Gaussian free field and chi>0, of different angles with each other and with the domain boundary.


Introduction.
1. 1. Overview. The Schramm-Loewner evolution SLE κ (κ > 0) is the canonical model for a conformally invariant probability measure on noncrossing, continuous paths in a proper simply connected domain D in C. SLE κ was introduced by Oded Schramm [Sch00] 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 [LSW04, Smi05, CN07, SS09, Mil10, Smi10, CS09, CDCH + 13, HK11]. 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 whole-plane 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 [She, Dub09b, MS10, SS10, HBB10, IK10, She10, MS12a, MS12b, MS12c, MS13] with different angles.
Recall that chordal SLE κ is self-intersecting for κ > 4 and space-filling for κ ≥ 8 [RS05]. The dimension in (1.1) for κ ∈ (4, 8) was first predicted by Duplantier-Saleur [DS89] in the context of the contours of the FK model. The almost sure Hausdorff dimension of SLE κ is 1 + κ 8 for κ ∈ (0, 8) and 2 for κ ≥ 8 [Bef08] and, by SLE duality, the outer boundary of an SLE κ process for κ > 4 stopped at a positive and finite time is described by a certain SLE κ process [Zha08, Zha10, Dub09a, MS12a, MS12c, MS13]. Thus (1.1) for κ ≥ 8 states that the double point dimension is equal to the dimension of the outer boundary of the path. We note that chordal SLE κ does not have triple points for κ ∈ (4, 8) and the set of triple points is countable for κ ≥ 8; see Remark 5. 3.
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 [Bef08] (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 [MS13,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 dim H (I j ) = 1 8κ 4 + κ + 2ρ − 2j 2 + ρ 4 + κ − 2ρ + 2j 2 + ρ (1.6) 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 dim H (L j ) = 1 2κ κ − 2j 2 + ρ 2 + j 2 + ρ almost surely on {L j = ∅}. (1.7) and dim H (R j ) = 1 2κ κ + 2ρ − 2j(2 + ρ) 2 − ρ + j(2 + ρ) almost surely on {R j = ∅}. 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 [MS13].
The proofs of Theorem 1.1 and Theorem 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 Dub09b,MS12a,MS12c,MS13]. We will refer to these paths simply as "GFF flow lines." (An overview of this theory is provided in Section 2.2.) The duality statement which is relevant for the cut-set (see Figure 2.5) 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 (1.9) θ cut = π.
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 [MS12c] (1.10) θ double = π κ − 2 2 − κ 2 .
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. THEOREM 1. 6. Fix κ > 0 and ρ ∈ ((−2) ∨ ( κ 2 − 4), κ 2 − 2). Let η be an SLE κ (ρ) process with a single force point located at 0 + . Almost surely, (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 Section 2.2. See, in particular, Figure 2.4. By [MS13, 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 [MS13,Proposition 4.11], applying Theorem 1.6 with an angle gap of (1.14) 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 remark that an alternative proof to the lower bound of Theorem 1.6 for κ ∈ (8/3, 4) is given in [WW13] using the relationship between the SLE κ (ρ) processes for these κ values and the Brownian loop soups. We obtain as a corollary (when ρ = 0) the following which was first proved in [AS08].
COROLLARY 1.7. Fix κ ∈ (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 Theorem 1.5 and Theorem 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.
Outline. The remainder of this article is structured as follows. In Section 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 Section 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 Section 4. Finally, in Section 5 we will complete the proof of Theorem 1.1.

Preliminaries.
We will give an overview of the SLE κ and SLE κ (ρ) processes in Section 2.1. Next, in Section 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 Section 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 Section 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 . 2.1. 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., [Wer04b,Law05]. 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 is the swallowing time of z defined by sup{t ≥ 0 : min s∈[0,t] |g s (z) − W s | > 0}. 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]) [RS05]. 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 [RS05]. The dimension of the path is 1 + κ 8 for κ ∈ [0, 8] and 2 for κ > 8 [Bef08].
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 [LSW03, 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 [MS12a, 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 [MS12a, 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: LEMMA 2.1. Suppose that η is an SLE κ (ρ L ; ρ R ) process in H from 0 to ∞ with force points located at (x L ; x R ).
then η can hit and bounce off of (x k,R , x k+1,R ). Moreover, η ∩ (x k,R , x k+1,R ) has empty interior. PROOF 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 √ κB 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 [MS12a,Theorem 1.3]; this is explained in [MS13, 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 2 − 4] (κ > 4). The latter means that, for any time t, the outer boundary of the range of η up to time t is almost surely contained in η([t, ∞)).

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 [Wer04a]. It is shown in [SW05, Theorem 6 and 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 [SW05,Equation 9], a special case of which we will describe and make use of in Section 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 1 implies that the reweighted process almost surely does not hit (−∞, x L ). GFF. We are now going to give a brief overview of the coupling between SLE and the GFF. We refer the reader to [MS12a, Sections 1 and 2] as well as [MS12b, Section 2] for a more detailed overview. Throughout, we fix κ ∈ (0, 4) and κ = 16/κ > 4.

SLE and the
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 (2.9) h = ∑ n α n f n 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 zeroboundary GFF on D and the function F in D which is harmonic and is equal to f on ∂D. See [She07] for a detailed introduction. Let Suppose that η is an SLE κ (ρ L ; ρ R ) process in H from 0 to ∞ with force points (x L ; x R ), let (g t ) be the associated Loewner flow, W its driving function, and x 2,L −λ(1+ρ 1,L +ρ 2,L ) −λ ::: λ : Fig 2.1: 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 [MS12a, Figures 1.9 and 1.10]. The boundary data for the coupling of SLE κ (ρ) with many force points arises as the obvious generalization of the above. 2: 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 [MS12a, Figures 1.9 and 1.10]. The boundary data for the coupling of SLE κ (ρ ) with many force points arises as the obvious generalization of the above.
. Then the conditional law of (h + φ 0 )| H\K τ given K τ is equal to the law of h • f τ + φ τ . In this coupling, η is almost surely determined by h [SS10, The left (resp. right) boundary η L (resp. η R ) of η is given by the flow line of h with angle π 2 (resp. − π 2 ) starting from −i and targeted at i; these paths can be drawn if 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 . Dub09b,MS12a]. For κ ∈ (0, 4), η has the interpretation as being the flow line of the (formal) vector field e i(h+φ 0 )/χ [She10] starting from 0; we will refer to η simply as a flow line of h + φ 0 . See Figure 2.1 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, 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 [MS12a, Figures 1.9 and 1.10]. When κ = 4, η has the interpretation of being the level line of h + φ 0 [SS10]. Finally, when κ > 4, η has the interpretation of being a "tree of flow lines" which travel in the opposite direction of η [MS12a,MS13]. 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 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 [MS12a, 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 [MS12a, Theorem 1.1]. This is why the signs of the boundary data in Figure  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 Figure 2.3. 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 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 Figure 2 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 (W, V L , V R ) of an SLE κ (ρ L ; ρ R ) process with force points located at (x L ≤ 0 ≤ x R ) as n → ∞. The same likewise holds in the setting of multi-force-point SLE κ (ρ) processes.
We show in Lemma 2.3 that with positive probability, η gets within distance of γ(T) before leaving A( ).
from 0 to ∞ with force points located at (x L ; x R ) with x 1,L = 0 − and x 1,R = 0 + (possibly by taking ρ 1,q = 0 for q ∈ {L, R}). Assume that ρ 1,L , ρ 1,R > −2. Suppose that γ : [0, T] → R is any deterministic simple curve in H starting from 0 and otherwise does not hit ∂H. Fix > 0, let A( ) be the neighborhood of γ([0, T]), and define stopping times PROOF. See Figure 2.6 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 P Section 2] and then conformally mapping back, we may assume without loss of generality that ρ 1,L = ρ 1,R = 0. Let U be a Jordan domain which contains γ([0, T]) and is contained in A( ). Assume, moreover, that ∂U ∩ [x 2,L , x 2,R ] is an interval, say [y L , y R ], 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 .

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 [Law05, Section 4.7].

PROOF. See
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 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 η| (0, σ 1 ] almost surely does not hit 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 and define stopping times Then we have that PROOF. See Figure 2.8. 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.

2.3.
Radon-Nikodym Derivative. Following [Dub09a, 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. Let c = (D, z 0 , x L , x R , z ∞ ) be a configuration consisting of a Jordan domain D in C with + r + 2 marked points on ∂D. An SLE κ (ρ L ; ρ R ) process η with configuration c is given by the image of an SLE κ (ρ L ; ρ R ) process η in H under a conformal transformation ϕ taking H to D with ϕ(0) = z 0 , ϕ(∞) = z ∞ , and which takes the force points of η to those of η.
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 [LW04] for more on the Brownian loop measure). Also, The following result is proved in [Dub09a,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 (2.14) PROOF. We are first going to prove the result in the case that 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 [MS12a, Proposition 3.2]. Since η (resp. η) is almost surely determined by h (resp. h) [MS12a, 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 f (η) : Lemma 13], 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. Let ν U c,h (·) (resp. ν U c,h (·)) denote the law of h| U (resp. h| U ). Then we have that Rearranging, we see that c (· | η) (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).
LEMMA 2. 8. Assume that we have the same setup as in Lemma 2.7 with D = H, D ⊆ H, U ⊆ H bounded, and z 0 = 0. Fix ζ > 0 and suppose that the distance between U and H\ D is at least ζ, the force points of c, c in U are identical, the corresponding weights are also equal, and the force points which are outside of U are at distance at least ζ from U. There exists a constant C ≥ 1 depending on U, ζ, κ, and the weights of the force points such that Recall from (2.13) that the terms in 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 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 [LSW03, 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 whereK 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.

2.4.
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; the out-radius of D with respect to w. By the Schwarz lemma and the Koebe one-quarter theorem, As a consequence, where the right-hand inequality above holds for |ζ| ≤ 1/2. Finally, we state the Beurling estimate [Law05, 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 τ D = inf{t ≥ 0 : B(t) ∈ ∂D}. 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
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 Let G be the extension of g τ to C\(K ∪ K) which is obtained by Schwarz reflection. By (2.15), we have On the event E δ,r , we run a Brownian motion started from the midpoint of the line segment [1, η(τ )]. Then this Brownian motion has uniformly positive (though δdependent) 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 Figure 3.1). 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 .
and let η be an SLE κ (ρ L ; ρ 1,R , ρ 2,R ) process with force points (x L ; x R , 1). Let E δ,r be the event 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, 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: LEMMA 3.4. Let η be a continuous curve in H starting from 0 with continuous Loewner driving function W and let (g t ) be the corresponding family of conformal maps. For each t ≥ 0, let O L t (resp. O R t ) be the leftmost (resp. rightmost) point of g t (η([0, t])) in R. There exists a universal constant C ≥ 1 such that the following is true. Fix ϑ > 0 and let σ be the first time that η exits ϑT. Then 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. By [Law05, Remark 3.50] we have that Combining (3.10) with (3.11) and (3.12) gives (3.7). The bounds (3.8) and (3.9) are proved similarly.
Suppose C > 0; we will set its value later in the proof. For each 1 ≤ k ≤ 1 Cϑ , we let 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 where p 2 = p 2 ( 1 2 ) 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 Therefore it suffices to prove 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 Figure 3.2). 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 : is bounded from below by a positive universal constant times the probability that a Brownian motion starting from z ϑ 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 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 V L t , V R t denote the evoltuion of x L , x R under g t , respectively. For u ≥ 0, define T L u = inf{t ≥ 0 : W t − V L t = u} (as in the statement of Lemma 3.5). Then it is sufficient to prove P[σ 1 < T L 0 ] ≤ α+o(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 u ≥ 0, define τ L u = inf{t ≥ 0 : M L t = u}. Note that τ L 0 = T L 0 . 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 Since ρ 1,R + ρ 2,R > κ 2 − 4 it follows that ρ L < κ 2 − 2. 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 By Lemma 3.5, we know that 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 Combining (3.17) with (3.21) and (3.22) we get that P[σ 1 < T L 0 ] ≤ α+o(1) , as desired.
Recall that (see for example [MP10, 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 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].

3). Finally, we let
The following is the main input into the proof of the lower bound.
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 If, moreover, y ≥ 1 and 1 2 m+2 < |x − y| ≤ 1 2 m+1 , then we have that In each of the above, the constants in depend only on δ, κ and ρ.
Therefore Lemma 2.8 implies there exists C 1 ≥ 1 so that This proves (3.26) in the case that n = m + 1. We now suppose that n ≥ m + 2. 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 Figure 3.5). 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 x j | [0,σ x j ] for j = m + 2, . . . , n − 1 is independent of η x k | [0,σ x k ] for k = 1, . . . , m (see Figure 3.5). 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 where the constants depend only on δ, κ, and ρ. 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 Figure 3.5). 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 x m+1 | [0,σ x m+1 ) before leaving B(x, 1 2 m )\B(x, m+2 ).
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, (Lemma 3.8 and Lemma 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 |z − w| β 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 − 1 2 ) n + 1 be the center of the jth such interval for j = 1, . . . , −1 n . Let C n be the subset of D n = {z j,n : j = 1, . . . , −1 n } for which E n (z) occurs. Let I n (z) = [z − n 2 , z + n 2 ] be the interval with center z and length n . Finally, we let C = k≥1 n≥k z∈C n I n (z).
It is easy to see that C ⊆ η R + .
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 Figure 2.5 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. As explained in Figure 2.5, the right boundary of η is equal to the flow line η R of h with angle − π 2 starting from −i. In particular, η R is an SLE κ ( κ 2 − 2; κ − 4 + κ 4 ρ ) process with force points ((−i) − ; (−i) + ) where κ = 16 κ ∈ (0, 4). The intersection of η with the counterlcockwise segment S of ∂ ([−1, 1] 2 ) from −i to i coincides with η R ∩ S. 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 Section 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 Section 4.2 we will prove the one point estimate which we will use in Section 4.3 to prove the upper bound. Finally in Section 4.4 we will complete the proof by establishing the lower bound.

Derivative estimate.
Recall from Section 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 [VL09, 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 and ξ = ξ(r) = r 2 8 κ.
(In the notation of [VL09], a = 2/κ.) Then we have that [VL09, Proposition 6.1]: This martingale also appears in [SW05, 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 , and z ∈ H such that arg(z) ∈ (δ, π − δ). Let P be the law of η weighted by M. We have that, 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 PROOF. Note that (4.5) and (4.6) are proved in [VL09, Equation (6.9)], so we will not repeat the arguments here. Following [VL09], we define the radial parametrization (i.e., by log conformal radius) u(t) by Υ t = Υ u(t) = e −4t/κ and write η(t) = η(u(t)) and Θ t = Θ u(t) . Then Θ t satisfies the SDE (see [VL09, Section 6.3]) where W is a P -Brownian motion. The process Θ almost surely does not hit {0, π} (see [Law05,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 [Law05,Lemma 1.28]). Moreover, as t → ∞, the law of Θ 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 Θ| [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 Θ 0 , also stopped upon leaving ( δ 2 , π − δ 2 ). Fix 0 ≤ t ≤ T. Then a Brownian motion starting from Θ 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 Θ 1 , Θ 2 are solutions of (4.9) where Θ 1 0 = δ and Θ 2 0 = π − δ. We assume further that the Brownian motions driving Θ, Θ 1 , and Θ 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 ( Θ 1 , Θ, Θ 2 ) such that Θ 1 t ≤ Θ t ≤ Θ 2 t for all t ≥ 0 almost surely. Note that after Θ 1 first hits Θ 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 Θ 1 t and the stationary distribution is at most With this particular choice of T, we have that This proves (4.7). For (4.8), note that, under P , η has the same law as a radial SLE κ (ρ) in H from 0 to z with a single boundary force point located at ∞ of weight ρ = κ − 6 − rκ ≥ κ 2 − 2 (see [SW05, Theorem 3 and Theorem 6]). Define The endpoint continuity of the radial SLE κ (ρ) processes with ρ > −2 [MS13,Theorem 1.12] implies that P [ σ R < ∞] → 0 as R → ∞, as desired.
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.
Fix a constant C > 1 and suppose that ζ is a stopping time for η such that 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 From (4.7), we know that P [Θ 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 On the left side, η 1 (resp. η 2 ) is a flow line of a GFF on H with the indicated boundary data with angle 0 (resp. θ ∈ (π − 2λ/χ, 0)) starting from x 1 (resp. x 2 > x 1 ). Note that η 1 (resp. η 2 ) is an SLE κ (−θχ/λ) (resp. SLE κ (2, −θχ/λ − 2)) process. The force point for η 1 is located at x 2 and the force points for η 2 are located at x 1 and x − 2 . By Figure 2.4, the conditional law of η 2 given η 1 drawn up to any stopping time is also an SLE κ (2, −θχ/λ − 2) process. Shown is the event G δ (z) that η 1 hits ∂B(z, ), say for the first time at ζ 1 , before exiting B(0, R 0 ) where R 0 > 0 is a large, fixed constant, the harmonic measure of the left (resp. right) side of η 1 stopped upon hitting ∂B(z, ) is not too small, and that η 2 also hits ∂B(z, ). We estimate the probability of G δ (z) by combining Lemma 4.2 with Theorem 3.1.
where the flow line with angle θ stays to the right of the flow line with angle 0 [MS12a, Theorem 1.5]. Let Let h be a GFF on H with boundary data as illustrated in Figure 4.1. That is, Let η 1 (resp. η 2 ) be the flow line of h starting from x 1 (resp. x 2 ) with angle 0 (resp. θ). Fix δ ∈ (0, π 2 ) and let z ∈ D ∩ H with arg(z) ∈ (δ, π − δ). For i = 1, 2, let ζ i be the first time that η i hits ∂B(z, ) and let Θ 1 t be the process as in Lemma 4.2 for η 1 .
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 [−20x 2 , 20x 2 ] 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 Θ 1 / ∈ (δ, π − δ) for some δ > 0. This is the purpose of the following estimate.
For each n ≥ n z , on the event {ζ n < ∞}, let ϕ n : H\η([0, ζ n ]) → H be the unique conformal map with ϕ n (η(ζ n )) = 0, ϕ n (∞) = ∞, and satisfies Im(ϕ n (z)) = 1. Note that ϕ n (B(z, 2 −n−3) )) ⊆ B(ϕ n (z), 1) by [Law05, Corollary 3.25]. Therefore it follows from (4.18) that Iterating (4.19) and taking p(δ) = (q(δ)) 1/3 proves the lemma. 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 Figure 4.3 for an illustration.
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 Suppose that Q ∈ D m and suppose that n ∈ N with n ≤ m. Then the function Q → R given by w → Θ w ζ w,n is positive and harmonic. Consequently, it follows from the Harnack inequality [Law05, 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 Kδ z(Q),m occurs. Thus letting E δ Q,m = ∪ w∈Q E δ w,m we have that ,m ] 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 4p(Kδ) < 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. PROPOSITION 4.6. Suppose that h is a GFF on H with piecewise constant boundary conditions which change values a finite number of times. Let η 1 (resp. η 2 ) be the flow line of h starting from x 1 = 0 (resp. x 2 > 0) with angle 0 (resp. θ ∈ (π − 2λ/χ, 0)). We have that where A is as in (4.13). PROOF. We are going to prove the proposition assuming that the boundary data is as in Lemma 4.3. This suffices by absolute continuity for GFFs. Fix 0 < < δ 2 < δ < π 4 . For each t > 0, we let H 1 t be the unbounded 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 Since the above holds for every n, we therefore have that H 2−A+2ξ (I ,δ ) = 0 almost surely. Since ξ > 0 was arbitrary, we have that dim H (I ,δ ) ≤ 2 − A almost surely, as desired. 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. Figure 4.7.

4.4.2.
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 with angle 0 on its right side. We will now use the events introduced in Section 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 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 Figure 4.1 and Figure 4.2 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.
(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 Lemma 2.3 and Lemma 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). 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 (4.28) ϕ j,k z = ϕ j+1,z • · · · • ϕ k,z for each 0 ≤ j ≤ k and ϕ k z = ϕ 0,k z .
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 hits ∂B(i, 10e − β ) before leaving B(i, e 2β ) is |g (i)e − β | α+O β (1)o β (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 OF PROPOSITION 4.9. This follows by combining Lemma 4.10 with Lemma 4.12.
LEMMA 4. 13. Fix δ ∈ (0, π 2 ) and z, w ∈ D ∩ H distinct with arg(z), arg(w) ∈ (δ, π − δ) and let m be the smallest integer such that V m−1 (z) ∩ V m−1 (w) = ∅. Let P w be the event that η 1,1 hits V m (w) before hitting V m (z). There exists β 0 > 1 such that for every β > β 2 > β ≥ β 0 we have that 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.
for all k ≥ m.

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 P[E m+1,n | E m+1 , F k (w)]1 E k (w),P w P[E n−m−1 ]1 E k (w),P w (4.34) 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. Thus by conformally mapping back (see Figure 4.7) 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.
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 exists 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 integer such that V m−1 (z) ∩ V m−1 (w) = ∅. Then P[E n (z), E n (w)] ≤ Ce β(1+ )mA P[E n (z)]P[E n (w)] for all n ≥ n 0 . 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 (z) and let P z be the event in which the roles of z and w are swapped. We have that We are going to bound the first summand; the second is bounded analogously. We have,  (possibly increasing β 0 ). The same likewise holds when we swap the roles of P w and P z . Combining (4.35)-(4.38) gives the result.
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 C ⊆ η 1 ∩ η 2 ∩ H.
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.

5.
Proof of Theorem 1.1. We will first work towards proving (1.1) for κ ∈ (4, 8); let κ = 16 κ ∈ (2, 4). It suffices to compute the almost sure Hausdorff dimension of the double points of the chordal SLE κ ( κ 2 − 4; κ 2 − 4) processes. Indeed, this follows since the conditional law of an SLE κ process given its left and right boundaries is independently that of an SLE κ ( κ 2 − 4; κ 2 − 4) in each of the bubbles which lie between these boundaries (recall Figure 2.5). In order to establish this result, we are going to make use of the path decomposition developed in [MS12c] which was used to prove the reversibility of SLE κ for κ ∈ (4, 8). This, in turn, makes use of the duality results established in [MS12a, Section 7]. For the convenience of the reader, we are going to review the path decomposition here.
Throughout, we suppose that h is a GFF on the horizontal strip T = −λ+ π 2 χ λ− 3π 2 χ w = η 1 z (τ 1 z ) is an SLE κ (− κ 2 ; κ − 4) process from 0 to ∞ (Figure 2.4). Similarly, η 2 0 is an SLE κ (− κ 2 ; κ 2 − 2) process in H from 0 to ∞ (Figure 2.1) 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 (Figure 2.4). In particular, by the main result of [MS12b], 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 Figure 5.2. Consequently, we can use Theorem 1.5 to compute the almost sure dimension of the intersection of the latter.
intersection of two GFF flow lines with an angle gap of θ double (recall (1.10)) as given in Theorem 1.5. Figure 5.2 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 Figure 5.2. 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 [MS12c, Lemma 3.3]. (The κ − 4 force point lies between η 1 0 and η 2 0 .) By the main result of [MS12b], 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 Figure 5.3, it consequently follows from Theorem 1.5 that (5.1) dim H (η 1 0 ∩ η 2 0 ) = 2 − (12 − κ )(4 + κ ) 8κ almost surely since this is the almost sure dimension of η 1 0 ∩ η 2 0 (using the notation of Fig-ure 5.3). Thus to complete the proof, we just have to argue that dim H (P ∩ (z)) is also given by this value. 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. We consequently have that the image under ϕ of the region between η 1 1 and η 2 1 is equal in distribution to P(z) as described before the lemma statement. Since dim H (η 1 1 ∩ η 2 1 ) is also almost surely given by the value in (5.1), the desired result follows. where z 1 ∈ ∂T is fixed. The conditional law of η given P(z 1 ) is independently that of an SLE κ ( κ 2 − 4; κ 2 − 4) in each of the components C of T\P(z 1 ) starting from the first point of C hit by η and exiting at the last. Fix z 2 on the boundary of a component C of T\P(z 1 ). Then we can consequently form the set P(z 2 ) which describes the interface between the set of points that η , viewed as a path in C, hits before and after hitting z 2 . The intersection of the left and right boundaries of P(z 2 ) consists of double points of η . Moreover, the conditional law of η given both P(z 1 ) and P(z 2 ) is independently that of an SLE κ ( κ 2 − 4; κ 2 − 4) in each of the components of T\(P(z 1 ) ∪ P(z 2 )). Consequently, we can iterate this procedure to eventually explore the entire trajectory of η (and, as we will explain in Lemma 5.2, the double points of η ). We will use this in Lemma 5.2 to reduce the double point dimension to computing the intersection dimension of GFF flow lines with an angle gap of θ double (recall (1.10)).

PROOF. See
Let D be the set of double points of η . To complete the proof of Theorem 1.1, we will show that every double point of η is in fact in some P ∩ (z). To this end, we explore the trajectory of η as follows. Let (d j ) j∈N be a sequence that traverses N × N in diagonal order, i.e. d 1 = (1, 1), d 2 = (1, 2), d 3 = (2, 1), etc. Let (z 1,k ) k∈N be a countable dense subset of ∂T, and set z 1 = PROOF OF THEOREM 1.1 FOR κ ∈ (4, 8). Lemma 5.1 and Lemma 5.2 together imply that dim(D) = 2 − (12 − κ )(4 + κ )/(8κ ) almost surely, as desired.
We finish by proving Theorem 1.1 for κ ≥ 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 [RS05]. 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 [MS13, Theorem 1.1 and Theorem 1.13] and [Bef08] 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)} < ∞.
Clearly, ω ∈ γ(z k ). This completes the proof for κ ≥ 8. 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.