Boundary Arm Exponents for SLE$(\kappa)$

We derive boundary arm exponents for SLE. Combining with the convergence of critical lattice models to SLE, these exponents would give the alternating half-plane arm exponents for the corresponding lattice models.


Introduction
Schramm-Loewner evolution (SLE) was introduced by Oded Schramm [Sch00] as the candidates for the scaling limits of interfaces in 2D critical lattice models.It is a one-parameter family of random fractal curves in simply connected domains from one boundary point to another boundary point, which is indexed by a positive real κ.Since its introduction, it has been proved to be the limits of several lattice models: SLE 2 is the limit of Loop Erased Random Walk and SLE 8 is the limit of the Peano curve of Uniform Spanning Tree [LSW04], SLE 3 is the limit of the interface in critical Ising model and SLE 16/3 is the limit of the interface in FK-Ising model [CDCH + 14], SLE 4 is the limit of the level line of discrete Gaussian Free Field [SS09] and SLE 6 is the limit of the interface in critical Percolation [Smi01].
In the study of lattice models, arm exponents play an important role.Take percolation for instance, Kesten has shown that [Kes87] in order to understand the behavior of percolation near its critical point, it is sufficient to study what happens at the critical point, and many results would follow from the existence and values of the arm exponents.To be more precise, consider critical percolation with fixed mesh equal to 1, and for n ≥ 2, consider the the event E n (z, r, R) that there exist n disjoint crossings of the annulus A z (r, R) := {w ∈ C : r < |w − z| < R}, not all of the same color.People would like to understand the decaying of the probability of E n (z, r, R) as R → ∞.It turns out that this probability decays like a power in R, and the exponent is called plane arm exponents.There are another related quantities, called half-plane arm exponents.In this case, consider critical percolation in the upper-half plane H, and for n ≥ 1, x ∈ R, define H n (x, r, R) to be the event that there exist n disjoint crossings of the semi-annulus A + x (r, R) := {w ∈ H : r < |w − x| < R}.After the identification between SLE 6 and the limit of critical percolation on triangular lattice [Smi01], one could derive these exponents via the corresponding arm exponents for SLE 6 [SW01]: where α n := (n 2 − 1)/12, α + n := n(n + 1)/6.In this paper, we derive boundary arm exponents for SLE κ .Combining with the identification between the limit of critical lattice model and SLE curves, these exponents for SLE would imply the arm exponents for the corresponding lattice models.
Fix κ > 4 and let η be an SLE κ in H from 0 to ∞. Suppose that y ≤ 0 < ε ≤ x and let T be the first time that η swallows the point x which is almost surely finite when κ > 4. We first define the crossing event H 2n−1 (resp.Ĥ2n ) that η crosses between the ball B(x, ε) and the half-infinite line (−∞, y) at least 2n − 1 times (resp.at least 2n times) for n ≥ 1.To be precise with the definition, we need to introduce a sequence of stopping times.Set τ 0 = σ 0 = 0. Let τ 1 be the first time that η hits the ball B(x, ε) and let σ 1 be the first time after τ 1 that η hits (−∞, y).For n ≥ 1, let τ n be the first time after σ n−1 that η hits the connected component of ∂ B(x, ε) \ η[0, σ n−1 ] containing x + ε and let σ n be the first time after τ n that η hits (−∞, y).Define H 2n−1 (ε, x, y) to be the event that {τ n < T }.Define Ĥ2n (ε, x, y) to be the event that {σ n < T }.In the definition of H 2n−1 (ε, x, y) and Ĥ2n (ε, x, y), we are particular interested in the case when x is large.Roughly speaking, the event H 2n−1 (ε, x, y) means that η makes at least (2n − 1) crossings between B(x, ε) and (−∞, y).Imagine that η is the interface in the discrete model, then H 2n−1 (ε, x, y) interprets the event that there are 2n − 1 arms going from B(x, ε) to far away place.The event Ĥ2n (ε, x, y) means that η makes at least 2n crossings between B(x, ε) and (−∞, y).Imagine that η is the interface in the discrete model, then Ĥ2n (ε, x, y) interprets the event that there are 2n arms going from B(x, ε) to far away place.See Figure 1.1(a).Next, we define the crossing event H 2n (resp.Ĥ2n+1 ) that η crosses between the half-infinite line (−∞, y) and the ball B(x, ε) at least 2n times (resp.at least 2n + 1 times) for n ≥ 0. Set τ 0 = σ 0 = 0. Let σ 1 be the first time that η hits (−∞, y) and τ 1 be the first time after σ 1 that η hits the connected component of ∂ B(x, ε) \ η[0, σ 1 ] containing x + ε.For n ≥ 1, let σ n be the first time after τ n−1 that η hits (−∞, y) and τ n be the first time after σ n that η hits the connected component of ∂ B(x, ε) \ η[0, σ n ] containing x + ε.Define H 2n (ε, x, y) to be the event that {τ n < T }.Define Ĥ2n+1 (ε, x, y) to be the event that {σ n+1 < T }.In the definition of H 2n (ε, x, y) and Ĥ2n+1 (ε, x, y).we are interested in the case when x is of the same size as ε and y is large.Roughly speaking, the event H 2n (ε, x, y) means that η makes at least 2n crossings between (−∞, y) and B(x, ε).Imagine that η is the interface in the discrete model, then H 2n (ε, x, y) interprets the event that there are 2n arms going from B(x, ε) to far away place.The event Ĥ2n+1 (ε, x, y) means that η makes at least 2n + 1 crossings between (−∞, y) and B(x, ε).Imagine that η is the interface in the discrete model, then Ĥ2n+1 (ε, x, y) interprets the event that there are 2n + 1 arms going from B(x, ε) to far away place.See Figure 1.1(b).
Note that in the definition of H 2n−1 and Ĥ2n , we start from τ 1 and In the definition of H 2n and Ĥ2n+1 , we start from σ 1 and The two sequences of stopping times are defined in different ways.Readers may wander why we do not define the events using the same sequence of stopping times.We realize that the definition using the same sequence of stopping times causes ambiguity.Therefore, we decide to define these events in the above way.The advantages of the current definition will become clear in the proofs.
We define the arm exponents as follows.Set α + 0 = 0.For n ≥ 1 and κ ∈ (0, 8), define For n ≥ 1 and κ ≥ 8, define The crossing events H 2n−1 (ε, x, y) and H 2n (ε, x, y) are defined as above.Then, for any y ≤ 0 < ε ≤ x and n ≥ 1, we have where the constants in depend only on κ and n.In particular, fix some δ > 0, we have where the constants in depend only on κ, n and δ .
By a similar proof, we could obtain a similar result as Theorem 1.1 for SLE κ (ρ) curve in the case that x coincides with the force point.The exponents and a complete proof can be found in [Wu16b, Section 3], where the conditions are loosen such that the force point may not be equal to x.One may also study the arm exponents for κ ∈ (0, 4].Whereas, when κ ≤ 4, the SLE curve does not touch the boundary, thus the above definition of the crossing events is not proper for κ ≤ 4. In Section 4, we have Theorem 4.4 for the crossing events between a small circle and a half-infinite strip, where the arm exponents are defined in the same way as in (1.1).The proof of Theorem 4.4 also works for SLE κ (ρ) when x coincides with the force point.
It is worthwhile to spend some more words on the relation between α + n and α+ n .In fact, we can also define the crossing events Ĥn (ε, x, y) for κ ∈ [0, 4] and κ ≥ 8.When κ ≤ 4, the SLE curve does not touch the boundary, thus the exponent α+ n coincides with α + n−1 .When κ ≥ 8, the SLE curve is space-filling, thus the exponent α+ n coincides with α + n+1 .Whereas, when κ ∈ (4, 8), the exponent α+ n is distinct from α + n in general.In terms of discrete model, both α + n and α+ n interpret the boundary n-arm exponents, but their boundary conditions are different.It is explained in [SW01] that combining the following three facts would imply the arm exponents for the discrete model: (1) Identification between SLE κ and the limit of the interface in critical lattice model; (2) The arm exponents of SLE κ ; (3) Crossing probabilities enjoy (approximate) multiplicativity property.For critical Ising and FK-Ising model on Z 2 with Dobrushin boundary conditions, the convergence to SLE 3 and SLE 16/3 respectively is derived in [CS12, CDCH + 14], and the multiplicativity is derived in [CDCH13].Therefore, we could derive the arm exponents for these two models.See more details in [Wu16b,Wu16a].Moreover, the formula of α + 2n−1 in (1.1) was predicted by KPZ in [Dup03,Equations (11.42)

Preliminaries
Notations.We denote by f g if f /g is bounded from above by universal finite constants, by f g if f /g is bounded from below by universal positive constants, and by f g if f g and f g.
Let Ω be an open set and let V 1 ,V 2 be two sets such that V 1 ∩ Ω = / 0 and V 2 ∩ Ω = / 0. We denote the extremal distance between V 1 and V 2 in Ω by d Ω (V 1 ,V 2 ), see [Ahl10, Section 4] for the definition.

H-hull and Loewner chain
We call a compact subset K of H an H-hull if H \ K is simple connected.Riemann's Mapping Theorem asserts that there exists a unique conformal map We call such g K the conformal map from H \ K onto H normalized at ∞.The limit hcap(K) := lim |z|→∞ z(g K (z) − z) exists and is called the half-plane capacity of K.
Lemma 2.1.Fix x > 0 and ε > 0. Let K be an H-hull and let g K be the conformal map from H \ K onto H normalized at ∞. Assume that x > max(K ∩ R).
Denote by γ the connected component of H ∩ (∂ B(x, ε) \ K) whose closure contains x + ε.Then g K (γ) is contained in the ball with center g K (x + ε) and radius 3(g K (x + 3ε) − g K (x + ε)).Hence g K (γ) is also contained in the ball with center g K (x + 3ε) and radius 8εg K (x + 3ε). (2.1) We will prove (2.1) by estimates on the extremal distance: By the conformal invariance and the comparison principle [Ahl10, Section 4.3], we can obtain the following lower bound.
On the other hand, we will give an upper bound.Recall a fact for extremal distance: for x < y and r > 0, the extremal distance in H between [y, ∞) and a connected set S ⊂ H with x ∈ S ⊂ B(x, r) is maximized when , by the above fact, we have the following upper bound.
Combining the lower bound with the upper bound, we have This implies (2.1) and completes the proof.
Lemma 2.2.Fix z ∈ H and ε > 0. Let K be an H-hull and let g K be the conformal map from H \ K onto H normalized at ∞. Assume that dist(K, z) ≥ 16ε.
Then g K (B(z, ε)) is contained in the ball with center g K (z) and radius 4ε|g K (z)|.
Proof.By Koebe 1/4 theorem, we know that dist(g Therefore h(B(g K (z), d)) contains the ball B(z, ε), and this implies that B(g K (z), d) contains the ball g K (B(z, ε)) as desired.
Loewner chain is a collection of H-hulls (K t ,t ≥ 0) associated with the family of conformal maps (g t ,t ≥ 0) obtained by solving the Loewner equation: for each z ∈ H, where (W t ,t ≥ 0) is a one-dimensional continuous function which we call the driving function.Let T z be the swallowing time of z defined as sup{t ≥ 0 : Here we spend some words about the evolution of a point y ∈ R under g t .We assume y ≤ 0, the case of y ≥ 0 can be analyzed similarly.There are two possibilities: if y is not swallowed by K t , then we define Y t = g t (y); if y is swallowed by K t , then we define Y t to the be image of the leftmost of point of K t ∩ R under g t .The process Y t is decreasing in t, and it is uniquely characterized by the following equation: In this paper, we may write g t (y) for the process Y t .Consider two points x ≥ 0 ≥ y in R. By the above fact, we have Therefore, the quantity g t (x) − g t (y) is increasing in t.We will use this fact in the paper without reference.

SLE processes
An SLE κ is the random Loewner chain (K t ,t ≥ 0) driven by W t = √ κB t where (B t ,t ≥ 0) is a standard onedimensional Brownian motion.In [RS05], the authors prove that (K t ,t ≥ 0) is almost surely generated by a continuous transient curve, i.e. there almost surely exists a continuous curve η such that for each t ≥ 0, H t is the unbounded connected component of H\η[0,t] and that lim t→∞ |η(t)| = ∞.
We can define an SLE κ (ρ L ; ρ R ) process with two force points (x L ; x R ) where x L ≤ 0 ≤ x R .It is the Loewner chain driven by W t which is the solution to the following systems of SDEs: The solution exists up to the first time that W hits V L or V R .When ρ L > −2 and ρ R > −2, the solution exists for all times t ≥ 0, and the corresponding Loewner chain is almost surely generated by a continuous curve which is almost surely transient ([MS12, Section 2]).There are two special values of ρ: κ/2 − 2 and κ/2 − 4. When ρ R ≥ κ/2 − 2, then the curve will never hits [x R , ∞).When ρ R ≤ κ/2 − 4, then the curve will almost surely accumulates at x R at finite time.See [Dub09,Lemma 15].
From Girsanov Theorem, it follows that the law of an SLE κ (ρ L ; ρ R ) process can be constructed by reweighting the law of an ordinary SLE κ .
Then M is a local martingale for SLE κ and the law of SLE κ weighted by M (up to the first time that W hits one of the force points) is equal to the law of SLE κ (ρ L ; ρ R ) with force points (x L ; x R ).
Proof.Since the quantity g t (x) − g t (y) is increasing in t, we have g T (x) − g T (y) ≥ (x − y).This implies the upper bound.We only need to show the lower bound.To this end, we will compare η with SLE κ (ν) with force point x − y and show that the law of (g T (x) − g T (y))/(x − y) is stochastically dominated by a random variable whose law depends only κ, ν.By the scaling invariance of SLE κ (ν), we may assume x − y = 1.Let η be an SLE κ (ν) with force point 1, and define W , gt , T accordingly.Define Ṽt to be the image of the leftmost point of η[0,t] ∩ R under gt .Set Define the stopping time τ = inf{t : Jt = −y}.Note that J0 = 0, J T = 1 and J is continuous, we have that 0 ≤ τ ≤ T .
Given η[0, τ], the process ( η(t + τ), 0 ≤ t ≤ T − τ), under the map has the same law as (η(t), 0 ≤ t ≤ T ) after a linear time-change.Therefore, given η[0, τ], we have Since gτ (1) − Ṽτ ≥ 1, we may conclude that the quantity (g T (x) − g T (y)) is stochastically dominated from above by ( g T (1) − Ṽ T ).To complete the proof, it is sufficient to show where P denotes the law of SLE κ (ν) with force point 1.Define the event It is clear that P[ F] is strictly positive and depends only on κ and ν, thus This implies (2.3) and completes the proof.
Proof.Since the quantity g t (x) − g t (y) is increasing in t, we have g σ (x) − g σ (y) ≥ (x − y).This implies the upper bound.We only need to show the lower bound.We may assume that x − y = 1.We first argue that The proof of (2.4) is similar to the proof of Lemma 2.4.Let η be an SLE κ (ν) with force point 0 + .Define W , g accordingly and let σ be the first time that η hits (−∞, −1).Let Ṽt be the evolution of the force point.Define , τ := inf{t : Jt = x}.

Estimate on the derivative
Proposition 3.1.Fix κ > 0 and let η be an SLE κ in H from 0 to ∞.Let O t be the image of the rightmost point of For λ ≥ 0, define Then we have where the constants in depend only on κ and λ , b.
Attention that, in Proposition 3.1, we use the stopping time τε instead of τ ε which is defined to be the first time that η hits B(1, ε).Due to Koebe 1/4 thoerem, these two times are very close: Due to technical reason, we only prove the conclusion in Proposition 3.1 for the time τε , but this is sufficient for our purpose later in the paper.Lemma 3.2.Fix κ > 0 and ν ≤ κ/2−4.Let η be an SLE κ (ν) in H from 0 to ∞ with force point 1.Denote by W the driving function, V the evolution of the force point.Let O t be the image of the rightmost point of K t ∩R under g t .Set ϒ t = (g t (1) − O t )/g t (1) and σ (s) = inf{t : where the constants in depend only on κ, ν, β .
Proof.Since 0 ≤ J t ≤ 1, we only need to show the upper bounds.Define X t = V t −W t .We know that where B is a standard 1-dimensional Brownian motion.By Itô's formula, we have that .
Proof of Proposition 3.1.Let O t be the image of the rightmost point of η[0,t] ∩ R under g t .Define Then M is a local martingale for η, and from Lemma 2.3, the law of η weighted by M is the law of SLE κ (ν) with force point 1.Set β = u 1 (λ ) + λ − b.Then we have where P * is the law of SLE κ (ν) with force point x and η * , J * , τ * ε , T * 0 are defined accordingly, and the last relation is due to (3.3).
Remark 3.3.Fix κ > 0 and let η be an SLE κ .For x > ε > 0, let u 1 (λ ) and b be as in Proposition 3.1.By the scaling invariance of SLE, we have where the constants in depend only on κ, and λ , b. Taking λ = b = 0, we have This implies that (1.3) holds for n = 1.

From 2n − 1 to 2n
Lemma 3.4.Fix κ > 4 and let η be an SLE κ .For y < 0 < x, define where c is the constant decided in Lemma 2.5.For λ ≥ 0, define Then we have, for λ ≥ 0 and b ≤ u 2 (λ ), where the constants in and depend only on κ and λ , b. Proof.Define Then M is a local martingale for η and the law of η weighted by M is the law of SLE κ (ν) with force point x.By the definition of u 2 , we can also write .
Proof of Lemma 3.6, Upper Bound.Let η be an SLE κ and define We stop the curve at time σ .Let η be the image of η[σ , ∞) under the centered comformal map f := g σ − W σ .Then η is an SLE κ .Define H2n−1 for η.
• By Lemma 2.1, we know that f (γ) is contained in the ball with center f (x + 3ε) and radius 8ε f (x + 3ε).
Combining these two facts, we know that, given η[0, σ ] with σ < T , the event H 2n (ε, x, y) implies the event H2n−1 (8ε f (x + 3ε), f (x + 3ε), 0).If f (x + 3ε) ≥ 8ε f (x + 3ε), by the assumption hypothesis, we have the above upper bound is trivially true.Therefore, the above upper bound always holds.Then To apply Lemma 3.4, we only need to note that T is the first time that η swallows x which happens before the first time that η swallows x + 3ε.Note further that Thus, by Lemma 3.4, we have This completes the proof of the upper bound.
Proof of Lemma 3.6, Lower Bound.Let η be an SLE κ and assume the same notations as in the proof of the upper bound.Define F = {dist(η[0, σ ], x) ≥ cε}, where c is the constant decided in Lemma 2.5.We stop the curve at time σ .Let η be the image of η[σ , ∞) under the centered comformal map f := g σ − W σ .Then η is an SLE κ .Define H2n−1 for η.
• On the event F, by Koebe 1/4 Theorem, we know that f (B(x, ε)) contains the ball with center f (x) and radius c f (x)ε/4.
Proof of Lemma 3.7, Upper Bound.If ε ≤ x ≤ 64ε, by the assumption hypothesis we have which gives the upper bound in (1.3) for 2n + 1.
In the following, we assume that x > 64ε.Let η be an SLE κ .Define T to be the first time that η swallows x.For ε > 0, let τ ε be the first time that η hits B(x, ε).Define O t to be the image of the rightmost point of η[0,t] ∩ R under g t .Define τε = inf{t : We stop the curve at time τ64ε .Let η be the image of η[ τ64ε , ∞) under the centered conformal map f := g τ64ε −W τ64ε .Then η is an SLE κ .Define the event H2n for η.
Given η[0, τ64ε ], consider the event H 2n+1 (ε, x, y).We wish to control the image of the ball B(x, ε) and the image of the half-infinite line (−∞, y) under f .We have the following observations.
• At time τ64ε , there are two possibilities for the image of y under f : if y is not swallowed by η[0, τ64ε ], then f (y) = g τ64ε (y) −W τ64ε is the image of y under f ; if y is swallowed by η[0, τ64ε ], then the image of y under f is the image of leftmost point of η[0, τ64ε ] ∩ R under f , in this case, we still write f (y) = g τ64ε (y) −W τ64ε as explained in Section 2.
Combining these two facts, we know that, given η[0, τ64ε ], H 2n+1 (ε, x, y) implies H2n (4 f (x)ε, f (x), f (y)).By the assumption hypothesis, we have For fixed x and y, the quantity g t (x) − g t (y) is increasing in t, thus g t (x) − g t (y) ≥ x − y.Plugging in the above inequality, we have By Proposition 3.1 and (3.4), we have which completes the proof.
Proof of Lemma 3.7, Lower Bound.Let η be an SLE κ .Define T to be the first time that η swallows x.For ε > 0, let τ ε be the first time that η hits B(x, ε).We stop the curve at time τ ε .Let η be the image of η[τ ε , ∞) under the centered conformal map f := g τ ε −W τ ε .Then η is an SLE κ .Define the event H2n for η.
Given η[0, τ ε ], consider the event H 2n+1 (ε, x, y).We wish to control the image of the ball B(x, ε) and the image of the half-infinite line (−∞, y) under f .We have the following observations.

Proof of Theorems 1.1 and 1.2
Proof of Theorem 1.1.Combining Remark 3.3 and Lemmas 3.7 and 3.6 implies the conclusion.
Proof of Theorem 1.2.We have the following observations.
• By the same arguments in Section 3.3, we could prove that, assume (1.6) holds for 2n − 1 with n ≥ 1, then (1.7) holds for 2n where (3.8) should be replaced by • By the same arguments in Section 3.2, we could prove that, assume (1.7) holds for 2n with n ≥ 1, then (1.6) holds for 2n + 1 where (3.7) should be replaced by Combining these three facts, we obtain the conclusion.

Definitions and Statements
In this section, we assume κ ∈ (0, 4], let η be a chordal SLE κ curve, and let g t be the corresponding Loewner maps.Since η does not hit the boundary other than its end points, H n and Ĥn defined in Section 1 are empty sets.So we need to modify their definitions.For y ∈ R and r > 0, we define half strips: and write L ± y = L ± y;π .A crosscut in a domain D is an open simple curve in D, whose end points approach boundary points of D. Suppose S is a relatively closed subset of H such that ∂ S ∩ H is a crosscut of H. Then we use ∂ + H S (resp.∂ − H S) to denote the curve ∂ S ∩ H oriented so that S lies to the left (resp.right) of the curve.For example, ∂ − H L − y;r is from y to ∞; and for x ∈ R, ∂ + H B(x, r) is from x − r to x + r.Let ξ j : [0, T j ] → C, j = −1, 1, and η : [0, T ) → C be three continuous curves.For j = −1, 1, define increasing functions R j (t) = max({0} ∪ {s ∈ [0, T j ] : ξ j (s) ∈ η([0,t])}) for t ∈ [0, T ).Let τ 0 = 0.After τ n is defined for some n ≥ 0, we define , where we set inf / 0 = ∞ by convention, and if any τ n 0 = ∞, then τ n = ∞ for all n ≥ n 0 .Definition 4.1.If τ n 0 < ∞ for some n 0 ∈ N, then we say that η makes (at least) n 0 well-oriented (ξ −1 , ξ 1 )-crossings.
Remark 4.2.The above name comes from the fact that the orientation-preserving reparametrizations of ξ 1 , ξ −1 , η do not affect the event.
(ii) For any R > 0 and n ∈ N, there is a constant C n,R depending only on κ, n, R such that Remark 4.5.Using (4.1), we see that, if x − y ≥ 12n and 2 5n ε < x − y, then So we get the same upper bound as in the case κ > 4.

Comparison principle for well-oriented crossings
Let D be a simply connected domain.We say that η : [0, T ) → D is a non-self-crossing curve in D if η(0) ∈ ∂ D, and for any t 0 ≥ 0, there is a unique connected component D t 0 of D \ η[0,t 0 ] such that η(t 0 + •) is the image of a continuous curve in U under a continuous map from U onto D t 0 , which is an extension of a conformal map from U onto D t 0 .For example, an SLE curve is almost surely a non-self-crossing curve.
Remark 4.7.The assumption that η is non-self-crossing forces η(τ n + •) to stay in the closure of the remaining domain D τ n .We need assumption (iii) to prevent η(τ n + •) to sneak into the region bounded by the crosscut ξ(−1) n+1 (( R(−1) n+1 (τ n ), 1)) of D τ n through one of its endpoints without hitting the crosscut.This assumption is certainly satisfied if η is an SLE curve.
Remark 4.8.The lemma also holds if we do not assume that ξ −1 and ξ−1 are crosscuts of D, but assume that they are the same curve in D.

Estimates on half strips
Given a nonempty H-hull K, Let a K = min(K ∩R) and b K = max(K ∩R).
Lemma 4.12.Let x 0 , y 0 ∈ R. Let K be an Proof.Let L be the unbounded component of L − y 0 \ K. Let y 1 = sup ℜ(g K (γ)).From (4.6) we see that g K = f −1 K decreases the imaginary part of points in H.So we have g K (γ) ⊂ L − y 1 .Let x 1 = g K (x 0 ).First, we prove that x 1 > y 1 .Choose z 1 ∈ g K (γ) such that y 1 = ℜz 1 .Suppose x 1 ≤ y 1 .Then z 1 ∈ R for otherwise z 1 is the image of γ ∩ ∂ K under g K , which must lie to the left of the image of x 0 .Let γ v denote the vertical open line segment (y 1 , z 1 ).It disconnects x 1 from ∞ in H \ g K (L).Thus, f K (γ v ) is a crosscut in H \ (K ∪ L), which connects f K (z 1 ) ∈ γ with f K (y 1 ) ≥ x 0 , and separates Here the equality holds because f K (γ v ) disconnects K from ∞ in H \ L (here we use the fact that L is the unbounded component of L − y 0 \ K); the first inequality holds because H \ L − y 0 ⊂ H \ L; and the second inequality holds because Thus, hm(ih, H \ L − y 0 ; [y 0 , x 0 ]) ≤ hm(g K (ih), H \ L − y 1 ; [y 1 , y 1 + iπ]).Combining the above inequalities with (4.8) and letting h → ∞, we get Then we get g L − 0 Let K t , 0 ≤ t ≤ t 0 , be chordal Loewner hulls driven by W t , 0 ≤ t ≤ t 0 .Recall that every K t is an H-hull with hcap(K t ) = 2t.From (2.2) it is easy to see that sup{ℜz : z ∈ K t 0 } ≤ max{W t : 0 ≤ t ≤ t 0 }, sup{ℑz : z ∈ K t 0 } ≤ 4t 0 . (4.9) From [LSW01, Theorem 2.6] and [Zha08, Lemma 5.3], we know that Proof.Let m = (x + y)/2.Then R is symmetric w.r.t.{ℜz = m}.So g R (m + iπ) = m.By conformal invariance and comparison principle of harmonic measures, for any h > π, we get Letting h → ∞, we get m − g R (x + iπ) ≤ m − x, and so g R (x + iπ) ≥ x.Similarly, Letting h → ∞, and using Lemma 4.11 (applied to right half strips) and Proof.Let l = min{W t : 0 ≤ t ≤ t 0 } and r = max{W t : 0 ≤ t ≤ t 0 }.From (4.9), we know that From the above lemma, we get c K t 0 ≥ c R ≥ y − 2 > l, which contradicts (4.10).So the proof is finished.

Estimate on the derivative
where the constants in depend only on κ, λ , b.

Fig. 1. 1 :
Fig. 1.1:The explanation of the definition of the crossing events.The gray part is the ball B(x, ε).
[Law14]4)].Relation to previous results.The formula of α + n and α n for κ = 6 was obtained in [LSW01, SW01].The exponent α +1 is related to the Hausdorff dimension of the intersection of SLE κ with the real line which is 1 − α + 1 when κ > 4.This dimension was obtained in[AS08].The most important ingredients in proving Theorem 1.1 is the Laplace transform of the derivatives of the conformal map in SLE evolution, which was obtained in[Law14].Acknowledgment.The authors acknowledge Hugo Duminil-Copin, Christophe Garban, Gregory Lawler, Stanislav Smirnov, Vincent Tassion, Brent Werness, and David Wilson for helpful discussions.Hao Wu's work is supported by the NCCR/SwissMAP, the ERC AG COMPASP, the Swiss NSF.Dapeng Zhan's work is partially supported by NSF DMS-1056840.