Up-to-constants bounds on the two-point Green's function for SLE curves

The Green's function for the chordal Schramm-Loewner evolution $SLE_\kappa$ for $0<\kappa<8$, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.


Introduction
The Schramm-Loewner evolution (SLE κ ) is a conformally invariant family of probability measures on curves originally given by Schramm as a candidate for the scaling limit of lattice models in statistical physics. The chordal Green's function gives the normalized probability that the path goes through a point and the two-point Green's functions gives the correlations for this quantity. While the one-point function is known (up to an arbitrary multiplicative constant in the definition), and the existence of the two-point function has been established, the exact form of the two-point function is not known. Estimates for the two-point function have proved to be important in analyzing fractal properties of the SLE curves, in particular the Hausdorff dimension and the Minkowski content. The goal of this paper is to give up-to-constant bounds valid for all pairs of points in a domain. It is still open to give a closed form for the function.
We start by reviewing the definition of SLE and giving the relevant known results. See [2] for more details. Suppose that γ : (0, ∞) → H = {x + iy : y > 0} is a curve with γ(0+) ∈ R and γ(t) → ∞ as t → ∞. Let H t be the unbounded component of H \ γ(0, t]. Using the Riemann mapping theorem, one can see that there is a unique conformal transformation g t : H t −→ H satisfying g t (z) − z → 0 as z → ∞. For any a > 0, it can be parametrized so that as z → ∞, The conformal maps g t satisfy the chordal Loewner equatioṅ where U t = g t (γ(t)) is a continuous real-valued function. The Schramm-Loewner evolution (SLE κ ) is obtained by choosing a = 2/κ and U t to be a standard (one-dimensional) Brownian motion. In this paper we will consider only 0 < κ < 8 and let a = 2/κ > 1/4. We write For z ∈ H\{0}, the function t → g t (z) is well defined up to time T z := sup{t : Im[g t (z)] > 0}. Rohde and Schramm [7] showed that for κ < 8 the Loewner equation above generates a random curve γ, which is also called SLE κ , and they showed in a weak sense that the dimension of the path is If κ ≥ 8, the curve exists but is plane filling and is not relevant for this paper. If 0 < κ ≤ 4, the paths are simple with γ(0, ∞) ⊂ H while there are double points and γ(0, ∞) ∩ R = ∅ for 4 < κ < 8. Moreover, if H t denotes the unbounded component of H \ γ(0, t], then Their starting point to compute (2) was to assume that there exists a function G and a constantĉ such that lim where γ = γ(0, ∞). Although did not establish the limit, they did note that if such a function exists, then the conformal Markov property of SLE κ implies that must be a local martingale. From this one can determine the only possible value of d is that given in (2), and the function G must be a multiple of We call G (with this choice of constant) The SLE κ Green's function.
In [1] it was proved that the Hausdorff dimension of the path is indeed d, and in [4] it was established that the d-dimensional Minkowski content of γ[0, t] is finite and nonzero. In [3], the limit was shown to exist if we replace distance with the conformal radius of z in the domain H \ γ. More recently, [4] established the existence of the limit as given although the value of the constantĉ is unknown.
The two-point Green's function is defined by lim ǫ→0,δ→0 The existence of the limit with conformal radius replacing distance was established in [5] and the limit with distance was proved in [4]. As shown in [4], if Θ(Y ) denotes the d-dimensional respectively. Unlike the one-point case, no exact expression has been given for G(z, w). The goal of this paper is we give up-to-constants functions by proving the following theorem.
Two important estimates exist in the literature now. In [5], and implicitly in [1] although it was not phrased in this way, it was shown that if V is a bounded domain in H bounded away from the real line, then where ≍ V indicates that the implicit constant depends on V . In [6], it was shown that there exists c such that for all z, w, G(z, w) ≥ c G(z) G(w).
While we have defined the Green's function in terms of SLE in H, it can easily be extended to simply connected domains D. To be more precise, suppose that D is a simply connected domain and w 1 , w 2 are distinct points in ∂D. Let F : H → D be a conformal transformation of H onto D with F (0) = w 1 , F (∞) = w 2 . Then the distribution of is that of SLE κ in D from w 1 to w 2 . Although the map F is not unique, the scaling invariance of SLE κ in H shows that the distribution is independent of the choice. The Green's functions G D (F (z); w 1 , w 2 ), G D (F (z), F (w); w 1 , w 2 ) can be defined by conformal covariance, and the corresponding limits (3) and (6) hold. We can write Here Υ D (F (z)) = |F ′ (z)| −1 /2 denotes (1/2) times the conformal radius of D with respect to F (z) and S D (F (z); w 1 , w 2 ) = sin arg[z]. If ∂ 1 , ∂ 2 denote the two components of ∂D \{w 1 , w 2 }, then Here, and throughout this paper, hm will denote harmonic measure; that is, hm D (z, K) is the probability that a Brownian motion starting at z exits D at K.
Using the Schwarz lemma and the Koebe (1/4)-theorem, we see that If γ(t) is an SLE κ curve with transformations g t and driving function U t , we write γ t = γ(0, t], γ = γ ∞ . If z ∈ H and t < T z , we let It is easy to check that if t < T z , then Υ t (z) as given in (9) is the same as Υ Ht (z). Also, if z ∈ γ, then Υ(z) := Υ Tz− (z) = Υ D (z) where D denotes the connected component of H \ γ containing z. Similarly, if w 1 , w 2 are distinct boundary points on a simply connected domain D and z ∈ D, we define We extend the definition (5) as follow. If D is a simply connected domain with distinct w 1 , w 2 ∈ ∂D, we define Under this definition G(z) = G H (z; 0, ∞). The Green's function satisfies the conformal covariance rule The local martingale M t (z) is not a martingale because it "blows up" at time t = T z . If we stop it before that time, it is actually a martingale. To be precise, suppose that Then for every ǫ > 0, M t∧τ (z) is a martingale. The following is proved in [3] (the proof there is in the upper half plane, but it immediately extends by conformal invariance).
Let us sketch the proof of the Theorem 1.1. By scaling, it suffices to prove the theorem for w = x w + iy w with |w| = 1, in which case the conclusion can be written as Here and for the reminder of this paper we write ≍ to indicate that quantities are bounded by constants where the constants depend only on κ. Let us give a heuristic description of this estimate to show where this comes from. The goal of this paper is to justify this heuristic. Let ǫ be very small and let E z , E w denote the events that dist(γ, z) < ǫ and dist(γ, w) < ǫ, respectively.
• The hardest part of the proof is to show that if |z − w| ≍ 1, then E z and E w are independent events up to constants, that is, Therefore, as ǫ ↓ 0, • Suppose |z − w| is small and y w ≤ 2|z − w|. Again, let E ′ be the event that the path gets within distance 2|z − w| of w. In this case Given E ′ , E z and E w are conditionally independent up to a multiplicative constant. If Therefore, as ǫ ↓ 0, 2 Proof of the theorem In [5] it is shown that for each z, w, there exist ǫ z , δ w such that if ǫ < ǫ z , δ < δ w , When estimating P{∆(z) ≤ ǫ} there are two regimes. The interior or bulk regime, where ǫ ≤ Im(z) can be estimated using Proposition 1.2 since in this case ∆(z) ≍ Υ(z). However for the boundary regime ǫ > Im(z), one needs the following estimate.
Proof. The bound P{σ < ∞} ≍ y 4a−1 can be found in a number of places. A proof which includes a proof of the first inequality can be found in [6]. The first inequality is Lemma 2.10 of that paper.
In particular, the lemma implies that if η : (0, 1) → H is a curve with η(0+), η(1−) ∈ (0, ∞) and η = η(0, 1), then One way to estimate the right-hand side is in terms of (Brownian) excursion measure (see [5, 4.1] for definitions and similar estimates). We recall that if D is a simply connected domain and V 1 , V 2 are two arcs in ∂D, then the excursion measure (of the set of excursions from V 1 to V 2 in D) is given by where H ∂D denotes the boundary Poisson kernel (normal derivative of the Green's function).
We can also write this as where φ j is the harmonic function on D with boundary value 1 V j and ∂ n denotes normal derivative. These formulas assume that V 1 , V 2 are smooth; however, this quantity is a conformal invariant so one can define this for nonsmooth boundaries. A standard calculation shows that if diam(η) ≤ dist(0, η), and H denotes the unbounded component of H \ η, then Then monotonicity of the excursion measure implies that The upshot of this is that if we can find such an η ′ , then We will prove Theorem 1.1 in a sequence of propositions. We assume |z| ≤ |w| and let It will be useful to define a quantity that allows us to consider the boundary and interior cases simultaneously. Let and let Φ(z) = Φ ∞ (z). Note that Φ 0 (z) = |z| 4a−1 , and scaling implies that the distribution of Φ(rz) is the same as that of r 4a−1 Φ(z). Since 4a − 1 > 2 − d, we see that The next lemma combines the interior and boundary estimates into one estimate.
Lemma 2.2. There exist 0 < c 1 < c 2 < ∞ such that for all z ∈ H and 0 < ǫ ≤ 1, Proof. Let z = x+iy. By scaling we may assume that |z| = 1 and hence Φ 0 (z) = 1, S(z) = y. Let ∆ = ∆ ∞ (z), Φ = Φ ∞ (z). Proposition 1.2 and Lemma 2.1 imply that The hardest step in estimating the two-point Green's function is to show that if two points are not very close to each other, then the events that the paths get close to the two points are independent at least up to a multiplicative constant. The next proposition gives a precise version of this statement in terms of the quantity Φ(z). Proposition 2.3. There exists c < ∞ such that if |z| ≤ 4|w|, and 0 < ǫ z , ǫ w ≤ 1, then The proof is similar to proofs in [5]. The details are somewhat technical so let us sketch the basic strategy. The idea is to show that if one is going to get very close to both z and w, then one is likely to get very close to one of them first without getting too close to the other and then one goes to the other point. In other words, one does not keep going back and forth between smaller and smaller neighborhoods of z and w. The way that one establishes this is to fix a curve I between z and w and consider excursions of the SLE paths from I. What one shows is that if γ is already very close to z, then it is unlikely that γ will get even closer to z and return to I. There are two different possibilities. Suppose that I t is a crosscut of H t contained in I and γ(t) ∈ I t . If z is in the bounded component of H t \ I t , then S Ht (z; γ(t), ∞) is small, and the SLE path does not want to get closer to z. If z is in the unbounded component of H t \ I t , then the SLE path can get closer to z, but then it is unlikely to return to I t . The proof makes this idea precise.
To prove Proposition 2.3 we start with a lemma that gives an upper bound for the probability that an SLE path gets close to a point and subsequently returns to a given crosscut. It is a generalization of Lemmas 4.10 and 4.11 of [5], and we use ideas from those proofs. Before stating the lemma, we set up some notation. Suppose η : (0, 1) → H is a simple curve with η(0+) = 0, η(1−) > 0 and write η = η(0, 1). Let V 1 , V 2 denote respectively the bounded and unbounded components of H \ η and assume that z = x z + iy z ∈ V 1 , w = x w + iy w ∈ V 2 . Recall that H t is the unbounded component of H \ γ t . We will let I t be a decreasing collection of subarcs of η that are crosscuts of H t separating z and w. To be more specific, one can show (see [5,Appendix A]) that there is a collection of open subarcs {I t : t < T z ∧ T w } of η with the following properties.
• I t ⊂ H t . Moreover, H t \ I t has two connected components, one containing z and the other containing w.
• If s < t, then I t ⊂ I s . Moreover, if γ(s, t] ∩ I s = ∅, then I t = I s .
We will now show that Let H = Hσ, I = Iσ, g = gσ, U = Uσ. By (15), Let H * be the component of H \ C containing w. On the eventσ < ρ, H * is unbounded. Using simple connectedness of H, we can see that there is a subarc l ∈ ∂H * ∩ C that is a crosscut of H and that separates w from I in H. Since l does not separate w from ∞, g(l) is a crosscut of H that does not separate U from ∞; for ease let us assume that its endpoints are on (−∞, U]. Since l separates w from I, l also separates I from ∞ in H. Therefore g(l) separates g(I) from U and ∞ in H. We use excursion measure to estimate the probability that γ[σ, ∞) returns to I. The excursion measure between g(I) and [U, ∞) in H \ g(I) is bounded above by the excursion measure between g(I) and g(l) in H \ (g(I) ∪ g(l)) which by conformal invariance equals the excursion measure between I and l in H \(I ∪l). This in turn is bounded above by the excursion measure between C and ∂D in {ζ ∈ H : 1 < |ζ| < |w|} which is O(1/ |w|). Given this, we can use (13) to see that the probability that an SLE κ path from U to ∞ in H hits g(I) is O(|w| −(4a−1)/2 ). Using conformal invariance, we conclude that which gives (16). We noted above that if j ≤ k 0 , then P{ρ <σ < ∞} = 0.
We will now show that if j > k 0 , The proposition then follows by summing over j. Consider the event Using (15), we see that Let H = H ρ . On the event E j , there is a subarc l of H ∩ C that is a crosscut of H with one endpoint equal to γ(ρ) such that l disconnects w from ∞ in H. Using this and the relationship between S and harmonic measure, we see that S ρ (w) is bounded above by the probability that a Brownian motion starting at w reaches C without leaving H. Using (14), we see that on the event E j , dist(w, ∂H) ≤ 2 −j/(4a−1) |w|. Using the Beurling estimate, we see that the probability a Brownian motion starting at w reaches distance |w|/2 from w without leaving H is O(2 −j/2(4a−1) ). Given this, the probability that is reaches C without leaving H is bounded above by O(1/| |w|). Therefore, on the event E j , Using the strong Markov property and (15), we see that which combined with (18) gives (17).
Proof of Proposition 2.3. By scaling, we may assume that |z| ≤ 1/2, |w| = 2. We will consider crosscuts of H t that are contained in the unit circle.
To be more precise, we consider a decreasing collection of arcs {I t : t < T z ∧ T w } with the following properties.
• For each t, I t is a crosscut of H t that separates z from w in H t .
• If t > s, then I t ⊂ I s . Moreover, if γ(s, t] ∩ I s = ∅, then I t = I s .
We define a sequence of stopping times as follows.
If one of the stopping times takes on the value infinity, then all the subsequent ones are set equal to infinity.
The key estimate, which we now establish, is the following.
Note thatÎ is a crosscut of H with one endpoint equal to zero. We consider separately the cases whereζ is in the bounded or unbounded component of H \Î.
In this case, we an argue as in the previous paragraph to see that the probability SLE κ in H hitsĝ(V ) is bounded above by c Φ 1/4 r. For the other case we assume that Im(ζ ′ ) ≥ ∆ 1/4 |ζ ′ | for someζ ′ ∈ĝ(V ). Using the Poisson kernel in H, we can see that the probability that a Brownian motion starting atζ ′ hitsÎ before leaving H is bounded below by a constant times diam(Î) From this we conclude that diam(Î) ≤ c ∆ 1/4 |ζ ′ |.
We appeal to Lemma 2.4 to say that the probability that SLE κ in H hitsĝ(V ) and then returns toÎ is bounded above by a constant times Given (19), the remainder of the proof proceeds in the same way as [5, Section 4.4] so we omit this.
Proof. By scaling it suffices to prove the result when ǫ = 1. This can be done as the proof of the previous proposition, so we omit the details. The key step is to choose an appropriate splitting curve I 0 . We can choose I 0 either to be a half-circle with endpoints on R or a vertical line. We choose I 0 so that I 0 separates z and w and dist(z, I 0 ), dist(w, I 0 ) ≥ 1/4. Therefore, G(z, w) ≍ y 4a−1 w (q/y w ) 2−d q 2(d−2) (y z ∧ q) q q 2 4a−1 , y w ≥ q, G(z, w) ≍ q 4a−1 q 2(d−2) (y z ∧ q) y w q 2 4a−1 , y w ≤ q.
If q ≤ y w ≤ 2q we can use either expression. If y w ≤ 2q, then y w ∧ q ≍ y w , y z ∧ q ≍ y z , S(w) ∨ q ≍ q and we can write If y w ≥ 2q, then y z ≍ y w , y z ∧ q ≍ q, S(w) ∨ q ≍ y w , and we can write

Open problems
The obvious open problem is to determine the value of the Green's function G(z, w). One can use the argument of Rohde and Schramm to determine a partial differential equation satisfied by G, see [5], but it is unknown whether or not there is an explicit solution.
As a starting point, we can ask the following questions.
• Does there exist c < ∞ such that for any n and z 1 , ..., z n ∈ H, • Suppose V is a compact subset of H with dist(0, R) > 0. Is it true that G(z 1 , ...z n ) ≍ V,n n i=1 |z i − z i+1 | d−2 ?