On the smoothness of the partition function for multiple Schramm-Loewner evolutions

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1 On the smoothness of the partition function for multiple Schramm-Loewner evolutions Mohmammad Jahangoshahi Gregory F. Lawler August Abstract We consider the measure on multiple chordal Schramm-Loewner evolution (SLE κ ). We estabalish a derivative estimate and use it to give a direct proof that the partition function is C 2 if κ < 4. 1 Introduction The (chordal) Schramm-Loewner evolution with parameter κ > 0 (SLE κ ) is a measure on curves connecting two distinct boundary points z w of a simply connected domain D. As originally defined by Schramm 6 this is a probability measure on paths. If κ 4 the measure is supported on simple curves that do not touch the boundary. Although Schramm 6 originally defined SLE κ as a probability measure if κ 4 and z w are locally analytic boundary points it is natural to consider SLE κ as a finite measure µ(z w) with partition function that is as a measure with total mass Ψ D (z w) = H D (z w) b. Here H denotes the boundary Poisson kernel normalized so that H H (0 x) = x 2 and b = (6 κ)/2κ is the boundary scaling exponent. If f : D f(d) is a conformal transformation then H D (z w) = f (z) f (w) H f(d) (f(z) f(w)). There are several reasons for considering SLE κ as a measure with a total mass. First SLE κ is known to be the scaling limit of various two-dimensional discrete models that are considered as measures with partition functions and hence it is natural to consider the (appropriately normalized) partition function in the scaling limit. Second the restriction property or boundary perturbation can be described more naturally for the nonprobability measures; see (1) below. This description leads to one way to define SLE κ in multiply connected domains or as is important for this paper for multiple SLE κ in a simply connected domain. See 3 9 for more information. We write µ # D (z w) = µ D(z w)/ψ D (z w) for the probability measure which is well defined even for rough boundaries. The definition of the measure on multiple SLE κ paths immediately gives a partition function defined as the total mass of the measure. The measure on multiple SLE κ paths γ = (γ 1... γ n ) Research supported by National Science Foundation grant DMS

2 has been constructed in Even though the definition in 2 is given for the so-called rainbow arrangement of the boundary points it can be easily extended to the other arrangements One can see that unlike SLE κ measure on single curves conformal invariance and domain Markov property do not uniquely specify the measure when n 2. This definition makes it unique by requiring the measure to satisfy the restriction property which is explained in Section 2. Study of the multiple SLE κ measure involves characterizing the partition function. For n = 2 the partition function is explicitly given in terms of the hypergeometric function. For n 3 the goal is to characterize the partition function by a particular second-order PDE. However it does not directly follow from the definition that the partition function is C 2. There are two main approaches to address this problem. One approach is to show that the PDE system has a solution and use it to describe the partition function. In 1 it is shown that a family of integrals taken on a specific set of cycles satisfy the required PDE system. In 8 conformal field theory and partial differential equation techniques such as Hörmander s theorem are used to show that the partition function satisfies the PDE system. The other approach which is the one we take in this work is to directly prove that the partition function is C 2. Then Itô s formula can be used next to show that the partition function satisfies the PDEs. The basic idea of our proof is to interchange derivatives and expectations in expressions for the partition function. This interchange needs justification and we prove an estimate about SLE κ to justfiy this. Here we summarize the paper. We finish this introduction by reviewing examples of partition functions for SLE κ. Definitions and properties of multiple SLE κ and the outline of the proof are given in Section 2. Section 3 includes an estimate for SLE κ using techniques similar to the ones in 4. Proof of Lemma 2.3 which explains estimates for derivatives of the Poisson kernel is given in Section Examples SLE κ in a subset of H. Let κ 4 and suppose D H is a simply connected domain such that K = H \ D is bounded and dist(0 K) > 0. Also assume that γ is parameterized with half-plane capacity. By the restriction property we have dµ D (0 ) { c } dµ H (0 ) (γ) = 1{γ D = } exp 2 m H(γ K) (1) where m H (γ D) denotes the Brownian loop measure of the loops that intersect both γ and K and (6 κ)(3κ 8) c = 2κ is the central charge. We normalize the partition functions so that Ψ H (0 ) = 1. For an initial segment of the curve γ t let g t : H \ γ t H be the unique conformal transformation with g t (z) = z + o(1) as z. Then t g t (z) = a g t (z) U t 2

3 where a = 2/κ and U t is a standard Brownian motion. Suppose γ t K = and let D t = g t (D \ γ t ). One can see that m H (γ t K) = a 6 t 0 SΦ s (U s )ds where S denotes the Schwarzian derivative and Φ s (U s ) = H Ds (U s ). It follows from conditioning on γ t that { c } M t = exp 2 m H(γ t K) Ψ Dt (U t ) is a martingale. We assume the function V (t x) = Ψ Dt (x ) is C 2 for a moment. Therefore we can apply Itô s formula and we get ac 12 V (t U t) SΦ t (U t ) + t V (t U t ) xxv (t U t ) = 0. Straightforward calculation shows that V (t x) = H Dt (x ) b is C 2 and satisfies this PDE. Here b is the boundary scaling exponent b = 6 κ 2κ. Other examples. Similar ideas were used in 2 to describe the partition function of two SLE κ curves with a PDE. Differentiability of the partition function was justified using the explicit form of the solution in terms of the hypergeometric function. The PDE system in 9 characterizes the partition function of the annulus SLE κ. That PDE is more complicated and one cannot find an explicit form for the solution. In fact it is not easy to even show that the PDE has a solution. Instead it was directly proved that the partition function is C 2 and Itô s formula was used to derive the PDE. 2 Definitions and Preliminaries We will consider the multiple SLE κ measure only for κ 4 on simply connected domains D and distinct locally analytic boundary points x = (x 1... x n ) y = (y 1... y n ). The measure is supported on n-tuples of curves γ = (γ 1... γ n ) where γ j is a curve connecting x j to y j in D. If n = 1 then µ D (x 1 y 1 ) is SLE κ from x 1 to y 1 in D with total mass H D (x 1 y 1 ) b whose corresponding probability measure µ # D (x 1 y 1 ) = µ D (x 1 y 1 )/H D (x 1 y 1 ) b is (a time change of) SLE κ from x 1 to y 1 as defined by Schramm. Definition If κ 4 and n 1 then µ D (x y) is the measure absolutely continuous with respect to µ D (x 1 y 1 ) µ D (x n y n ) with Radon-Nikodym derivative c n Y (γ) := I(γ) exp m K j (γ) 2. j=2 3

4 Here c = (6 κ)(3κ 8)/2κ is the central charge I(γ) is the indicator function of the event {γ j γ k = 1 j < k n} and m K j (γ) denotes the Brownian loop measure of loops that intersect at least j of the paths γ 1... γ n. Brownian loop measure is a measure on (continuous) curves η : 0 t γ C with η(0) = η(t η ). Let ν # (0 0; 1) be the law of the Brownian bridge starting from 0 and returning to 0 at time 1. Brownian loop measure can be considered as the measure ( ) 1 m = area 2πt 2 dt ν # (0 0; 1) on the triplets (z t η η) where η(t) = tη 1/2 η(t/t η ) for t 0 1. For a domain D C we denote the restriction of m to the loops η D by m D. One important property of m D is conformal invariance. More precisely if f : D f(d) is a conformal transformation then f m D = m f(d) where f m D is the pushforward measure. Note that if σ is a permutation of {1... n} and γ σ = (γ σ(1)... γ σ(n) ) then Y (γ) = Y (γ σ ). The partition function is the total mass of this measure Ψ D (x y) = µ D (x y). We also write which can also be written as Ψ D (x y) = Ψ D (x y) n j=1 H D(x j y j ) b Ψ D (x y) = EY where the expectation is with respect to the probability measure µ # D (x 1 y 1 ) µ # D (x n y n ). Note that Ψ D (x y) is a conformal invariant f Ψ D (x y) = Ψ f(d) (f(x) f(y)) and hence is well defined even if the boundaries are rough. Since SLE κ is reversible 7 interchanging x j and y j does not change the value. To compute the partition function we use an alternative description of the measure µ D (x y). We will give a recursive definition. For n = 1 µ D (x 1 y 1 ) is the usual SLE κ measure with total mass H D (x 1 y 1 ) b. Suppose the measure has been defined for all n-tuples of paths. Suppose x = (x x n+1 ) y = (y y n+1 ) are given and write an (n + 1)-tuple of paths as γ = (γ γ (n+1) ). The marginal measure on γ induced by µ D (x y) is absolutely continuous with respect to µ D (x y ) with Radon-Nikodym derivative H D(x n+1 y n+1 ) b. Here D is the component of D \ γ containing x n+1 y n+1 on its boundary. (If there is no such component then we set H D(x n+1 y n+1 ) = 0 and µ D (x y) is the zero measure.) 4

5 Given γ the curve γ n+1 is chosen using the probability distribution µ # D(z n+1 y n+1 ). One could try to use this description of the measure as the definition but it is not obvious that it is consistent. However one can see that the first definition satisfies this property using the following lemma. Lemma 2.1. Let γ denote (n + 1)-tuple of paths as γ = (γ γ (n+1) ) and let D be the connected component of D \ γ containing the end points of γ (n+1) on its boundary. Then n+1 m K j (γ) = j=2 n m K j (γ ) + m D (γ (n+1) D \ D). j=2 Proof. Let Kj 1(γ) denote the set of loops in K j(γ) that intersect γ (n+1) and let Kj 2 (γ) denote the set of loops that do not intersect γ (n+1). Then m K 1 2(γ) = m D (γ (n+1) D \ D). (2) Note that K 1 j (γ) is equivalent to the set of loops in D that intersect γ(n+1) and at least j 1 paths from γ. Moreover K 2 j (γ) is equivalent to the set of loops that intersect least j paths from γ but do not intersect γ (n+1). Therefore K j (γ ) = K 1 j+1(γ) K 2 j (γ). Now the result follows from this the fact that Kn+1 2 (γ) = and (2). We can also take the marginals in a different order. For example we could have defined the recursive step above as follows. The marginal measure on γ n+1 induced by µ D (x y) is absolutely continuous with respect to µ D (x n+1 y n+1 ) with Radon-Nikodym derivative Ψ D(x y ) where D = D \ γ. (It is possible that D has two separate components in which case we multiply the partition functions on the two components.) We will consider boundary points on the real line. We write just H Ψ Ψ µ µ # for H H Ψ H Ψ H µ H µ # H ; and note that Ψ(x y) = E Y = Ψ(x y) n y j x j 2b where the expectation is with respect to the probability measure If n = 1 then Y 1 and Ψ(x y) = 1. j=1 µ # (x 1 y 1 ) µ # (x n y n ). 5

6 For n = 2 and γ = (γ 1 γ 2 ) then HD\γ EY γ 1 1(x 2 y 2 ) b =. H D (x 2 y 2 ) The right-hand side is well defined even for non smooth boundaries provided that γ 1 stays a positive distance from x 2 y 2. In particular EY = E E(Y γ 1 ) (HD\γ ) 1(x 2 y 2 ) b = E 1. H D (x 2 y 2 ) If 8/3 < κ 4 then c > 0 and Y > 1 on the event I(γ) so the inequality EY 1 is not obvious. More generally if γ = (γ γ n+1 ) Using this we see that Ψ D (x y) 1. EY γ = Y (γ HD\γ (x n+1 y n+1 ) b ) Y (γ ). H D (x n+1 y n+1 ) For n = 2 if x 1 = 0 y 1 = y 2 = 1 and x 2 = x with 0 < x < 1 we have (see for example 2 (3.7)) Γ(2a) Γ(6a 1) Ψ(x y) = φ(x) := Γ(4a) Γ(4a 1) xa F (2a 1 2a 4a; x) (3) where F = 2 F 1 denotes the hypergeometric function and a = 2/κ. finding E H H\γ 1(x 1) b. In fact this calculation is valid for κ < 8 if it is interpreted as E H H\γ 1(x 1) b ; H H\γ 1(x 1) > 0. This is computed by It will be useful to write the conformal invariant (3) in a different way. If V 1 V 2 are two arcs of a domain D let E D (V 1 V 2 ) = H D (z w) dz dw. V 1 V 2 This is π times the usual excursion measure between V 1 and V 2 ; the factor of π comes from our choice of Poisson kernel. Note that 1 0 dr ds 1 E H (( 0 x 1) = x (s r) 2 = dr x r = log(1/x) Hence we can write (3) as φ (exp { E H (( 0 x 1)}). More generally if x 1 < y 1 < x 2 < y 2 ( }) Ψ(x y) = φ (exp { E H (x 1 y 1 x 2 y 2 )}) = φ exp 6 { y1 y2 x 1 x 2 dr ds (s r) 2

7 and if D is a simply connected subdomain of H containing x 1 y 2 x 2 y 2 on its boundary then ( { y1 y2 }) Ψ D (x y) = φ (exp { E D (x 1 y 1 x 2 y 2 )}) = φ exp H D (r s) dr ds. (4) This expression is a little bulky but it allows for easy differentiation with respect to x 1 x 2 y 1 y 2. At this point we can state the main proposition. Proposition 2.2. Ψ and Ψ are C 2 functions. It clearly suffices to prove this for Ψ. The idea is simple we will write the partition function as an expectation and differentiate the expectation by interchanging the expectation and the derivatives. This interchange requires justification and this is the main work of this paper. We will use the following fact which is an analogue of derivative estimates for positive harmonic functions. The proof is straightforward but we delay it to Section 4. Lemma 2.3. There exists c < such that for every x 1 < y 1 < x 2 < y 2 the following holds. Suppose D H is a simply connected domain whose boundary contains an open real neighborhood of x 1 y 1 and suppose that Then if z 1 z 2 {x 1 y 1 } δ := min { x 1 y 1 dist {x 1 y 1 } H \ D} > 0. x 1 z1 H D (x 1 y 1 ) c δ 1 H D (x 1 y 1 ). z1 z 2 H D (x 1 y 1 ) c δ 2 H D (x 1 y 1 ). Suppose D H is a simply connected domain whose boundary contains open real neighborhoods of x 1 y 1 and x 2 y 2 and suppose that δ := min {{ w 1 w 2 ; w 1 w 2 and w 1 w 2 {x 1 x 2 y 1 y 2 }} dist {x 1 y 1 x 2 y 2 } H \ D}. Then if z 1 {x 1 y 1 } z 2 {x 2 y 2 } z1 z 2 ΨD (x y) c δ 2 ΨD (x y). Moreover the constant can be chosen uniformly in neighborhoods of x 1 y 1 x 2 y 2. We will also need to show that expectations do not blow up when paths get close to starting points. We prove this lemma in Section 3. Let jk (γ) = dist { {x k y k } γ j} (γ) = min j k jk(γ). x 2 7

8 Lemma 2.4. If κ < 4 then for every n and every (x y) there exists c < such that for all ɛ > 0 and all j k E Y ; ɛ c ɛ 12 κ 1. In particular E Y 2 m= Proof. It suffices to show that for each j k 2 2m E Y ; 2 m < 2 m+1 <. E Y ; jk ɛ c ɛ 12 κ 1 and by symmetry we may assume j = 1 k = 2. If we write γ = (γ 1 γ 2 γ ) then the event { 12 ɛ} is measurable with respect to (γ 1 γ 2 ) and EY γ 1 γ 2 Y (γ 1 γ 2 ). Hence it suffices to prove the result when n = 2. This will be done in Section 3; in that section we consider κ < 8. For n = 1 2 it is clear that Ψ is C from the exact expression so we will assume that n 3. By invariance under permutation of indices it suffices to consider second order derivatives involving only x 1 x 2 y 1 y 2. We will assume x j < y j for j = 1 2 and x 1 < x 2 (otherwise we just relabel the vertices). The configuration x 1 < x 2 < y 1 < y 2 is impossible for topological reasons. If x 1 < x 2 < y 2 < y 1 we can find a Móbius transformation taking a point y (y 2 y 1 ) to and then the images would satisfy y 1 < x 1 < x 2 < y 2 and this reduces to above. So we may assume that x 1 < y 1 < x 2 < y 2. Case I: Derivatives involving only x j y j for some j. We assume j = 1. We will write x = (x x ) y = (y y ) γ = (γ 1 γ ) and let D be the connected component of H \ γ containing x y on the boundary. Then EY γ = Y (γ HD (x y) b ) = Y (γ ) Q D (x y) b H(x y) where Q D (x y) is the probability that a (Brownian) excursion in H from x to y stays in D. Hence Ψ(x y) = E Y (γ ) Q D (x y) b. Let δ = δ(γ ) = dist{{x y} γ }. Using Lemma 2.3 we see that x Q D (x y) b c δ 1 Q D (x 1 y 1 ) b xy Q D (x y) b + xx Q D (x y) b c δ 2 Q D (x 1 y 1 ) b. (Here c may depend on x y but not on D). Hence E Y (γ ) x Q D (x y) b c E Y (γ ) δ(γ ) 1 Q D (x y) b 8

9 and if z = x or y E Y (γ ) xz Q D (x y) b c E Y (γ ) δ(γ ) 2 Q D (x y) b. Since E Y (γ ) δ(γ ) 2 Q D (x y) b = E E ( Y δ 2 γ ) = EY δ 2 EY 2 < the interchange of expectation and derivative is valid x Ψ(x y) = E Y (γ ) x Q D (x y) b xz Ψ(x y) = E Y (γ ) xz Q D (x y) b. Case 2: The partial z1 z 2 where z 1 {x j y j } z 2 {x k y k } with j k. We assume j = 1 k = 2. We will write x = (x 1 x 2 x ) y = (y 1 y 2 y ) γ = (γ 1 γ 2 γ ). We will write D = D \ γ and let D 1 D 2 be the connected components of D containing {x 1 y 1 } and {x 2 y 2 } on the boundary. It is possible that D 1 = D 2 or D 1 D 2. If D 1 D 2 then If D 1 = D 2 = D then EY γ = Y (γ ) Q D1 (x 1 y 1 ) b Q D2 (x 2 y 2 ) b. EY γ = Y (γ ) Q D1 (x 1 y 1 ) b Q D2 (x 2 y 2 ) b ΨD ((x 1 x 2 ) (y 1 y 2 )) where Ψ D is defined as in (4). In either case we have written EY γ = Y (γ ) Φ(z; γ ) where z = (x 1 y 1 x 2 y 2 ) and we can use Lemma 2.3 to see that z1 z 2 Φ(z; γ ) c (γ z) 2 Φ(z γ ) (γ z) = dist{γ {x 1 y 1 x 2 y 2 }}. As in the previous case we can now interchange the derivatives and the expectation. 3 Estimate In this section we will derive an estimate for SLE κ κ < 8. While the estimate is valid for all κ < 8 the result is only strong enough to prove our main result for κ < 4. We follow the ideas in 4 where careful analysis was made of the boundary exponent for SLE. Let g t denote the usual conformal transformation associated to the SLE κ path γ from 0 to parametrized so that t g t (z) = a g t (z) U t (5) where a = 2/κ and U t = W t is a standard Brownian motion. Throughout we assume that κ < 8 so that D = D = H \ γ is a nonempty set. If 0 < x < y < we let Φ = Φ(x y) = H D(x y) H H (x y) 9

10 where H denotes the boundary Poisson kernel. If x and y are on the boundary of different components of D (which can only happen for 4 < κ < 8) then H D (x y) = 0. As usual we let b = 6 κ 2κ = 3a 1. 2 As a slight abuse of notation we will write Φ b for Φ b 1{Φ > 0} even if b 0. Proposition 3.1. For every κ < 8 and δ > 0 there exists 0 < c < such that for all δ x < y 1/δ and all 0 < ɛ < (y x)/10 E Φ b ; dist({x y} γ) < ɛ c ɛ 6a 1. It is already known that P {dist({x y} γ) < ɛ} ɛ 4a 1 and hence we can view this as the estimate E Φ b dist({x y} γ) < ɛ c ɛ 2a. Using reversibility 5 7 and scaling of SLE κ we can see that to prove the proposition it suffices to show that for every δ > 0 there exists c = c δ such that if δ x < 1 E Φ b ; dist(1 γ) < ɛ c ɛ 6a 1. This is the result we will prove. Proposition 3.2. If κ < 8 there exists c < such that if γ is an SLE κ curve from 0 to 0 < x < 1 Φ = Φ(x 1) 0 < ɛ 1/2 E Φ b ; dist(γ 1) < ɛ (1 x) c x a (1 x) 4a 1 ɛ 6a 1. We will relate the distance to the curve to a conformal radius. In order to do this we will need 1 to be an interior point of the domain. Let D t be the unbounded component of K t = H \ ( x γ t { z : z γ t } and let T = T 1 = inf{t : 1 Dt }. Then for t < T the distance from 1 to Dt is the minimum of 1 x and dist(1 γ t ). In particular if t < T and ɛ < 1 x then dist(γ t 1) ɛ if and only if dist(1 Dt ) < ɛ. We define Υ t to be 4(1 x) 1 times the conformal radius of 1 with respect to Dt and Υ = Υ. Note that Υ 0 = 1 and if dist(1 Dt ) ɛ(1 x) then Υ ɛ. It suffices for us to show that E Φ b ; Υ < ɛ c ɛ 6a 1. We set up some notation. We fix 0 < x < 1 and assume that g t satisfies (5). Let X t = g t (1) U t Z t = g t (x) U t Y t = X t Z t K t = Z t X t 10

11 and note that the scaling rule for conformal radius implies that The Loewner equation implies that Υ t = Y t (1 x) g t (1). dx t = a X t dt + db t dz t = a Z t dt + db t t g t(1) = a g t(1) X 2 t t Υ t = Υ t a X 2 t t g t(x) = ag t(x) a X t Z t Let D t be the unbounded component of H \ γ t and let Z 2 t = aυ t 1 X 2 t Φ t = H D t (x 1) H D0 (x 1) = x2 g t(x) g t(1) Yt 2 t Y t = a Y t X t Z t. 1 K t K t. where we set Φ t = 0 if x is not on the boundary of D t that is if x has been swallowed by the path (this is relevant only for 4 < κ < 8). Note that Φ = Φ and Itô s formula implies that t Φ b t = Φ b t ab Xt 2 ab Zt 2 + 2ab ( ) = ab Φb t 1 2 Kt X t Z t Xt 2 K t { t ( ) Φ b Ks t = exp ab ds}. d 1 X t = 1 X 2 t dx t + 1 X 3 t 0 X 2 s K s d X t = 1 X t 1 a X 2 t and the product rule gives 1 a d1 K t = 1 K t Xt 2 dt a dt 1 dw t = 1 K t X t Z t X t Xt 2 dt 1 X t dw t (1 a) akt dt 1 K t X t dw t. which can be written as dk t = 1 K t X 2 t a + a 1 dt + 1 K t dw t. K t X t As in 4 we consider the local martingale Mt = (1 x) 1 4a Xt 1 4a g t(1) 4a 1 = (1 x) 1 4a (1 K t ) 4a 1 Υ 1 4a t which satisfies dm t = 1 4a X t M t dw t M 0 = 1 11

12 If we use Girsanov and tilt by the local martingale we see that dk t = 1 K t a Xt 2 3a dt + 1 K t dwt. K t X t where Wt is a standard Brownian motion in the new measure P. We reparametrize so that log Υ t decays linearly. More precisely we let σ(t) = inf{t : Υ t = e at } and define ˆX t = X σ(t) Ŷt = Y σ(t) etc. Since ˆΥ t := Υ σ(t) = e at and a ˆΥ t = t ˆΥt = a ˆΥ t 1 ˆX 2 t 1 ˆK t ˆK t σ(t) we see that Therefore σ(t) = ˆX t 2 ˆK t 1 ˆK t t ˆΦ b t := Φ b σ(t) { ab = exp 1 ˆK } { t s ds = e 0 ˆK abt exp ab s 0 } 1 ds ˆK s d ˆK t = a 3a ˆK t ( = ˆK t aˆkt dt + ˆK t (1 ˆK t ) dbt ) 3a dt + 1 ˆK t ˆK t db t. for a standard Brownian motion B t (in the measure P ). Let λ = 2a 2 and { } { a(7a 1) N t = e λt a(3a 1) t ˆΦb t ˆKa t = exp t exp } 1 ds ˆK s ˆK a t. Itô s formula shows that N t is a local P -martingale satisfying 1 dn t = N t a ˆK t dbt ˆK N 0 = x a t One can show it is a martingale by using Girsanov to see that d ˆK t = 2a 4a ˆK t dt + ˆK t (1 ˆK t ) d B t where B t is a Brownian motion in the new measure P. By comparison with a Bessel process we see that the solution exists for all time. Equivalently we can say that ˆM t := ˆM t N t 12

13 is a P-martingale with ˆM 0 = x a. (Although Mt is only a local martingale the time-changed version ˆM t := Mσ(t) is a martingale.) Using (3) we see that E Φ b γ σ(t) c ˆKa t ˆΦb t. If ɛ = e at then E Φ b ; σ(t) < = c E E(Φ b 1{σ(t) < } γ σ(t) ) c E ˆKa t ˆΦb t ; σ(t) < = c e λt e (1 4a)at (1 x) 4a 1 1 ˆM 0 E ˆMt (1 ˆK t ) 1 4a ; σ(t) < = c e a(1 6a)t x a (1 x) 4a 1 Ẽ (1 ˆK t ) 1 4a = c ɛ 6a 1 x a (1 x) 4a 1 Ẽ (1 ˆK t ) 1 4a. So the result follows once we show that Ẽ (1 ˆK t ) 1 4a < is uniformly bounded for t t 0. The argument for this proceeds as in 4. If we do the change of variables ˆK t = 1 cos Θ t /2 then Itô s formula shows that ( dθ t = 4a 1 ) cot Θ t dt + db t. 2 This is a radial Bessel process that never reaches the boundary. It is known that the invariant distribution is proportional to sin 8a 1 θ and that it approaches the invariant distribution exponentially fast. One then computes that the invariant distribution for ˆK t is proportional to x 4a 1 (1 x) 4a 1. In particular (1 ˆK t ) 1 4a is integrable with respect to the invariant distribution. 4 Proof of Lemma 2.3 We prove the first part of lemma 2.3 for x 1 = 0 y 1 = 1. Other cases follow from this and a Möbius transformation sending x 1 y 1 to 0 1. Lemma 4.1. There exists c < such that if D is a simply connected subdomain of H containing 0 1 on its boundary then where δ = dist({0 1} D H). x H D (0 1) + y H D (0 1) c δ 1 H D (0 1) xx H D (0 1) + xy H D (0 1) + yy H D (0 1) c δ 2 H D (0 1) Proof. Let g : D H be a conformal transformation with g(0) = 0 g(1) = 1 g (0) = 1. Then if x < δ y 1 < δ H D (x y) = g (x) g (y) g(y) g(x) 2. (6) 13

14 In particular g (0) g (1) = H D (0 1) H H (0 1) = 1 and hence g (1) 1. Using Schwartz reflection we can extend g to be a conformal transformations of disks of radius δ about 0 and 1. By the distortion estimates (the fact that a 2 2 a 3 3 for schlicht functions) we have g (0) 4 δ 1 g (0) 4 δ 1 g (0) 18 δ 2 g (0) 18 δ 2 and similarly g (1) 4 δ 1 g (1) and g (1) 18 δ 1 g (1). right-hand side of (6) we get the result. By direct differentiation of the Lemma 4.2. There exists c < such that if x 1 < y 1 0 < 1 x 2 < y 2 Ψ D (x y) is as in (4) and z 1 {x 1 y 1 } z 2 {x 2 y 2 } then z1 ΨD (x y) + z2 ΨD (x y) c δ 1 Ψ D (x y) z1 z 2 ΨD (x y) c δ 2 ΨD (x y) where δ := min {{ w 1 w 2 ; w 1 w 2 and w 1 w 2 {x 1 x 2 y 1 y 2 }} dist {x 1 y 1 x 2 y 2 } H \ D}. Proof. Let Ψ D (x y) = φ (u D (x y)). where u D (x y) = e E D(xy) E D (x y) = y1 y2 Using the Harnack inequality we can see that for j = 1 2 x 1 x 2 H D (r s) dr ds. H D (x s) H D (x j s) H D (r y) H D (r y j ) if x x j δ/2 y y j δ/2. From this we see that y2 x 2 H D (z 1 s) ds + y1 Let z 1 be x 1 or y 1 and let z 2 be x 2 or y 2. Then x 1 H D (r z 2 ) ds c δ 1 E D (x y) H D (z 1 z 2 ) c δ 2 E D (x y). z1 ΨD (x y) = φ (u D (x y)) z1 u D (x y) z1 z 2 ΨD (x y) = φ (u D (x y)) z1 u D (x y) z2 u D (x y) + φ (u D (x y)) z1 z 2 u D (x y). y2 z1 u D (x y) = ± H D (z 1 s) ds u D (x y). z2 u D (x y) = ± x 2 y1 x 1 H D (r z 2 ) ds u D (x y). 14

15 This gives y1 y2 z2 z 1 u D (x y) = ± H D (r z 2 ) dr H D (z 1 s) ds ± H D (z 1 z 2 ) u D (x y) x 1 x 2 The result will follow if we show that z1 u D (x y) + z2 u D (x y) c δ 1 E D (x y) u D (x y) z2 z 1 u D (x y) c δ 2 E D (x y) u D (x y). z2 z 1 u D (x y) u D (x y) (1 x) φ (1 x) φ(x) c δ 2 E D (x y). φ (x) φ(x) are uniformly bounded for x > x 0. Recall that u(x) = c x a F (x) where F (x) = 2 F 1 (2a 1 2a 4a; x). We recall that F is analytic in the unit disk with power series expansion where the coefficients b j satisfy F (x) = 1 + b n x n n=1 b n = C n 4a O(n 1 ). We therefore get asymptotic expansions for the coefficients of the derivatives of F. The important thing for us is that if κ < 8 then 4a 1 > 1 and we have as x 1 In other words the quantities F (1 x) = O(1) F (x) = o(x 1 ) F (x)o(x 2 ). F (x) (1 x) F (x) F (x) (1 x) 2 F (x) F (x) are uniformly bounded for 0 x < 1. If g(x) = x a F (x) then a g (x) = g(x) x + F (x) F (x) ( a g (x) = g(x) x + F ) (x) 2 a F (x) x 2 + F (x) F (x) F (x) 2 F (x) 2. Therefore for every x 0 > 0 the quantities φ(x) (1 x) φ (x) φ(x) (1 x) 2 φ (x) φ(x) are uniformly bounded for x 0 < x < 1. 15

16 References 1 J. Dubédat (2006). Euler integrals for commuting SLEs J. Stat. Phys M. Kozdron G. Lawler (2007). The configurational measure pn mutuallly avoiding SLE paths in Universality and Renormalization: From Stochastic Evolution to Renormalization of Quantum Fields I. Binder D Kreimer ed. Amer. Math. Soc G. Lawler (2009). Schramm-Loewner Evolution in Statistical Mechanics S. Sheffield T Spenceer ed. IAS/Park City Mathematics Series G. Lawler (2015). Minkowski content of the intersection of a Schramm-Loewner evolution (SLE) curve with the real line J. Math Soc. Japan J. Miller S. Sheffield (2016). Imaginary geometry III: reversibilty of SLE κ for κ (4 8) Annals of Math O. Schramm (2000). Scaling limits for loop-erased random walks and uniform spanning trees Israel J. Math D. Zhan (2008). Reversibility of chordal SLE Annals of Prob Eveliina Peltola Hao Wu (2017). Global Multiple SLE κ for κ 4 and Connection Probabilities for Level Lines of GFF Preprint in arxiv: G. Lawler (2011). Defining SLE in multiply connected domains with the Brownian loop measure arxiv: G. Lawler (2009). Partition Functions Loop Measure and Versions of SLE Journal of Statistical Physics