aa r X i v : . [ m a t h . P R ] N ov Duality of Schramm-Loewner Evolutions
Julien Dub´edat ∗ October 30, 2018
Abstract
In this note, we prove a version of the conjectured duality for Schramm-Loewner Evolutions, byestablishing exact identities in distribution between some boundary arcs of chordal SLE κ , κ >
4, andappropriate versions of SLE ˆ κ , ˆ κ = 16 /κ . Schramm-Loewner Evolutions (or SLE), introduced by Schramm in 1999, are probability distributions,parameterized by κ > κ . In particular, when κ ≤
4, the trace isa.s. a simple curve; this is no longer the case if κ > κ islocally absolutely continuous w.r.t. to (some version) of SLE ˆ κ , ˆ κ = 16 /κ . This was suggested by Duplantier.In the case ( κ, ˆ κ ) = (8 , κ, ˆ κ ) = (6 , / κ : the SLE κ ( ρ ) processes ( ρ = ρ , . . . , ρ n ). They satisfy a domainMarkov property when keeping track of n marked points z , . . . , z n (in addition of the origin and the targetof chordal SLE). The influence of z i on the SLE trace is quantified by the real parameter ρ i ; this influenceis attractive for ρ i > ρ i < H , going from 0 to infinity. In the phase 4 < κ < τ when the trace hits some point in (1 , ∞ ). The boundary arc straddling 1 is the boundary arc seen by 1 attime τ − . Theorem 1.
Consider a chordal
SLE κ in ( H , , ∞ ) , < κ < ; let D be the leftmost visited point on (1 , ∞ ) . Conditionally on D , the boundary arc straddling 1 is distributed as an SLE ˆ κ ( − ˆ κ , ˆ κ − , ˆ κ − in ( H , D, ∞ , , , D + ) , stopped when it hits (0 , . In the phase κ ≥
8, a.s. every point in H is visited by the trace. We isolate a boundary arc in a differentway. Let G be the leftmost point on ( −∞ ,
0) visited by the trace before τ . We consider the boundary of ∗ Partially supported by NSF grant DMS0804314 τ G , the hull of the SLE stopped when it first visits G ; this boundary is an arc between G and a point in(0 , Theorem 2.
Consider a chordal
SLE κ in ( H , , ∞ ) , κ ≥ . Let G be the leftmost visited point G on ( −∞ , before τ . Conditionally on G , the boundary of K τ G is distributed as an SLE ˆ κ ( ˆ κ , ˆ κ − , − ˆ κ , ˆ κ − in ( H , G, ∞ , G − , G + , , , stopped when it hits (0 , . The distributions of D and G are well known and easy to derive.In [3], it is shown that duality shares common features with reversibility and the question of definingmultiple SLE strands in a common domain. This local commutation property states that two SLE strandscan be grown in a domain to a positive size, in a way that does not depend on the order in which the SLE’sare growing. Such systems of commuting SLE’s are classified in [3]; in particular, two versions of SLE κ ,SLE ˆ κ can commute only if ˆ κ ∈ { κ, /κ } .In [16], Zhan proves reversibility of chordal SLE κ , κ ≤
4, i.e. that the range of the trace of an SLE κ in D going from x to y has the same distribution as the range of the trace of SLE going from y to x in D . This waspreviously known for κ ∈ { , / , , , } . The argument involves a sequence of coupling of an SLE κ ( D, x, y )with an SLE κ ( D, y, x ), such that each coupling in the sequence is absolutely continuous w.r.t. the trivial(independent) coupling, and the limiting coupling is exact (the ranges of the two traces are identical).Let γ , ˆ γ be traces of two SLE’s satisfying the local commutation condition. Then, for U, V disjoint opensubsets of the domain, one has a coupling of ( γ, ˆ γ ) which is “correct” on the time set { ( s, t ) : s ≤ τ, t ≤ ˆ τ } ,where τ , ˆ τ are stopping times for the two SLE’s, such that γ τ ⊂ U , ˆ γ ˆ τ ⊂ V . We construct a coupling of( γ, ˆ γ ), which is “correct” on the time set { ( s, t ) : γ [0 ,s ] ∩ ˆ γ [0 ,t ] = ∅ } . See Theorem 6 for a precise statement.The duality identities follow from applying Theorem 6 to appropriate pairs of commuting SLE’s, togetherwith some a priori geometric information on the traces. Plainly, many identities may be generated in thisfashion.The article is organized as follows. Section 2 recalls some absolute continuity properties of chordal SLE.Local commutation is discussed in Section 3. Maximal couplings of commuting SLE’s are constructed inSection 4. Geometric consequences (in particular duality) are drawn in Section 5. Some technical lemmasare postponed to Section 6. Acknowledgments.
I wish to thank Greg Lawler for comments on an earlier version of this article andOded Schramm for useful conversations.
In this section we consider some absolute continuity properties of chordal SLE, mostly based on [5]. ChordalSLE will also serve as a reference measure for variants we will study later.We adopt the following notation: c = ( D, x, y ) is a configuration where D is a simply connected domainand x, y are distinct boundary points. Unless there is an ambiguity, the configuration is simply denoted by D . The chordal SLE κ measure on c = ( D, x, y ) is denoted by µ c ( κ is fixed). It is seen as a measure onLoewner chains up to increasing time change; or as a configuration-valued continuous process (up to timechange); or as a measure on non self-traversing paths ([11]). This path (the SLE “trace”) is denoted by γ ,while the hull it generates is denoted by K . Let U be a subdomain of D , agreeing with D in a neighbourhoodof x , and not containing y on its boundary. Then µ UD denotes the measure on paths induced by chordal SLEstarting from x and stopped on exiting U ; this happens at a random time τ , at which the hull is K τ , the tipof the trace is γ τ , and the configuration c τ is ( D τ = D \ K τ , γ τ , y ). More generally, for τ a stopping time, γ τ denotes the trace stopped at τ (ie the process up to time τ ), µ τc the measure induced by stopping at τ .We will use γ to denote both the trace as a process and as a subset of D (the range of the process).2ater on, we will use tightness conditions, so we shall review some technical points now. Let ( D, x, y )be a (bounded) configuration, K a hull such that ( D \ K, x ′ , y ) is a configuration for some x ′ ∈ ∂K . By theRiemann mapping theorem, there is a conformal equivalence φ K : D \ K → D ; one can specify it uniquelyby requiring its 2-jet at y to be trivial ( φ K ( y ) = y , φ ′ K ( y ) = 1, φ ′′ K ( y ) = 0 if φ K extends smoothly at y ; thiscondition is coordinate independent, so one can first “straighten” the boundary at y ). One defines a topologyon hulls as follows: ( K n ) converges to K if φ K n converges to φ K uniformly on compact sets of D that are atpositive distance of K . This is a version of Carath´eodory convergence. A topology on chains ( K t ) t ≥ is givenby the condition: ( K nt ) t converges to ( K t ) t if for all T > K ′ a compact set of D at positive distance of K T , φ K nt converges to φ K t uniformly on [0 , T ] × K ′ . Then the Loewner equation maps continuously C ( R + , R )(with the usual topology of uniform convergence on compact sets) to the space of chains endowed with thistopology ([6] Section 4.7). Thus the induced measure on chains is a Radon measure. From [11], we knowthat the chain is a.s. generated by a continuous non self-traversing path γ . For clarity, we will think of SLEas a measure on such paths, with the topology on chains described above.To express densities, we need to define some conformal invariants. Let ( D, x, y ) be a configuration, z x , z y analytic local coordinates at the boundary ( z x mapping a neighbourhood of x in D to the neighbourhood of0 in the upper semidisk). The Poisson excursion kernel is defined as H D ( x, y ) = lim X → x,Y → y G D ( X, Y ) ℑ ( z x ( X )) ℑ ( z y ( Y ))where G D is the Green function in D (with Dirichlet boundary conditions); this depends on the choice of z x (or z y ) as a (real) 1-form. If D and D ′ agree in a neighbourhood of x , we choose the same local coordinate z x , so that H D ′ ( x, y ′ ) /H D ( x, y ) does not depend on a choice of local coordinate at x . Similarly for i, j = 1 , D ij , x i , y j ) such that D ij agrees with D i, − j in a neighbourhood of x i and with D − i,j in a neighbourhood of y j . Then the ratio: H D ( x , y ) H D ( x , y ) H D ( x , y ) H D ( x , y )is defined independently of any (coherent) choice of local coordinates. To simplify the notation, if c = ( D, x, y )is a configuration, we set H ( c ) = H D ( x, y ).There is a σ -finite measure µ loop on unrooted loops in C , the Brownian loop measure ([5, 8]). As in [4],let us denote m ( D ; K, K ′ ) = µ loop { δ : δ ⊂ D, δ ∩ K = ∅ , δ ∩ K ′ = ∅ } . In accordance with [5], set α = α κ = − κ κ , λ = λ κ = (6 − κ )(8 − κ )2 κ . Proposition 3. If c = ( D, x, y ) and c ′ = ( D ′ , x, y ′ ) are configurations agreeing in a neighbourhood U of x , ∂U ∩ ∂D = ∂U ∩ ∂D ′ a connected arc containing x at positive distance of y, y ′ , then µ Uc and µ Uc ′ are mutuallyabsolutely continuous, with density dµ Uc ′ dµ Uc ( γ ) = (cid:18) H ( c ′ τ ) H ( c ) H ( c τ ) H ( c ′ ) (cid:19) α exp( − λm ( D ; K τ , D \ D ′ ) + λm ( D ′ ; K τ , D ′ \ D )) uniformly bounded above and below.Proof. We will reduce the statement to two known cases.1. Assume that D = D ′ . Then the statement follows from Lemma 3.2 in [3]; see also [13]. Moreprecisely, consider the following situation: H is the upper half-plane, three boundary points x, y, y ′ aremarked; K is a hull around x , x ′ its tip, φ a conformal equivalence H \ K → H . Then H H ( x, y ) = ( x − y ) − , H H ( x, y ′ ) = ( x − y ′ ) − , computing in the natural local coordinate. It follows that H H \ K ( x ′ , y ) = φ ′ ( y )( φ ( y ) − φ ( x ′ )) ,3 H \ K ( x ′ , y ′ ) = φ ′ ( y ′ )( φ ( y ′ ) − φ ( x ′ )) , for an appropriate (common) local coordinate at x ′ . Then the ratio H H \ K ( x ′ , y ′ ) H H ( x, y ) H H \ K ( x ′ , y ) H H ( x, y ′ ) = φ ′ ( y ′ ) (cid:18) y ′ − x ′ φ ( y ′ ) − φ ( x ′ ) (cid:19) φ ′ ( y ) − (cid:18) y ′ − x ′ φ ( y ′ ) − φ ( x ′ ) (cid:19) − is independent of (coherent) choices. One concludes by identifying the density of an SLE κ ( H , x, y ) and anSLE κ ( H , x, y ′ ) with respect to the common reference measure SLE κ ( H , x, ∞ ).2. Assume that D ′ ⊂ D , D ′ and D agree in a neighbourhood of y = y ′ . Then the statement is a rephrasingof Proposition 5.3 in [5].3. The general case reduces to 1,2 as follows. Let y ′′ be a point on the connected boundary arc of x in ∂D ∩ ∂D ′ , which is not on ∂U ; and V the connected component of D ∩ D ′ having x on its boundary. Thenapply 1 to go from ( D, x, y ) to (
D, x, y ′′ ); then 2 to go from ( D, x, y ′′ ) to ( V, x, y ′′ ); then 1 to go from( V, x, y ′′ ) to ( D ′ , x, y ′′ ); then 2 to go from ( D ′ , x, y ′′ ) to ( D ′ , x, y ′ ). Cancellations occur due to the “inclusionexclusion” form of the ratios ( H ( c ′ τ ) H ( c ) /H ( c τ ) H ( c ′ )).For a general bound on densities, see Lemma 14. Following the discussion in Section 2.1 of [3], we phrase and then check a necessary condition for reversibility.Consider a configuration c = c , = ( D, x, y ), γ an SLE from x to y and ˆ γ an SLE from y to x . Denote c s,t = ( D \ ( K s ∪ ˆ K t ) , γ s , ˆ γ t ). Let U, ˆ U be disjoint neighbourhoods of x, y respectively; τ, ˆ τ denote firstexits of U, ˆ U by γ, ˆ γ respectively. Assume that γ, ˆ γ can be coupled so that one is the reversal of the other.Then an application of the Markov property for γ, ˆ γ shows that the distribution of γ τ conditional on ˆ γ ˆ τ is(stopped) SLE in c , ˆ τ = ( D \ ˆ K ˆ τ , x, ˆ γ ˆ τ ). Symmetrically, the conditional distribution of ˆ γ ˆ τ given γ τ is SLEin c τ, = ( D \ K τ , γ τ , y ). By integration, this gives the identity of measures: Z ˆ f (ˆ γ ˆ τ )( Z f ( γ τ ) dµ Uc , ˆ τ ( γ τ )) d ˆ µ ˆ Uc (ˆ γ ˆ τ ) = Z f ( γ τ )( Z ˆ f (ˆ γ ˆ τ ) d ˆ µ ˆ Uc τ, (ˆ γ ˆ τ )) dµ Uc ( γ τ ) (3.1)for arbitrary positive Borel functions f, ˆ f . This is the local commutation condition studied in [3]. Disinte-grating and inserting densities (that exist from absolute continuity properties) yields the condition: dµ Uc , ˆ τ dµ Uc ! ( γ τ ) = d ˆ µ ˆ Uc τ, d ˆ µ ˆ Uc ! (ˆ γ ˆ τ ) (3.2)almost everywhere in γ τ , ˆ γ ˆ τ . This is an identity between two (continuous) functions of the paths γ, ˆ γ . Fromthe above results on absolute continuity of SLE (Proposition 3), we see that both sides are indeed equal,and their common value is the explicit quantity: ℓ D ( γ τ , ˆ γ ˆ τ ) = (cid:18) H ( c τ, ˆ τ ) H ( c ) H ( c τ, ) H ( c , ˆ τ ) (cid:19) α exp( − λm ( D ; K τ , ˆ K ˆ τ )) (3.3)which is manifestly symmetric in γ τ , ˆ γ ˆ τ . (We use ℓ for likelihood ratio , somewhat abusively).Using the expression of ℓ as Radon-Nikodym derivatives of probability measures, we see that: Z ℓ D ( γ τ , ˆ γ ˆ τ ) d ˆ µ ˆ Uc (ˆ γ ˆ τ ) = Z d ˆ µ c τ, (ˆ γ ˆ τ ) = 1 ∀ γ τ Z ℓ D ( γ τ , ˆ γ ˆ τ ) dµ Uc ( γ τ ) = Z dµ c , ˆ τ ( γ τ ) = 1 ∀ ˆ γ ˆ τ (3.4)4his also applies to any pair of stopping times σ, ˆ σ dominated by τ, ˆ τ (ie σ is a stopping time for γ suchthat σ ≤ τ a.s.).Such local commutation identities (without relying on explicit densities) are proved in greater generalityin [3] under an infinitesimal commutation condition, which is easily checked in the present case.Let us also point out a (deterministic) tower property, dictated by compatibility with the SLE Markovproperty. Let ( D, x, y ) be a domain, ( K s ) a Loewner chain growing at x (with trace γ s ), and ( ˆ K t ) a Loewnerchain growing at x (with trace ˆ γ t ). Let c s,t = ( D \ ( K s ∪ ˆ K t ) , γ s , ˆ γ t ). Let us denote, for 0 ≤ s ≤ s ,0 ≤ t ≤ t : ℓ s ,t s ,t = ℓ c s ,t ( γ s , ˆ γ t )Then, for 0 ≤ s ≤ s ≤ s , 0 ≤ t ≤ t ≤ t : ℓ s ,t s ,t ℓ s ,t s ,t = ℓ s ,t s ,t , ℓ s ,t s ,t ℓ s ,t s ,t = ℓ s ,t s ,t . (3.5)For fixed γ , the first relation has to hold a.e. in γ to ensure compatibility of (3.4) with the Markov propertyof γ ; the second relation corresponds to the Markov property of ˆ γ . Alternatively, this can be checked directlyfrom the explicit expression (3.3), by telescopic cancellations and the restriction property of the loop measure µ loop ([8]). We now move to the general case (in simply connected domains) of local commutation, following Theorem7.1 of [3], which we rephrase in the present context.A configuration consists of a simply connected domain D with marked points: c = ( D, z , z , . . . , z n , z n +1 );the marked points are distinct and in some prescribed order on the boundary. The question is to classifypairs of SLE (with the SLE Markov property relatively to these configurations), one growing at z (hulls( K s ), trace γ ), one at z n +1 (hulls ( ˆ K t ), trace ˆ γ ), that satisfy local commutation (3.1), (3.2). As before, wedenote c s,t = ( D s,t = D \ ( K s ∪ ˆ K t ) , γ s , z , . . . , z n , ˆ γ t ) when this is still a configuration (ie before swallowingof any marked point).We take as reference measures a chordal SLE κ from z to z , and a chordal SLE ˆ κ from z n +1 to z , where z is another marked boundary point, used solely for normalization. Then: Theorem 4 ([3]) . Local commutation is satisfied iff: ˆ κ ∈ { κ, /κ } and there exists a conformally invariantfunction ψ on the configuration space and exponents ν ij such that P n +1 j =1 ν ,j = α κ , P ni =0 ν i,n +1 = α ˆ κ , and if Z ( c ) = ψ ( c ) Y ≤ i SLE measures, up toswallowing of a marked point. In the theorem, notice that M s = H c ( z , z ) − α κ Z ( c s, ) is defined via a choice of local coordinates at z, z , . . . , z n +1 , but not at z , where the evolution occurs (the choice of local coordinates is arbitrary butfixed under evolution). 5 good example of the situation is the following: ˆ κ = κ = 6, with four marked points ( z , z , z , z ), ψ the probability that there is a percolation crossing from ( xy ) to ( z z ).The absolute continuity properties of these SLE’s reduce to that of chordal SLE as in Lemma 13: let Z ( c ) = ψ ( c ) Q i 2, ˆ ρ = − ˆ κ , ˆ ρ i = − (ˆ κ/ ρ i = − (4 /κ ) ρ i , ρ + P i ρ i = κ − n − Z ( c ) = H ( z , z n +1 ) − κ κ Z ( c ) = H ( z , z n +1 ) − κ Y i H ( z , z i ) − ρi κ H ( z n +1 , z i ) − ρi κ Y i 1) + 2 β = 2 ν . Define a partition function Z ( c ) = H D ( x, y ) − κ κ H D ( z , z ) ν ψ ( u )6here u is the cross-ratio u = ( z − x )( z − y )( y − x )( z − z ) (in the upper half-plane) and ψ ( u ) = ( u (1 − u )) β F (2 β, β + 8 κ − 1; 2 β + 4 κ ; u )where F designates a solution of the hypergeometric equation with parameters 2 β, . . . (this equation isinvariant under u ↔ − u ). If the solution is chosen so that it is positive on the configuration space, thena computation shows that Z satisfies the condition of the theorem and drives two locally commuting SLE’sstarting from x, y . This covers for instance the following situations: a chordal SLE κ from x to y conditionednot to intersect the interval [ z , z ], 4 < κ < 8; a chordal SLE / from x to y conditioned not to intersecta restriction measure (with exponent ν ) from z to z ; and the marginal of a system of two SLE strands x ↔ y , z ↔ z ([2], Section 4.1; this corresponds to ν = α κ ). Let c = ( D, z , z , . . . , z n , z n +1 ) be a configuration, where D is a simply connected, bounded domain with n + 2 distinct marked points on the boundary in some prescribed order. We consider a system of two SLE’ssatisfying local commutation, one originating at z , the other at z n +1 . These two SLE’s have the SLE Markovproperty for domains with n + 2 marked points. The first one is absolutely continuous (up to a disconnectionevent) w.r.t. SLE κ ( D, z , z n ); the Loewner chain is ( K s ), the trace γ , the measure µ c . The second one isabsolutely continuous (up to a disconnection event) w.r.t. SLE ˆ κ ( D, z n +1 , z n ); the Loewner chain is ( ˆ K t ),the trace ˆ γ , the measure ˆ µ c . We also denote c s,t = ( D \ ( K s ∪ ˆ K t ) , γ s , z , . . . , z n , ˆ γ t ).From Theorem 4 and the following discussion, we know that ˆ κ ∈ { κ, /κ } and there is a positiveconformally invariant function ψ on configurations, weights ν ij , such that for τ, ˆ τ a pair of stopping times(before disconnection events) ℓ c ( γ τ , ˆ γ ˆ τ ) = (cid:18) Z ( c τ, ˆ τ ) Z ( c , ) Z ( c τ, ) Z ( c , ˆ τ ) (cid:19) exp( − λm ( D ; K τ , ˆ K ˆ τ ))= (cid:18) dµ τc , ˆ τ dµ τc (cid:19) ( γ τ ) = d ˆ µ ˆ τc τ, d ˆ µ ˆ τc ! (ˆ γ ˆ τ ) (4.6)where Z ( c ) = ψ ( c ) Q i The measure Θ ηc = L ( µ c ⊗ ˆ µ c ) is a coupling of µ c , ˆ µ c .Proof. The statement can be rephrased as ∀ ˆ γ, E ( L ) = 1 ∀ γ, ˆ E ( L ) = 1where E , ˆ E refer to integration w.r.t. dµ c ( γ ), d ˆ µ ˆ c (ˆ γ ) respectively (in other terms E ( . ) = E ⊗ ˆ E ( . | ˆ F ∞ )). Thesituation is completely symmetric, so we shall consider only the distribution of the second marginal; that is,we have to check that given any ˆ γ , E ( L ) = 1.For this, the relevant expression of the density is: L = Q ≤ i ≤ N ℓ i +1 ,J ( i ) i, . Notice that ˆ γ ˆ τ J ( i ) is ( F τ i ∨ ˆ F ∞ )-measurable. This implies that a term ℓ i +1 ,J ( i ) i, is ( F τ n ∨ ˆ F ∞ )-measurable for any i < n . Thus for fixed n : E n Y i =0 ℓ i +1 ,J ( i ) i, |F τ n ! = " n − Y i =0 ℓ i +1 ,J ( i ) i, E ( ℓ n +1 ,J ( n ) n, |F τ n )and E ( ℓ n +1 ,J ( n ) n, |F τ n ) = 1 by (4.7); the point is that given ( F τ n ∨ ˆ F ∞ ), γ τ n and ˆ γ ˆ τ J ( n ) are fixed. This issaying that M n = n − Y i =0 ℓ i +1 ,J ( i ) i, is a (discrete time, bounded) martingale in the filtration ( F τ n ∨ ˆ F ∞ ) n ≥ . In particular, E ( L ) = E ( M N ) = M = 1. 9e can actually get a more precise statement. Let τ be an arbitrary F -stopping time; let n = inf { i ∈ N : τ i ≥ τ } , a random integer. We assume that dist( γ τ , ∂ ) ≥ η a.s., so that ( n, ∈ G . Then τ n is a stoppingtime approximating τ (as η ց γ τ n , ˆ γ ˆ τ J ( n ) ) under Θ η . Then γ τ n hasdistribution µ τ n c . Moreover: E ⊗ ˆ E ( L |F τ n ∨ ˆ F ˆ τ J ( n ) ) = Y i Let µ c , ˆ µ c be SLE measures in a configuration ( D, z , . . . , z n +1 ) satisfying local commutation.Then there exists a coupling Θ of µ c , ˆ µ c which is maximal in the following sense:for any F -stopping time τ , τ ≤ σ , let ˆ τ be the ( F τ ∨ ˆ F t ) t -stopping time: ˆ τ = sup { t ≥ K t ∩ ( K τ ∪ ˆ ∂ ) = ∅ } . Then under Θ , the pair ( γ τ , ˆ γ ˆ τ ) has the following distribution: γ τ is distributed according to µ τc , and condi-tionally on γ τ , ˆ γ ˆ τ is distributed according to ˆ µ ˆ τc τ, . The symmetric statement holds. roof. For any η > 0, the two marginal distributions of Θ η are fixed Radon measures (in the topologyof Carath´eodory convergence of Loewner chains). Thus the family (Θ η ) η> is tight, and Prokhorov’s the-orem ensures existence of subsequential limits. Let ( η k ) k be a sequence η k ց η k has aweak limit Θ. Then Θ is a coupling of µ c and ˆ µ c . We can consider a probability space with sample(( γ , ˆ γ ) , . . . , ( γ k , ˆ γ k ) , . . . ) such that the distribution of ( γ k , ˆ γ k ) is Θ η k and ( γ k , ˆ γ k ) → ( γ, ˆ γ ) a.s., where thedistribution of ( γ, ˆ γ ) is Θ.Let τ be an F -stopping time; we approximate τ in a convenient way. Firstly, τ can be approximated by τ ′ taking values in some discrete countable sequence ( t i ) i ≥ (eg dyadic times). Hence there are Borel sets B i such that B i is a Borel function of K t i and τ ′ = inf { t i : K t i ∈ B i } . Replace the Borel set B i by a largeropen set U i such that the measure of U i \ B i is very small. Then τ ′′ = inf { t i : K t i ∈ U i } is a stopping timeequal to τ ′ with probability arbitrarily close to 1. Finally, let τ ′′′ = τ ′′ ∧ sup { t : dist( γ t , ∂ ) ≥ ε ′ } for somefixed ε ′ > 0. Let us assume for now that τ is of type τ ′′′ . This gives a common stopping rule for all thechains K k. : stop the first time that K kt i is in U i or at distance ε of ∂ . We denote τ k this stopping time forthe chain K k. . In particular, τ k → τ a.s. (using that the U i ’s are open).For η > τ kn is an approximation of τ k as above: τ kn = inf { τ i : τ i ≥ τ k } ; then K kτ kn − ⊂ K kτ k ⊂ K kτ kn ⊂ ( K kτ kn − ) η k . It is easy to see that τ kn → τ , τ kn − → τ , K kτ kn → K τ and γ kτ kn → γ τ (since γ τ = ∩ s> K τ + s \ K τ ) as k → ∞ .We have seen that the conditional distribution of ˆ γ ˆ τ J ( n ) k is ˆ µ ˆ τ J ( n ) c τn, . Notice that ˆ τ J ( n ) occurs after first entrancein ( K kτ kn ) η k and before entrance in ( K kτ kn ) η k .For fixed ε > K kτ kn ⊂ ( K τ ) ε for k large enough. The configuration ˆ c kτ kn , converges in the Carath´eodorytopology to c τ, (with also convergence of γ kτ kn to γ τ ); this implies weak convergence of the conditionaldistribution of ˆ γ k stopped when entering ( K τ ) ε to the corresponding stopped SLE in c τ, . This gives thecorrect conditional distribution of ˆ γ stopped when entering ( K τ ) ε , conditional on γ τ . One concludes bytaking ε ց τ ′′′ as above (this will be enough to drawgeometric consequences). A general stopping time τ ≤ σ is the limit of a sequence of stopping times τ ′′′ m ; foreach m , the conditional distribution of ˆ γ stopped upon entering ( K τ ′′′ m ) ε is correct. One concludes by taking m → ∞ and then ε ց n (pairwise) commuting SLE’s. Let us discuss this casebriefly.Consider a configuration c = ( D, z , . . . , z n , z n +1 , . . . , z n + m ), with n SLE’s starting at z , . . . , z n , drivenby the same partition function Z . One can reason as above (sampling the SLE’s at discrete times τ i , . . . , τ ni n ).In a maximal coupling, one can stop the first SLE at a stopping time τ , the second at τ (first time it ceasesto be defined or meets K τ ), . . . , the n -th at τ n (first time it ceases to be defined or meets ∪ n − i =1 K iτ i ) andget the appropriate joint distribution. This works for any permutation of indices.Let us describe the local coupling in this case: let U , . . . , U n be disjoint neighbourhoods of z , . . . , z n , µ . , . . . , µ n. the commuting SLE measures, Z their common partition function. Let K i be the hull of thestopped i -th SLE; c ε ...ε n , ε i ∈ { , } , is the configuration where the i -th SLE has grown (until stopped) if ε i = 1. Consider the density L = dµ nc ... dµ nc ... · dµ n − c ... dµ n − c ... · · · dµ c ... dµ c ... = Z ( c ... ) Z ( c ... ) n − Z ( c ... ) . . . Z ( c ... ) exp − λ n X j =2 m ( D, ∪ j − i =1 K i , K j ) Then it is clear from the first expression that the first marginal of L ( µ c ⊗ · · · ⊗ µ nc ) is µ c (integrating out11 n , then K n − , . . . ); the second expression shows that the construction is symmetric (for a discussion ofthe loop measure contribution, see Section 3.4 of [2]). We have proved (Theorem 6) existence of maximal couplings under a local commutation assumption. Onthe other hand, the systems of SLE’s satisfying this assumption are classified (Theorem 4). So we can nowapply the existence of maximal couplings to appropriate systems of commuting SLE’s to extract informationon the geometry of SLE curves. Reversibility for κ ∈ (0 , 4] is proved in [16]. We review the result for the reader’s convenience. Theorem 7. If κ ≤ , SLE is reversible; any maximal coupling Θ of chordal SLE κ in ( D, x, y ) with SLE κ in ( D, y, x ) is the coupling of SLE with its reverse trace.Proof. Let ( D, x, y ) be a configuration, µ c the chordal SLE measure from x to y , ˆ µ c the chordal SLE measurefrom y to x . They satisfy local commutation, hence there exists a maximal coupling Θ.Take a countable dense sequence of F -stopping times ( τ m ) (e.g., capacity of the hull reaches a rationalnumber); denote simply by τ an element in this sequence. Then in the maximal coupling Θ, the conditionaldistribution of ˆ γ ˆ τ is SLE in ( D \ K τ , y, γ τ ) stopped upon hitting K τ . For κ ≤ 4, the SLE trace intersectsthe boundary only at its endpoints. Hence ˆ γ ˆ τ = γ τ . This proves that under Θ, the intersection γ ∩ ˆ γ is a.s.dense in γ . Since both γ, ˆ γ are a.s. closed, γ ⊂ ˆ γ ; since ˆ γ is simple, removing a point disconnects it, but γ is connected. Hence the occupied sets of γ, ˆ γ are equal. Again, as the paths are simple, the occupied setdetermines the parameterized trace. Hence in any maximal coupling Θ, ˆ γ = γ r (the reverse trace) a.s.; thisdetermines the coupling uniquely.Besides local commutation, the argument uses only qualitative properties of the paths. So we can phraseat no additional cost: Corollary 8. Let κ ≤ , µ c , ˆ µ c a system of commuting SLE in the configuration c = ( D, z , z , . . . , z n , z n +1 ) .Assume that µ c , ˆ µ c are supported on simple paths that meet the boundary of D only at z , z n +1 . Then γ r and ˆ γ are identical in distribution. A simple example of the situation is as follows: let ( z , . . . , z ) be four marked points on the arc ( z z ).Then we can consider chordal SLE from z to z weighted by any, say, bounded above and below functionof the cross-ratio of ( z , . . . , z ) in D \ γ . This plainly preserves both local commutation and reversibility.Another setup where the corollary applies is the following: c = ( D, z , z , . . . , z n +1 ) with points incounterclockwise order. Let ρ , . . . , ρ n be such that ρ + · · · + ρ i ≥ ≤ i < n and ρ + · · · + ρ n = 0.Then the traces of SLE ( ρ, − 2) starting from z and SLE ( − , − ρ ) starting from z n +1 are the reverse ofeach other in distribution. This describes the scaling limit of the zero level line of a discrete free field ([12])with piecewise constant boundary conditions (with jump at z i proportional to ρ i ). A version with markedpoints on both sides of z also holds.One also obtains reversibility identities for the pairs of commuting SLE’s (aiming at each other) withfour marked points described at the end of Section 3.2. By degenerating two points into one, this describesthe reversal of SLE κ ( ρ ), κ ≤ ρ ≥ κ − 2. For instance, if κ = 8 / 3, one can represent an SLE / ( ρ ) in( H , , , ∞ ) as the limit of a chordal SLE / in ( H , , ∞ ) conditioned not to intersect a restriction measurewith exponent ν = ν ( ρ ) from 1 to z ≫ κ , it12s unclear whether there is a simple probabilistic interpretation, but one still gets an exact (if unwieldy ingeneral) description of the reversal. Corollary 9. Let κ ≤ , ρ ≥ κ − , ( D, x, y ) a configuration. Then SLE κ ( ρ ) in ( D, x, y, x + ) and in ( D, y, x, y − ) have the same occupied set in distribution, where x, x + , y − , y are in this order on the boundary.Proof. We sketch the argument. The result follows from reversibility in the regular situation with four markedpoints described at the end of Section 3.2. Indeed, if x, z , z , y are in this order on the boundary, one has apair of commuting SLE’s starting at x, y with common partition function: Z ( c ) = H D ( x, y ) − κ κ H D ( z , z ) ν ψ ( u )where u is the cross-ratio u = ( z − x )( z − y )( y − x )( z − z ) (in the upper half-plane), κ β ( β − 1) + 2 β = 2 ν and ψ ( u ) = ( u (1 − u )) β F (2 β, β + 8 κ − 1; 2 β + 4 κ ; u ) . When κ ≤ ρ = κβ ≥ κ − 2, the processes do not hit [ z , z ], by comparison arguments. Thus the localcommutation extends to a maximal coupling, and in this coupling the occupied sets coincide.The first SLE is the martingale transform of chordal SLE κ in ( D, x, y ) by the martingale (in upperhalf-plane coordinates): t (cid:18) g ′ t ( z ) g ′ t ( z )( g t ( z ) − g t ( z )) (cid:19) ν ψ ( u t )where u t is the cross-ratio at time t . Take z = y − ε . Then the leading term of the expansion of themartingale as ε ց t (cid:18) g ′ t ( z ) g ′ t ( y )( g t ( z ) − g t ( y )) (cid:19) ν (cid:18) ( g t ( z ) − X t ) g ′ t ( y )( g t ( y ) − X t )( g t ( y ) − g t ( z )) (cid:19) β so that this limiting process is identified from Lemma 12 as SLE κ ( ρ ) in ( D, x, y, z ), ρ = κβ . A symmetricresult holds for the other SLE.The same arguments can be used to establish reversibility of systems of multiple SLE’s considered in [2](this also follows from the symmetry of the density of the system w.r.t. independent chordal SLE’s whenthe pairing of endpoints is fixed). The question of SLE duality is to describe boundaries of SLE κ , κ > 4, in terms of SLE ˆ κ , ˆ κ = 16 /κ .There are various parametric situations we can consider. Let us start with the simplest setting: aconfiguration c = D ( x, z , y, z ) has four marked points x, y, z , z on the boundary. We consider two SLE’s(inducing the measures µ c , ˆ µ c , with traces γ , ˆ γ ), see Table 1 ([ κ ] represents an SLE κ “seed”, the other entriesare the ρ parameters). Table 1: x z y z [ κ ] ρ − κ ρ − ˆ κ ˆ ρ [ˆ κ ] ˆ ρ The additional conditions for local commutation are ρ + ρ = ( κ − ρ i = − κ ρ i , consequentlyˆ ρ + ˆ ρ = (ˆ κ − ρ = ρ . We need to put conditions on ρ so that13aths have a correct geometry. Take ρ ∈ [ κ − , κ − κ > 4. Consequently, ρ ∈ [ κ − , κ − ρ , ˆ ρ ∈ [ ˆ κ − , ˆ κ − z , z ] at y (see Lemma 15).To restrict even more the situation, take ˆ ρ = ˆ κ − ρ = ˆ κ − z , x ), nor ( z , z ) except at y ; and it hits ( x, z ) since ˆ ρ = ˆ κ − < ˆ κ − Proposition 10. In a maximal coupling Θ of µ c , ˆ µ c , the range of ˆ γ is contained in that of γ . If ρ = κ − , ˆ γ is the right boundary of K ; if ρ = κ − , ˆ γ is the left boundary of K .Proof. As before, take ˆ τ a stopping time for the second SLE. The first SLE in c , ˆ τ is defined until it exitsat ˆ γ ˆ τ . More precisely, it is defined up to a time where it accumulates at ˆ γ ˆ τ and at no other point of theboundary arc [ z , z ] of c , ˆ τ (Lemma 15). But γ is continuous away from [ z , z ] in c ; so if ˆ τ is positive, γ stopped when exiting c , ˆ τ has a limit, which is ˆ γ ˆ τ . Hence ˆ γ ˆ τ is on γ . Taking countably many stoppingtimes, this shows that ˆ γ is included in (the range of) γ . Moreover, ordering is preserved: ˆ γ ˆ τ = γ t for some t ≤ τ , and for any t < ˆ τ , ˆ γ t / ∈ K τ .Set ρ = κ − 4. Then the range of γ is partitioned in points on ˆ γ , to its left, or to its right. Take astopping time τ . Then ˆ γ first hits K τ on the arc [ γ τ , z ]. If γ τ was to the right of ˆ γ , then ˆ γ would have tocircle γ τ and reenter in K τ , which would violate the ordering condition. Hence a generic point γ τ is on ˆ γ orto its left. This implies that ˆ γ is contained in the right boundary of the range of γ , which is a simple path.Since ˆ γ starts at y (where γ ends) and ends on ( x, z ), this shows that ˆ γ is the right boundary of the rangeof γ . Remark 11. The situation where ρ varies in [ κ − , κ − is of some independent interest and seems relatedto pivotal points questions. We consider now versions where the non simple SLE is actually chordal SLE κ , at the expense of somecomplication for the dual simple SLE ˆ κ . Proof of Theorem 1. Assume that κ ∈ (4 , κ , say in ( H , , ∞ ). The point 1 isswallowed at time τ ; D = γ τ is on (1 , ∞ ) with distribution given by: P ( D ∈ (1 , z )) = F ( z ) = c Z z u − κ ( u − κ − du where c = B (1 − /κ, /κ − − . In other words, D − has a Beta(1 − /κ, /κ − 1) distribution. The function F is such that t F (( g t ( z ) − W t ) / ( g t (1) − W t )) is a martingale. Let us disintegrate the SLE measure w.r.t. D (see [1] for related questions). It is easy to see that up to τ , the SLE conditional on D ∈ dz is themartingale transform of chordal SLE by: t ∂ z F (cid:18) g t ( z ) − W t g t (1) − W t (cid:19) = c g ′ t ( z ) g t (1) − W t ( g t ( z ) − W t ) − κ ( g t ( z ) − g t (1)) κ − ( g t (1) − W t ) − κ and this is readily identified with SLE κ ( κ − , − 4) in ( H , , ∞ , , z ) (Lemma 12). To get a regular situation,we split the point z into two points y and z , while setting x = 0, z = 1, z = ∞ . Consider the system ofcommuting SLE’s given by Table 2. Table 2: 4 < κ < x z y z z [ κ ] κ − − κ κ − − ˆ κ ˆ κ − κ ] ˆ κ − − ˆ κ z , z ] at y , while the second SLE will not hit [ z , z ] and exits [ x, z ] somewhere in( x, z ) (Lemma 15). Arguing as in Proposition 10, this shows that ˆ γ is the right boundary of K . Finally,one takes z ց y , so that the first SLE κ becomes chordal SLE κ conditional on D = y . This yields Theorem1. When κ ≥ 8, the trace is a.s. space filling, and we have to proceed differently to isolate a boundary arc. Proof of Theorem 2. Consider now the case of a chordal SLE κ in ( H , , ∞ ), κ ≥ κ ≤ γ τ = 1 a.s. There is a leftmost point G on ( ∞ , 0) visited by the trace before τ . We are interested in theboundary of K τ G , a simple curve from G to some point in (0 , G is given by: P ( G ∈ ( z, c Z z ( − u ) − /κ (1 − u ) κ − du where c = B (1 − /κ, − /κ ) In other words, G is such that G/ ( G − 1) has a Beta(1 − /κ, − /κ ) distribution(generalised arcsine distribution). The disintegrated SLE measure w.r.t. G is again SLE κ ( − , κ − 4) in( H , , ∞ , G, G . To get a regular situation, we need to split the point G into three points z , y, z ; we also set x = 0, z = 1, z = ∞ . Consider the system of two commuting SLE’s in ( H , y, z , x, z , z )given by Table 3. Table 3: κ ≥ z y z x z z − − κ κ − κ ] κ − ˆ κ [ˆ κ ] ˆ κ − − ˆ κ ˆ κ − − ˆ κ The first SLE exits at y , the second one exits in ( x, z ) (Lemma 15). As in Proposition 10, this showsthat one can couple the two SLE’s such that the second one is the boundary arc of the first one between y and a point of ( x, z ). Taking z ր y , z ց y gives Theorem 2.At the expense of some complications, one can consider more symmetric situations. Let ( D, x, y, z, z ′ , y ′ , x ′ )be a configurations (points are in that order). There is a system of four commuting SLE’s attached to thisconfiguration (where a + b = 2), see Table 4. Table 4: x y z z ′ y ′ x ′ [ κ ] a ( κ − − κ − κ b ( κ − 4) 22 a ( κ − − κ − κ b ( κ − 4) [ κ ] − ˆ κ a (ˆ κ − 4) [ˆ κ ] 2 b (ˆ κ − − ˆ κ − ˆ κ a (ˆ κ − 4) 2 [ˆ κ ] b (ˆ κ − − ˆ κ SLE In this subsection, we phrase similar absolute continuity results for different versions of SLE. In the contextof duality, it is useful to consider SLE-type measures in a parametric family SLE κ ( ρ ) ([5, 1]), as acknowledged15n [1].An SLE κ ( ρ ), ρ = ρ , . . . , ρ n , in the configuration ( H , x, ∞ , z , . . . , z n ) is an SLE the driving process ofwhich satisfies the SDE: dW t = √ κdB t + n X i =1 ρ i W t − g t ( z i ) dt and W = x , up to swallowing of a z i . See Lemma 3.2 of [3] for homographic change of coordinates. Inparticular, if P i ρ i = κ − 6, the point at infinity is used for normalization only.The following lemma is a change of measure result (see also [14]). Lemma 12. Consider an SLE κ starting from x in H , ρ = ρ , . . . , ρ n ; let Z it = g t ( z i ) − W t . Then: M t = Y i g ′ t ( z i ) α i | Z it | β i Y i 1) + 2 β i , η ij = κβ i β j . Before the swallowing of any marked point, M t /M is the density of an SLE κ ( ρ ) starting from ( x, z , . . . z n ) w.r.t. SLE κ , where ρ = κβ , . . . , κβ n .Proof. This is a standard computation relying on: dZ it = 2 Z it dt − √ κdB t dg ′ t ( z i ) g ′ t ( z i ) = − Z it ) dt d ( Z jt − Z it )( Z jt − Z it ) = − Z it Z jt dt so that: dM t M t = X i β i Z it (cid:18) Z it dt − √ κdB t (cid:19) + κ β i ( β i − Z it ) dt − α i ( Z it ) dt + X i Lemma 13. Let c = ( D, z , z , . . . , z n ) be a configuration consisting of a simply connected domain D with n + 1 marked points on the boundary; c ′ = ( D ′ , z , z ′ , . . . , z ′ n ) is another configuration agreeing with D in a neighbourhood U of z ; U is at positive distance of marked points other than z . Let µ Uc denote thedistribution of an SLE κ ( Z ) in c , stopped upon exiting U . Then: dµ Uc ′ dµ Uc ( γ ) = (cid:18) Z ( c ′ τ ) Z ( c ) Z ( c τ ) Z ( c ′ ) (cid:19) exp( − λm ( D ; K τ , D \ D ′ ) + λm ( D ′ ; K τ , D ′ \ D )) where c τ = ( D \ K τ , γ τ , z , . . . , z n ) , similarly for c ′ τ .Proof. One can proceed as follows: let µ Uc denotes chordal SLE κ in the configuration c (aiming at an auxiliarypoint z ), stopped upon exiting U . Then trivially: dµ Uc ′ dµ Uc = dµ Uc ′ dµ Uc ′ · dµ Uc ′ dµ Uc · dµ Uc dµ Uc The middle term is studied in Proposition 3, while the outer terms are, from the definition of SLE κ ( Z ): dµ Uc dµ Uc = M τ M = Z ( c τ ) Z ( c ) · H D ( z , z ) α H D τ ( γ τ , z ) α where τ is the first exit of U , and similarly for the other term. Under the assumptions above, M τ is uniformlybounded (see Lemma 14).Recall that Z ( c ) depends on a choice of local coordinates at the marked points as a 1-form; but the ratio Z ( c ′ τ ) Z ( c ) Z ( c τ ) Z ( c ′ ) does not depend on the choices. We give an upper bound on Radon-Nikodym derivatives that appear in the coupling argument. This is arough estimate that is sufficient for our purposes.A configuration c = ( D, x, y, z , . . . , z n ) consists in a bounded simply connected Jordan domain D , withdistinct marked points on its boundary; ∂ (resp. ˆ ∂ ) is the smallest connected boundary arc containing allmarked points except x (resp. y ); K (resp. ˆ K ) is a chain growing at x (resp. y ) generated by the continuoustrace γ (resp. ˆ γ ). We denote c s,t = ( D \ ( K s ∪ ˆ K t ) , γ s , z , . . . z n , ˆ γ t ); also Z ( c ) = ψ ( c ) Q i For any η > small enough, there exists C = C ( D, η ) > such that for all chains K, ˆ K , ≤ s ′ ≤ s, ≤ t ′ ≤ t with dist( K s , ˆ K t ) ≥ η , dist( K s , ∂ ) ≥ η , dist( ˆ K t , ˆ ∂ ) ≥ η , C − < ℓ s,ts ′ ,t ′ < C roof. From the identity: ℓ s,ts ′ ,t ′ = ℓ s ′ ,t ′ , ℓ s,t , ( ℓ s ′ ,t , ℓ s,t ′ , ) − , it is enough to prove the bound for s ′ = t ′ = 0. Frome.g. Corollary 2.8 in [9], it is enough to prove it in any reference Jordan domain, say the upper semidisk,with all marked points on the segment ( − , z .In the bounded domain D , the total mass of loops of diameter at least η in the loop measure µ loop isfinite; this gives uniform bounds above and below for the factor exp( − λm ( . . . )).Consider the set S of quadruplets ( K, x ′ , ˆ K, y ′ ) where K, ˆ K are compact subsets of D , K, ˆ K connected,with x ′ , y ′ on their respective boundaries, dist( K, K ′ ) ≥ η , dist( K, ∂ ) ≥ η , dist( ˆ K, ˆ ∂ ) ≥ η , and x ′ (resp. y ′ )corresponds to a single prime end on D \ ( K ∪ ˆ K ). (This last condition is always satisfied “at the tip”). The set S is compact (for Hausdorff convergence of compact subsets of D ). To such a quadruplet are associated fourconfigurations: c , = c , c , = ( D \ K, x ′ , y, z , . . . , z n ), c , = ( D \ ˆ K, x, y ′ , . . . ), c , = ( D \ ( K ∪ ˆ K, x ′ , y ′ , . . . ).Then the ratio Z ( c ) Z ( c ) Z ( c ) Z ( c ) defines a positive function on the compact set S . It is enough to prove that thisfunction is continuous.Let ( K n , x ′ n , ˆ K n , y ′ n ) n converge to ( K, x ′ , ˆ K, y ′ ). By Schwarz reflection across [ − , 1] and the Carath´eodoryconvergence theorem ([9], Theorem 1.8), the conformal equivalence φ n between D n = D \ ( K n ∪ ˆ K n ) and D = D \ ( K ∪ ˆ K ), extended by reflection and normalized by φ n ( z ) = z , φ n ( z ) > 0, converges locallyuniformly away from the unit circle and ( K ∪ K ∪ ˆ K ∪ ˆ K ) (here K is the conjugate of K ). The same holdsfor D n = D \ K n and D n = D \ ˆ K n .Fix small semidisks D ( z i , η/ 2) around the marked points z i ’s; the choice of D as the upper semidisk givesa choice of local coordinates at the z i ’s. Then the H D n.. ( z i , z j ) are numbers; they can be decomposed into:excursion harmonic measure in the semidisks D ( z i , η/ D ( z j , η/ C ( z i , η/ C ( z j , η/ φ n.. near the z i ’s. This provescontinuity of the H D n.. ( z i , z j ).The treatment of ratios of type H D n , ( x ′ n , z i ) /H D n , ( x ′ n , z j ) is similar: take a crosscut δ at positive distanceof K , separating it from ˆ K and the marked points. Then the Poisson excursion kernel can be decomposedw.r.t. the first crossing (by a Brownian motion starting near x ′ n ) of δ and the last crossing of C ( z i , η/ δ converges. This gives continuity of ratios of type H D n , ( x ′ n , z i ) /H D n , ( x ′ n , z j ). Assuming without loss of generality that there are at least 2 marked points z , z , the term H ( x, y ) . can be eliminated from the partition function.The only remaining thing to check is the convergence of the cross ratios. This is immediate for thosenot involving x, y , from the convergence of the φ n.. as above. This can be done also for those involving x, y ;though for our purposes it is enough to prove that all cross-ratios (between marked points) are uniformlybounded. By comparison arguments, it is enough to prove it for cross-ratios involving x − , x + , y − , y + (insteadof x, y, x ′ , y ′ ) , where these new points are on [ − , 1] and are such that the interval ( x − , x + ) (resp. ( y − , y + ))contains K n ∩ [ − , 1] (resp. ˆ K n ∩ [ − , n large enough, and no other marked point. This then reducesto the previous situation. SLE κ ( ρ ) We need to establish some simple qualitative properties of SLE κ ( ρ ) in, say, a reference configuration c =( H , , ∞ , z , . . . , z n ). In particular, we are interested in the position of the trace the first time a marked pointis swallowed.Assume that n = 2, 0 < z < z < ∞ . Then the SLE is well defined up to swallowing of z at time τ = τ z . There are several possibilities: τ = ∞ ; γ τ = z ; γ τ ∈ ( z , z ); γ τ = z ; γ τ ∈ ( z , ∞ ); or γ τ Y t = g t ( z ) − W t g t ( z ) − W t and ds = dt ( g t ( z ) − W t ) = − d log g ′ t ( z ). Then: dY s = (1 − Y s ) (cid:20) √ κdB s + ( ρ + 2 Y s + ρ + 2 − κ ) dt (cid:21) where B is a standard Brownian motion. This is a diffusion on [0 , g ′ t ( z ) is positive before τ , and goes to zero at t ր τ . A scale function of this diffusion is F : F ( y ) = Z y / u − κ (2+ ρ ) (1 − u ) κ (4+ ρ + ρ − κ ) du. It blows up at 0 if ρ ≥ κ − 2. This means that Y does not reach 0 in finite time, so that τ = τ a.s.(possibly infinite). If ρ < κ − ρ + ρ ≤ κ − 4, the scale function blows up at 1, not 0, meaning that τ < τ a.s.Assume that the trace has an accumulation point in [ z , z ) as t ր τ . Then ( Y t ) accumulates at 0 as t ր τ . This can be seen by interpreting ( g t ( z ) − W t ) as the limit of the probability divided by y that aBrownian motion started at iy , y ≫ 1, exits H \ K t on the boundary arc [ γ t , z ]. Consider a time wherethe trace is near an accumulation point in [ z , z ) In order to exit on [ γ t , z ], the Brownian motion has toget near z , and then move through a strait where the trace accumulates; this conditional probability iscontrolled by Beurling’s estimate.The following lemma gives conditions under which the exit point of an SLE κ ( ρ ) process (first disconnectionof a marked point) can be located (at least on a segment). The statement is in terms of accumulation points,which will be enough for our purposes. It is likely that a stronger statement (in terms of limits) holds. Lemma 15. Consider an SLE κ ( ρ ) in ( H , , ∞ , z , . . . , z n ) , < z < · · · < z n . Let ρ k = ρ + · · · + ρ k , ρ n = κ − .1. Assume that for some k , ρ i ≥ κ − for i < k and ρ i ≤ κ − for k ≤ i < n . Then a.s. as t ր τ , γ t accumulates at z k and at no other point in [ z , z n ] .2. Assume that for some k , ρ i ≥ κ − for i < k ; ρ k ∈ ( κ − , κ − ; and ρ i ≤ κ − for k < i < n . Thena.s. as t ր τ , γ t accumulates at a point in [ z k , z k +1 ] and at no point in [ z , z n ] \ [ z k , z k +1 ] .Proof. While the results are fairly intuitive from a simple Bessel dimension count, complete arguments area bit involved.1. a) Case k = n = 2. By a change of coordinates, one can send z to ∞ . Then the SLE is defined forall times, τ = τ = ∞ . This implies that the trace is unbounded (accumulates at z = ∞ ). Moreover, forany z ∈ ( z , z ), the (time changed) diffusion Y t = ( g t ( z ) − W t ) / ( g t ( z ) − W t ) goes to 1 as t ր τ = ∞ , bya study of its scale function. In particular, it does not accumulate at 0; hence the trace does not accumulatein [ z , z ). So the only point of accumulation of the trace in [ z , z ] is z .b) Case k = n ≥ 2. Again, we change coordinates so that z n = ∞ . Let ρ = κ − ρ ′ , ρ i = ρ ′ i − ρ ′ i − .By assumption, ρ ′ i ≥ i < n . Consider the SDE (notations as in Lemma 12): dZ t = 2 Z t dt − √ κdB t + n − X i =1 ρ i Z it dt = −√ κdB t + κ . dtZ t + n − X i =1 ρ ′ i Z i +1 t − Z it Z it Z i +1 t dt the last sum being nonnegative. By a stochastic domination argument (comparison with a Bessel process, δ = 2), this shows that τ = ∞ . Hence the process is defined for all times and the trace is unbounded.19ext we prove that there is no point of accumulation of the trace in [ z , z n − ]. Take a small neighbourhood U of [ z , z n − ]. Let σ n the first time the trace goes at distance n (an a.s. finite stopping time). Let U n bethe connected component of 1 in ( g σ n ( U ) − W t ) / ( g σ n ( z n − ) − W t ). By harmonic measure estimates, it iseasy to see that U n is contained in an arbitrarily small neighbourhood of 1 as n → ∞ , say D (1 , ε n ). By a),with probability 1 − o (1), an SLE κ ( ρ n − ) in ( H , , ∞ , 1) does not intersect D (1 , √ ε n ). On the other hand,the density of the SLE κ ( ρ ) starting with all marked points in D (1 , ε n ) w.r.t. to SLE κ ( ρ n − ) in ( H , , ∞ , o (1) on the event that the trace does not intersect D (1 , √ ε n ). This follows from an inspection ofthe densities (Lemma 12) and the fact that on the event { γ ∩ D (0 , √ ε n ) = ∅ } , ( g ′ t ( z ) /g ′ t (1) − 1) is smalluniformly in t and z ∈ [1 − ε n , ε n ]. Indeed, Brownian excursions starting from 1 and z couple with highprobability before exiting D (1 , √ ε n ).This proves that with probability 1 − o (1), the original SLE does not return to U after σ n . Notice thatone can insert a point z ′ n between z n − and z n with ρ n = 0 and the result still applies. This shows thatthere is no point of accumulation in [ z , z n ).c) Case n = 3, k = 2. We prove that the trace does not accumulate at z (similarly, at z ). By sending z at infinity, it is easy to see that the half-plane capacity of the hull at τ seen from z is finite a.s. (onecan even compute its Laplace transform). We have to rule out that the hull is unbounded while havingfinite half-plane capacity. It is enough to prove that the driving process stopped at τ stays bounded. Since Z t = g t ( z ) − W t goes to zero as t ր τ , it is enough to prove that R τ dsZ s is finite. Consider the SDE for Z t : dZ t = −√ κdB t + ρ + ρ + 2 + ε t Z t where ε t = ρ (1 − Z t /Z t ); ε t goes to zero as t ր τ (by studying the time changed diffusion ( Z t /Z t )). Onecan proceed with a comparison with Bessel processes. On the event { ε t ∈ [0 , ε ] , t ≥ t } , for t ≥ t , Z t isbetween a Bessel( δ − ε ) and a Bessel( δ ), δ = 1 + 2 ρ + ρ +2 κ ≤ − κ (both hit zero in finite time). Let t bethe first time the ratio of the two bounding Bessel processes X − and X + is 2; restart them at t from thesame position Z t , and define inductively t i , i > 1. One can think of restarting the majorizing Bessel processat a lower level at t as waiting for the Bessel to reach level Z t . This proves that R τt dsZ s ≤ R τ t dsX + s , whichis finite.d) General case. Send z k to infinity by a change of coordinate. The conditions on the ρ i ’s are rephrasedas: ρ , ρ + ρ , . . . , ρ + · · · + ρ k − ≥ κ − ρ n , ρ n + ρ n − , . . . , ρ n + · · · + ρ k +1 ≥ κ − σ n be the time of first exit of D (0 , n ) by γ . Rescale the processso that g t ( z ) (resp. g t ( z n ), W t ) is sent to − w t ) at t = σ n . If w t is away from ± 1, one can reasonas in b) from the result of c). If w is close to 1, say, one can rescale by sending w to 0 (and 1) is fixed. Theresulting process has density very close to 1 with a process of type b) as long as w t stays close to 1. When w t separates from 1, one can apply c) with a density argument.2. a) Case n = 2 , k = 1. It is easily seen by sending z (or z ) to infinity that the trace is defined fora finite time. Reasoning as in 1c) shows that the trace is bounded. Hence it accumulates somewhere in( z , z ), but not at z (and by symmetry z ).b) General case. Send z k to infinity (so that z k +1 < z , . . . , z k − being sent to infinitywhile z k +2 , . . . , z n are sent to z k +1 . It is easily seen that for that process, τ k +1 < τ . Consequently, this isalso the case for the original process, viz. the trace accumulates on [ z k , z n ] without accumulating on [ z , z k ).As in 1d), the situation is symmetric, so there is also no accumulation on ( z k +1 , z n ].20 eferences [1] J. Dub´edat. SLE( κ, ρ ) martingales and duality. Ann. Probab. , 33(1):223–243, 2005.[2] J. Dub´edat. Euler integrals for commuting SLEs. Journal Statist. Phys. , 123(6):1183–1218, 2006.[3] J. Dub´edat. Commutation relations for SLE. Comm. Pure Applied Math. , 60(12):1792–1847, 2007.[4] M. J. Kozdron and G. F. Lawler. The configurational measure on mutually avoiding SLE paths. 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