Half-plane capacity Recall: g t ( z ) = z + a − 1 z − 1 + a − 2 z − 2 + . . . hcap( K t ) := a − 1 is the “size” of K t . Lemma (additivity) hcap( K t + s ) = hcap( K t ) + hcap( g t ( K t + s \ K t )) . g t = z + a − 1 z − 1 + . . . z �→ z + b − 1 z − 1 + . . . g t ( K t + s \ K t ) K t 0 g t + s = z + ( a − 1 + b − 1 ) z − 1 + . . . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 13 / 39
Half-plane capacity Recall: g t ( z ) = z + a − 1 z − 1 + a − 2 z − 2 + . . . hcap( K t ) := a − 1 is the “size” of K t . Lemma (additivity) hcap( K t + s ) = hcap( K t ) + hcap( g t ( K t + s \ K t )) . Lemma (scaling) hcap( rK t ) = r 2 hcap( K t ) rK t z �→ rz K t Observe that � g t ( z ) := rg t ( z / r ) is the mapping out function of rK t and that 0 0 g t rg t ( · /r ) g t ( z ) = z + r 2 hcap( K t ) z − 1 + . . . � z �→ rz 0 0 N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 13 / 39
Half-plane capacity Recall: g t ( z ) = z + a − 1 z − 1 + a − 2 z − 2 + . . . hcap( K t ) := a − 1 is the “size” of K t . Lemma (additivity) hcap( K t + s ) = hcap( K t ) + hcap( g t ( K t + s \ K t )) . Lemma (scaling) hcap( rK t ) = r 2 hcap( K t ) Convention: Parametrize η such that hcap( K t ) = 2 t . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 13 / 39
Driving function and Loewner equation η simple curve in ( H , 0 , ∞ ) parametrized by half-plane capacity. Definition (Driving function) W ( t ) := g t ( η ( t )) Proposition (Loewner equation) If τ z = inf { t ≥ 0 : z ∈ K t } then 2 g t ( z ) = ˙ g t ( z ) − W ( t ) for t ∈ [0 , τ z ) , g 0 ( z ) = z ∈ H . g t η | [0 ,t ] 0 0 W ( t ) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 14 / 39
Schramm’s idea Key idea: study W instead of η . If η describes the conjectural scaling limit of certain discrete models, then W must be a multiple of a Brownian motion! g t η | [0 ,t ] 0 0 W ( t ) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 15 / 39
Definition of SLE κ in ( H , 0 , ∞ ) κ ≥ 0 and ( B ( t )) t ≥ 0 is a standard Brownian motion. Solve Loewner equation with driving function W = √ κ B 2 τ z = sup { t ≥ 0 : g t ( z ) well-defined } . g t ( z ) = ˙ g t ( z ) − W ( t ) , Define K t := { z ∈ H : τ z ≤ t } . Let η be the curve generating ( K t ) t ≥ 0 . K t = H \ { unbounded component of H \ η ([0 , t ]) } , η is well-defined: Rohde-Schramm’05, Lawler-Schramm-Werner’04. Definition (The Schramm-Loewner evolution in ( H , 0 , ∞ )) η is an SLE κ in ( H , 0 , ∞ ). ( B ( t )) t ≥ 0 ( g t ) t ≥ 0 ( K t ) t ≥ 0 ( η ( t )) t ≥ 0 N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 16 / 39
Definition of SLE κ in ( D , a , b ) b f � η η D a 0 Definition (The Schramm-Loewner evolution) Let � η be an SLE κ in ( H , 0 , ∞ ). Then η := f ( � η ) is an SLE κ in ( D , a , b ). Note that f is not unique since f ◦ φ r also sends ( H , 0 , ∞ ) to ( D , a , b ) if φ r ( z ) := rz for r > 0. SLE κ in ( D , a , b ) is still well-defined by scale invariance in law of SLE κ in ( H , 0 , ∞ ) (next slide). N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 17 / 39
Scale invariance in law of SLE κ Exercise (Scale invariance of SLE κ ) Let η be an SLE κ in ( H , 0 , ∞ ) and let r > 0 . Prove that t �→ r η ( t / r 2 ) has the law of an SLE κ in ( H , 0 , ∞ ) . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 18 / 39
Scale invariance in law of SLE κ Exercise (Scale invariance of SLE κ ) Let η be an SLE κ in ( H , 0 , ∞ ) and let r > 0 . Prove that t �→ r η ( t / r 2 ) has the law of an SLE κ in ( H , 0 , ∞ ) . η ( t ) = r η ( t / r 2 ) and argue that mapping out fcn � Hint: Let � g t of � η satisfy � � 2 ˙ g t ( z ) = rg t / r 2 ( z / r ) , � g t ( z ) = ∂ t � rg t / r 2 ( z / r ) = g t ( z ) − rW ( t / r 2 ) . � η | [0 ,t ] � z �→ rz η | [0 ,t/r 2 ] 0 0 � g t/r 2 g t z �→ rz 0 0 N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 18 / 39
Conformal invariance and domain Markov property Probability measure µ D , a , b on curves η modulo time reparametrization in ( D , a , b ) for each simply connected domain D ⊂ C , a , b ∈ ∂ D . 1 b η D a 1 Identify η and η ◦ φ if φ : I 1 → I 2 cts and strictly increasing. ∂ D Martin bdy of D . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 19 / 39
Conformal invariance and domain Markov property Probability measure µ D , a , b on curves η modulo time reparametrization in ( D , a , b ) for each simply connected domain D ⊂ C , a , b ∈ ∂ D . 1 Suppose η ∼ µ H , 0 , ∞ a.s. generated by Loewner chain. b η D a 1 Identify η and η ◦ φ if φ : I 1 → I 2 cts and strictly increasing. ∂ D Martin bdy of D . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 19 / 39
Conformal invariance and domain Markov property Probability measure µ D , a , b on curves η modulo time reparametrization in ( D , a , b ) for each simply connected domain D ⊂ C , a , b ∈ ∂ D . 1 Suppose η ∼ µ H , 0 , ∞ a.s. generated by Loewner chain. Conformal invariance (CI): If η ∼ µ D , a , b then φ ◦ η has law µ � b . a , � D , � b � b η φ ◦ η φ D � D a � a Conformal invariance 1 Identify η and η ◦ φ if φ : I 1 → I 2 cts and strictly increasing. ∂ D Martin bdy of D . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 19 / 39
Conformal invariance and domain Markov property Probability measure µ D , a , b on curves η modulo time reparametrization in ( D , a , b ) for each simply connected domain D ⊂ C , a , b ∈ ∂ D . 1 Suppose η ∼ µ H , 0 , ∞ a.s. generated by Loewner chain. Conformal invariance (CI): If η ∼ µ D , a , b then φ ◦ η has law µ � b . a , � D , � Domain Markov property (DMP): Conditioned on η | [0 ,τ ] for stopping time τ , the rest of the curve η | [ τ, ∞ ) has law µ D \ K τ ,η ( t ) , b . b b � b η η ([ τ, ∞ )) φ ◦ η φ D K τ = η ([0 , τ ]) � D D a � a a Conformal invariance Domain Markov property 1 Identify η and η ◦ φ if φ : I 1 → I 2 cts and strictly increasing. ∂ D Martin bdy of D . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 19 / 39
Conformal invariance and domain Markov property Probability measure µ D , a , b on curves η modulo time reparametrization in ( D , a , b ) for each simply connected domain D ⊂ C , a , b ∈ ∂ D . 1 Suppose η ∼ µ H , 0 , ∞ a.s. generated by Loewner chain. Conformal invariance (CI): If η ∼ µ D , a , b then φ ◦ η has law µ � b . a , � D , � Domain Markov property (DMP): Conditioned on η | [0 ,τ ] for stopping time τ , the rest of the curve η | [ τ, ∞ ) has law µ D \ K τ ,η ( t ) , b . Theorem (Schramm’00) The following statements are equivalent: µ D , a , b satisfies (CI) and (DMP). There is a κ ≥ 0 such that µ D , a , b is the law of SLE κ . 1 Identify η and η ◦ φ if φ : I 1 → I 2 cts and strictly increasing. ∂ D Martin bdy of D . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 19 / 39
Conformal invariance of percolation b � b a � a ⇓ ⇓ b � b µ � µ D,a,b a, � D, � b � D a � D a N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 20 / 39
Conformal invariance of percolation b � b a � a ⇓ ⇓ b � b φ µ � µ D,a,b a, � D, � b � D a � D a Conformal invariance: If η ∼ µ D , a , b then φ ◦ η has law µ � b . a , � D , � N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 20 / 39
Outline Lecture 1: Definition and basic properties of SLE, examples Lecture 2: Basic properties of SLE (today) Lecture 3: Imaginary geometry References: Conformally invariant processes in the plane by Lawler Lectures on Schramm-Loewner evolution by Berestycki and Norris Imaginary geometry I by Miller and Sheffield Key message today: The Loewner equation allows us to analyze SLE using stochastic calculus. N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 21 / 39
Domain Markov property of percolation b a N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 22 / 39
Domain Markov property of percolation Conditioned on η | [0 , 25] , the rest of the percolation interface has the law of a percolation interface in ( D \ K 25 , η (25) , b ). b D \ K 25 η (25) K 25 D a N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 22 / 39
Domain Markov property of the self-avoiding walk Number of length n self-avoiding paths on Z 2 from (0 , 0): µ n (1+ o (1)) . (0 , 0) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 23 / 39
Domain Markov property of the self-avoiding walk Number of length n self-avoiding paths on Z 2 from (0 , 0): µ n (1+ o (1)) . µ ∈ [2 . 62 , 2 . 68] is the connective constant of Z 2 . (0 , 0) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 23 / 39
Domain Markov property of the self-avoiding walk Number of length n self-avoiding paths on Z 2 from (0 , 0): µ n (1+ o (1)) . µ ∈ [2 . 62 , 2 . 68] is the connective constant of Z 2 . The self-avoiding walk (SAW) : W random path s.t. for w a self-avoiding path on discrete approximation ( D m , a m , b m ) to ( D , a , b ), P [ W = w ] = c µ −| w | , where | w | is the length of w and c is a renormalizing constant. b m 1 m W a m (0 , 0) D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 23 / 39
Domain Markov property of the self-avoiding walk Number of length n self-avoiding paths on Z 2 from (0 , 0): µ n (1+ o (1)) . µ ∈ [2 . 62 , 2 . 68] is the connective constant of Z 2 . The self-avoiding walk (SAW) : W random path s.t. for w a self-avoiding path on discrete approximation ( D m , a m , b m ) to ( D , a , b ), P [ W = w ] = c µ −| w | , where | w | is the length of w and c is a renormalizing constant. Conjecture: W ⇒ SLE 8 / 3 . b m 1 m W a m (0 , 0) D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 23 / 39
Domain Markov property of the self-avoiding walk Number of length n self-avoiding paths on Z 2 from (0 , 0): µ n (1+ o (1)) . µ ∈ [2 . 62 , 2 . 68] is the connective constant of Z 2 . The self-avoiding walk (SAW) : W random path s.t. for w a self-avoiding path on discrete approximation ( D m , a m , b m ) to ( D , a , b ), P [ W = w ] = c µ −| w | , where | w | is the length of w and c is a renormalizing constant. Conjecture: W ⇒ SLE 8 / 3 . Exercise: Given W | [0 , k ] the remaining path has the law of a SAW in ( D m \ W ([0 , k ]) , W ( k ) , b m ). b m W ( k ) a m (0 , 0) D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 23 / 39
SLE satisfies (CI) and (DMP) (CI): follows from the definition of SLE κ on general domains ( D , a , b ). b f � η η D a 0 N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 24 / 39
SLE satisfies (CI) and (DMP) (CI): follows from the definition of SLE κ on general domains ( D , a , b ). (DMP): sufficient to verify for ( H , 0 , ∞ ) and parametrization by half-plane capacity. η ( τ ) K τ 0 Want to prove: η | [ τ, ∞ ) has the law of an SLE κ in ( H \ K τ , η ( τ ) , ∞ ). N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 24 / 39
SLE satisfies (CI) and (DMP) (CI): follows from the definition of SLE κ on general domains ( D , a , b ). (DMP): sufficient to verify for ( H , 0 , ∞ ) and parametrization by half-plane capacity. Centered mapping out functions � g t ( z ) := g t ( z ) − W ( t ) satisfy 2 d � g t ( z ) − dW ( t ) , � . g t ( z ) = g 0 ( z ) = z . (CL) � η τ satisfy Exercise: Centered mapping out functions ( � g τ, t ) t ≥ 0 of � � g τ + t = � g τ, t ◦ � g τ . Exercise: Use previous exercise to argue that ( � g τ, t ) t ≥ 0 satisfies (CL) d w/driving function ( W ( τ + t ) − W ( τ )) t ≥ 0 = ( W ( t )) t ≥ 0 . η τ has the law of an SLE κ in ( H , 0 , ∞ ). The last exercise implies that � � η ( τ + t ) � g τ,t g τ η τ � η ( τ ) 0 0 0 � g τ + t N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 24 / 39
(CI) and (DMP) imply that η is an SLE Suppose ( µ D , a , b ) D , a , b satisfies (CI) and (DMP). Let η ∼ µ H , 0 , ∞ be param. by half-plane capacity; let W denote the driving fcn of η . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 25 / 39
(CI) and (DMP) imply that η is an SLE Suppose ( µ D , a , b ) D , a , b satisfies (CI) and (DMP). Let η ∼ µ H , 0 , ∞ be param. by half-plane capacity; let W denote the driving fcn of η . d = ( rW ( t / r 2 )) t ≥ 0 . (CI) ⇒ scale invariance ⇒ ( W ( t )) t ≥ 0 η | [0 ,t ] � z �→ rz η | [0 ,t/r 2 ] 0 0 � g t/r 2 g t z �→ rz 0 0 η ( t ) := r η ( t / r 2 ). Then η d Let � = � η . Mapping out fcn ( � g t ) t ≥ 0 of � η satisfy: � � 2 ˙ � � g t ( z ) = rg t / r 2 ( z / r ) , g t ( z ) = ∂ t rg t / r 2 ( z / r ) = g t ( z ) − rW ( t / r 2 ) . � N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 25 / 39
(CI) and (DMP) imply that η is an SLE Suppose ( µ D , a , b ) D , a , b satisfies (CI) and (DMP). Let η ∼ µ H , 0 , ∞ be param. by half-plane capacity; let W denote the driving fcn of η . d = ( rW ( t / r 2 )) t ≥ 0 . (CI) ⇒ scale invariance ⇒ ( W ( t )) t ≥ 0 (DMP) η ( s ) K s 0 (DMP): η | [ s , ∞ ) has law µ H \ K s ,η ( s ) , ∞ . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 25 / 39
(CI) and (DMP) imply that η is an SLE Suppose ( µ D , a , b ) D , a , b satisfies (CI) and (DMP). Let η ∼ µ H , 0 , ∞ be param. by half-plane capacity; let W denote the driving fcn of η . d = ( rW ( t / r 2 )) t ≥ 0 . (CI) ⇒ scale invariance ⇒ ( W ( t )) t ≥ 0 (DMP) ⇒ ( W ( t )) t ≥ 0 has i.i.d. increments. η s d η s is independent of η | [0 , s ] . By (DMP), � = η and � η s satisfy the centered The centered mapping out fcn ( � g s , t ) t ≥ 0 of � Loewner equation w/driving function ( W ( s + t ) − W ( s )) t ≥ 0 . d Combining the above, ( W ( s + t ) − W ( s )) t ≥ 0 = ( W ( t )) t ≥ 0 and is independent of W | [0 , s ] . � η ( s + t ) � g s,t g s η s � η ( s ) 0 0 0 � g s + t N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 25 / 39
(CI) and (DMP) imply that η is an SLE Suppose ( µ D , a , b ) D , a , b satisfies (CI) and (DMP). Let η ∼ µ H , 0 , ∞ be param. by half-plane capacity; let W denote the driving fcn of η . d = ( rW ( t / r 2 )) t ≥ 0 . (CI) ⇒ scale invariance ⇒ ( W ( t )) t ≥ 0 (DMP) ⇒ ( W ( t )) t ≥ 0 has i.i.d. increments. (CI) + (DMP) ⇒ W = √ κ B for some κ ≥ 0. N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 25 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. b a κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. b a κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. b a κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. b a κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. b a κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. b a κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phases of SLE Rohde-Schramm’05: SLE κ has the following phases: κ ∈ [0 , 4]: The curve is simple. κ ∈ (4 , 8): The curve is self-intersecting and has zero Lebesgue measure. κ ≥ 8: The curve fills space. κ ∈ [0 , 4] κ ∈ (4 , 8) κ ≥ 8 Figures by P. Nolin, W. Werner, and J. Miller N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 26 / 39
Phase transition at κ = 4 Lemma If κ ∈ [0 , 4] then η is a.s. simple (i.e., η ( t 1 ) � = η ( t 2 ) for t 1 � = t 2 ). If κ > 4 then η is a.s. not simple. We will deduce the lemma from the following result, where τ x = inf { t ≥ 0 : x ∈ K t } for x > 0 . Lemma If κ ∈ [0 , 4] then τ x = ∞ a.s. If κ > 4 then τ x < ∞ a.s. η η η ( τ x ) 0 0 x = η ( τ x ) x N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 27 / 39
Phase transition at κ = 4 Recall τ x = inf { t ≥ 0 : x ∈ K t } for x > 0. Lemma If κ ∈ [0 , 4] then τ x = ∞ a.s. If κ > 4 then τ x < ∞ a.s. 2 g t (1) − √ κ B ( t ) , w.l.o.g. x = 1; g t (1) = ˙ g t (1) = 1 , N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 28 / 39
Phase transition at κ = 4 Recall τ x = inf { t ≥ 0 : x ∈ K t } for x > 0. Lemma If κ ∈ [0 , 4] then τ x = ∞ a.s. If κ > 4 then τ x < ∞ a.s. 2 w.l.o.g. x = 1; g t (1) = ˙ g t (1) − √ κ B ( t ) , g t (1) = 1 , Y ( t ) = κ − 1 / 2 ( g t (1) − √ κ B ( t )) , g t η √ κY ( t ) 0 1 0 √ κB ( t ) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 28 / 39
Phase transition at κ = 4 Recall τ x = inf { t ≥ 0 : x ∈ K t } for x > 0. Lemma If κ ∈ [0 , 4] then τ x = ∞ a.s. If κ > 4 then τ x < ∞ a.s. 2 g t (1) − √ κ B ( t ) , w.l.o.g. x = 1; g t (1) = ˙ g t (1) = 1 , Y ( t ) = κ − 1 / 2 ( g t (1) − √ κ B ( t )) , τ 1 = inf { t ≥ 0 : Y ( t ) = 0 } , g t η √ κY ( t ) 0 1 0 √ κB ( t ) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 28 / 39
Phase transition at κ = 4 Recall τ x = inf { t ≥ 0 : x ∈ K t } for x > 0. Lemma If κ ∈ [0 , 4] then τ x = ∞ a.s. If κ > 4 then τ x < ∞ a.s. 2 g t (1) − √ κ B ( t ) , w.l.o.g. x = 1; g t (1) = ˙ g t (1) = 1 , Y ( t ) = κ − 1 / 2 ( g t (1) − √ κ B ( t )) , τ 1 = inf { t ≥ 0 : Y ( t ) = 0 } , � 4 � 2 κ Y ( t ) dt − dB ( t ) , so Y ( t ) is a dY ( t ) = κ + 1 -dim. Bessel process. g t η Y ( t ) √ κY ( t ) Y ( t ) √ κB ( t ) 0 dim ≥ 2 1 0 dim ∈ (1 , 2) κ ∈ (0 , 4] κ ∈ (4 , ∞ ) N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 28 / 39
Phase transition at κ = 4 Lemma If κ ∈ [0 , 4] then η is a.s. simple � g t η ( t ) (i.e., η ( t 1 ) � = η ( t 2 ) for t 1 � = t 2 ). If κ > 4 then η is a.s. not 0 κ ∈ [0 , 4] simple. η ( t ) τ x = inf { t ≥ 0 : x ∈ K t } , x > 0 . � g t η ( t 1 ) = η ( t 2 ) Lemma 0 If κ ∈ [0 , 4] then τ x = ∞ a.s. κ > 4 If κ > 4 then τ x < ∞ a.s. N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 29 / 39
Locality of SLE 6 Proposition η SLE 6 in ( D , x , y ) . Set τ := inf { t ≥ 0 : η ( t ) ∈ arc( � y , y ) } . Define � η and � τ in the same way for ( D , x , � y ) . d Then η | [0 ,τ ] = � η | [0 , � τ ] . η ( τ ) y y � D x N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 30 / 39
Locality of SLE 6 Proposition η SLE 6 in ( D , x , y ) . Set τ := inf { t ≥ 0 : η ( t ) ∈ arc( � y , y ) } . Define � η and � τ in the same way for ( D , x , � y ) . d Then η | [0 ,τ ] = � η | [0 , � τ ] . � y y x N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 30 / 39
Locality of SLE 6 Proposition η SLE 6 in ( D , x , y ) . Set τ := inf { t ≥ 0 : η ( t ) ∈ arc( � y , y ) } . Define � η and � τ in the same way for ( D , x , � y ) . d Then η | [0 ,τ ] = � η | [0 , � τ ] . η L y Want to prove: If η is an SLE 6 in ( H , 0 , ∞ ) then η has the law of an SLE 6 in ( H , 0 , y ) until hitting L . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 30 / 39
Locality of SLE 6 : Proof sketch η ∗ η Φ L ∗ L y Φ( ∞ ) g ∗ g t t t ◦ Φ ◦ g − 1 Φ t := g ∗ t W ( t ) W ∗ ( t ) η SLE 6 in ( H , 0 , ∞ ); g t mapping out function; W driving function. N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 31 / 39
Locality of SLE 6 : Proof sketch η ∗ η Φ L ∗ L y Φ( ∞ ) g ∗ g t t t ◦ Φ ◦ g − 1 Φ t := g ∗ t W ( t ) W ∗ ( t ) η SLE 6 in ( H , 0 , ∞ ); g t mapping out function; W driving function. Φ conformal map sending ( H , 0 , y ) to ( H , 0 , ∞ ). N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 31 / 39
Locality of SLE 6 : Proof sketch η ∗ η Φ L ∗ L y Φ( ∞ ) g ∗ g t t t ◦ Φ ◦ g − 1 Φ t := g ∗ t W ( t ) W ∗ ( t ) η SLE 6 in ( H , 0 , ∞ ); g t mapping out function; W driving function. Φ conformal map sending ( H , 0 , y ) to ( H , 0 , ∞ ). η ∗ ( t ) := Φ( η ( t )); g ∗ t map. out fcn; W ∗ ( t ) = Φ t ( W ( t )) driving fcn. b ′ ( t ) g ∗ b ( t ) = hcap( η ∗ ([0 , t ])) . ˙ t ( z ) = t ( z ) − W ∗ ( t ) , g ∗ N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 31 / 39
Locality of SLE 6 : Proof sketch η ∗ η Φ L ∗ L y Φ( ∞ ) g ∗ g t t t ◦ Φ ◦ g − 1 Φ t := g ∗ t W ( t ) W ∗ ( t ) η SLE 6 in ( H , 0 , ∞ ); g t mapping out function; W driving function. Φ conformal map sending ( H , 0 , y ) to ( H , 0 , ∞ ). η ∗ ( t ) := Φ( η ( t )); g ∗ t map. out fcn; W ∗ ( t ) = Φ t ( W ( t )) driving fcn. b ′ ( t ) g ∗ b ( t ) = hcap( η ∗ ([0 , t ])) . ˙ t ( z ) = t ( z ) − W ∗ ( t ) , g ∗ Want to show: η ∗ law of SLE 6 in ( H , 0 , ∞ ) until hitting L ∗ . N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 31 / 39
Locality of SLE 6 : Proof sketch η ∗ η Φ L ∗ L y Φ( ∞ ) g ∗ g t t t ◦ Φ ◦ g − 1 Φ t := g ∗ t W ( t ) W ∗ ( t ) η SLE 6 in ( H , 0 , ∞ ); g t mapping out function; W driving function. Φ conformal map sending ( H , 0 , y ) to ( H , 0 , ∞ ). η ∗ ( t ) := Φ( η ( t )); g ∗ t map. out fcn; W ∗ ( t ) = Φ t ( W ( t )) driving fcn. b ′ ( t ) g ∗ b ( t ) = hcap( η ∗ ([0 , t ])) . ˙ t ( z ) = t ( z ) − W ∗ ( t ) , g ∗ Want to show: η ∗ law of SLE 6 in ( H , 0 , ∞ ) until hitting L ∗ . √ 6 B ∗ ( b ( t ) / 2) for B ∗ std Brownian motion. Equivalently, W ∗ ( t ) = N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 31 / 39
Locality of SLE 6 : Proof sketch η ∗ η Φ L ∗ L y Φ( ∞ ) g ∗ g t t t ◦ Φ ◦ g − 1 Φ t := g ∗ t W ( t ) W ∗ ( t ) η SLE 6 in ( H , 0 , ∞ ); g t mapping out function; W driving function. Φ conformal map sending ( H , 0 , y ) to ( H , 0 , ∞ ). η ∗ ( t ) := Φ( η ( t )); g ∗ t map. out fcn; W ∗ ( t ) = Φ t ( W ( t )) driving fcn. b ′ ( t ) g ∗ b ( t ) = hcap( η ∗ ([0 , t ])) . ˙ t ( z ) = t ( z ) − W ∗ ( t ) , g ∗ Want to show: η ∗ law of SLE 6 in ( H , 0 , ∞ ) until hitting L ∗ . √ 6 B ∗ ( b ( t ) / 2) for B ∗ std Brownian motion. Equivalently, W ∗ ( t ) = Find dW ∗ by Itˆ o’s formula; prove and use ˙ Φ t ( W ( t )) = − 3Φ ′′ t ( W ( t )). N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 31 / 39
Restriction property Definition Let µ D , x , y for D ⊂ C simply connected and x , y ∈ ∂ D be a family of probability measures on curves η in D from x to y . Let η ∼ µ D , x , y for some ( D , x , y ) and let U ⊂ D be simply connected s.t. x , y ∈ ∂ U . The measures µ D , x , y satisfy the restriction property if η conditioned to stay in U has the law of a curve sampled from µ U , x , y . For which κ ≥ 0 does SLE κ satisfy the restriction property? y η U x D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 32 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction property? N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction property? Let � W be a simple random walk on discrete approximation ( D m , a m , b m ) to ( D , a , b ). b m � 1 W a m m D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction property? Let � W be a simple random walk on discrete approximation ( D m , a m , b m ) to ( D , a , b ). The loop-erased random walk (LERW) W is loop-erasure of � W . b m W � W a m D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction prop.? NO Let � W be a simple random walk on discrete approximation ( D m , a m , b m ) to ( D , a , b ). The loop-erased random walk (LERW) W is loop-erasure of � W . Let U m ⊂ D m be connected s.t. a m , b m ∈ U m . b m W � W a m D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction prop.? NO Let � W be a simple random walk on discrete approximation ( D m , a m , b m ) to ( D , a , b ). The loop-erased random walk (LERW) W is loop-erasure of � W . Let U m ⊂ D m be connected s.t. a m , b m ∈ U m . “LERW in ( D m , a m , b m ) conditioned to stay in U m ” � = “LERW in ( U m , a m , b m )”, since the latter requires � W ⊂ U m (not just W ⊂ U m ). b m W � W a m D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction prop.? NO Does the self-avoiding walk satisfy the restriction property? The self-avoiding walk (SAW) W is s.t. for any fixed self-avoiding path w on discrete approximation ( D m , a m , b m ) to ( D , a , b ), P [ W = w ] = c µ −| w | , where µ is the connective constant, | w | is the length of w , and c is a renormalizing constant. b m W a m D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of discrete models Does the loop-erased random walk satisfy the restriction prop.? NO Does the self-avoiding walk satisfy the restriction property? YES The self-avoiding walk (SAW) W is s.t. for any fixed self-avoiding path w on discrete approximation ( D m , a m , b m ) to ( D , a , b ), P [ W = w ] = c µ −| w | , where µ is the connective constant, | w | is the length of w , and c is a renormalizing constant. “SAW in ( D m , a m , b m ) cond. to stay in U m ” d = “SAW in ( U m , a m , b m )” b m W a m D N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 33 / 39
Restriction property of SLE 8 / 3 Proposition η SLE 8 / 3 in ( H , 0 , ∞ ) ; K ⊂ H s.t. H \ K simply conn., 0 , ∞ �∈ K. Then η cond. on η ∩ K = ∅ has the law of SLE 8 / 3 in ( H \ K , 0 , ∞ ) . η K 0 N. Holden (ETH-ITS Z¨ urich) SLE and imaginary geometry August 4, 2020 34 / 39
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