Preliminary Conformal transformation Applications Sketch proof Complex Analysis in Backward SLE Dapeng Zhan Michigan State University Everything is complex Saas-Fee, March, 2016 Based on a joint work with Steffen Rohde. Dapeng Zhan Complex Analysis in Backward SLE 1 / 32
Preliminary Conformal transformation Applications Sketch proof Preliminary Dapeng Zhan Complex Analysis in Backward SLE 2 / 32
Preliminary Conformal transformation Applications Sketch proof Preliminary Conformal transformation Dapeng Zhan Complex Analysis in Backward SLE 2 / 32
Preliminary Conformal transformation Applications Sketch proof Preliminary Conformal transformation Applications Dapeng Zhan Complex Analysis in Backward SLE 2 / 32
Preliminary Conformal transformation Applications Sketch proof Preliminary Conformal transformation Applications Sketch proof Dapeng Zhan Complex Analysis in Backward SLE 2 / 32
Preliminary Conformal transformation Applications Sketch proof Preliminary Conformal transformation Applications Sketch proof Dapeng Zhan Complex Analysis in Backward SLE 3 / 32
Preliminary Conformal transformation Applications Sketch proof Schramm’s SLE process is successful in describing random fractal curves, which are the scaling limit of some critical two-dimensional lattice models, which include critical percolation ([Smi01]), loop-erased random walk and uniform spanning tree ([LSW04]), critical Ising model and critical FK-Ising model ([CDCH+13]), and etc. The definition of SLE combines the Loewner’s differential equation with a random driving function: Brownian motion. Dapeng Zhan Complex Analysis in Backward SLE 4 / 32
Preliminary Conformal transformation Applications Sketch proof Schramm’s SLE process is successful in describing random fractal curves, which are the scaling limit of some critical two-dimensional lattice models, which include critical percolation ([Smi01]), loop-erased random walk and uniform spanning tree ([LSW04]), critical Ising model and critical FK-Ising model ([CDCH+13]), and etc. The definition of SLE combines the Loewner’s differential equation with a random driving function: Brownian motion. Backward SLE uses backward Loewner equation, which differs from the forward equation by a minus sign. The goal of the joint work was to study the backward SLE process as a whole instead of only the hulls at fixed capacity times. Prior to our work, S. Sheffield proved the existence of a coupling of a backward chordal SLE κ with a free boundary GFF such that real intervals [ x , 0] and [0 , y ] with the same quantum weight are welded by the backward SLE process. Dapeng Zhan Complex Analysis in Backward SLE 4 / 32
Preliminary Conformal transformation Applications Sketch proof It turns out that, with a few modifications, the standard tools used in forward SLE can also be used to study backward SLE, as long as we find the “correct” definition of the transformation of a backward Loewner process under a conformal map. Dapeng Zhan Complex Analysis in Backward SLE 5 / 32
Preliminary Conformal transformation Applications Sketch proof It turns out that, with a few modifications, the standard tools used in forward SLE can also be used to study backward SLE, as long as we find the “correct” definition of the transformation of a backward Loewner process under a conformal map. To explain the idea, let me briefly recall some notation. ◮ H := { z ∈ C : ℑ z > 0 } is the upper half plane. ◮ An H -hull is a bounded set K ⊂ H such that H \ K is a simply connected domain. ◮ For an H -hull K , g K is the unique conformal map from H \ K onto H such that g K ( z ) = z + c ( K ) + O (1 / z 2 ) as z → ∞ . Let f K = g − 1 K . z ◮ hcap( K ) := c ( K ) is called the H -capacity of K . We have hcap( ∅ ) = 0 and hcap( K 1 ) < hcap( K 2 ) if K 1 � K 2 . Dapeng Zhan Complex Analysis in Backward SLE 5 / 32
Preliminary Conformal transformation Applications Sketch proof The double of K : K doub is the union of K and the reflection of K about R . By Schwarz reflection principle, g K extends to a conformal map from C \ K doub onto C \ S K for some compact S K ⊂ R called the support of K . Dapeng Zhan Complex Analysis in Backward SLE 6 / 32
Preliminary Conformal transformation Applications Sketch proof The double of K : K doub is the union of K and the reflection of K about R . By Schwarz reflection principle, g K extends to a conformal map from C \ K doub onto C \ S K for some compact S K ⊂ R called the support of K . If K 1 ⊂ K 2 are two H -hulls, we define K 2 / K 1 = g K 1 ( K 2 \ K 1 ), which is also an H -hull. We call K 2 / K 1 a quotient hull of K 2 , and write K 2 / K 1 ≺ K 2 . If K 3 ≺ K 2 , then hcap( K 3 ) ≤ hcap( K 2 ), S K 3 ⊂ S K 2 , and there is a unique K 1 ⊂ K 2 s.t. K 3 = K 2 / K 1 . We write K 1 = K 2 : K 3 . Dapeng Zhan Complex Analysis in Backward SLE 6 / 32
Preliminary Conformal transformation Applications Sketch proof An H -Loewner chain is a strictly increasing family of H -hulls ( K t ) 0 ≤ t < T , which starts from K 0 = ∅ , and satisfies that � K t + ε / K t = { λ t } , 0 ≤ t < T , 0 <ε< T − t for some real continuous function λ t , 0 ≤ t < T . Dapeng Zhan Complex Analysis in Backward SLE 7 / 32
Preliminary Conformal transformation Applications Sketch proof An H -Loewner chain is a strictly increasing family of H -hulls ( K t ) 0 ≤ t < T , which starts from K 0 = ∅ , and satisfies that � K t + ε / K t = { λ t } , 0 ≤ t < T , 0 <ε< T − t for some real continuous function λ t , 0 ≤ t < T . If u is a continuously (strictly) increasing function with u (0) = 0, then K u − 1 ( t ) , 0 ≤ t < u ( T ), is also an H -Loewner chain, and is called a time-change of ( K t ). An H -Loewner chain is called normalized if hcap( K t ) = 2 t for each t . Every H -Loewner chain can be normalized by applying a time-change. Dapeng Zhan Complex Analysis in Backward SLE 7 / 32
Preliminary Conformal transformation Applications Sketch proof An H -Loewner chain is a strictly increasing family of H -hulls ( K t ) 0 ≤ t < T , which starts from K 0 = ∅ , and satisfies that � K t + ε / K t = { λ t } , 0 ≤ t < T , 0 <ε< T − t for some real continuous function λ t , 0 ≤ t < T . If u is a continuously (strictly) increasing function with u (0) = 0, then K u − 1 ( t ) , 0 ≤ t < u ( T ), is also an H -Loewner chain, and is called a time-change of ( K t ). An H -Loewner chain is called normalized if hcap( K t ) = 2 t for each t . Every H -Loewner chain can be normalized by applying a time-change. Example. We say that γ t , 0 ≤ t < T , an H -simple curve, if γ 0 ∈ R and γ t ∈ H for t > 0. An H -simple curve γ generates an H -Loewner chain: K t = γ (0 , t ], 0 ≤ t < T . Dapeng Zhan Complex Analysis in Backward SLE 7 / 32
Preliminary Conformal transformation Applications Sketch proof Let λ ∈ C ([0 , T ) , R ). The (forward) chordal Loewner equation driven by λ is 2 ∂ t g t ( z ) = , g 0 ( z ) = z . g t ( z ) − λ t For 0 ≤ t < T , let K t denote the set of z ∈ H such that the solution s �→ g s ( z ) blows up before or at time t . Then each K t is an H -hull with hcap( K t ) = 2 t and g K t = g t . We call g t and K t the chordal Loewner maps and hulls driven by λ . Chordal SLE κ is defined by taking λ t = √ κ B t , where κ > 0 and B t is a standard Brownian motion. Dapeng Zhan Complex Analysis in Backward SLE 8 / 32
Preliminary Conformal transformation Applications Sketch proof Let λ ∈ C ([0 , T ) , R ). The (forward) chordal Loewner equation driven by λ is 2 ∂ t g t ( z ) = , g 0 ( z ) = z . g t ( z ) − λ t For 0 ≤ t < T , let K t denote the set of z ∈ H such that the solution s �→ g s ( z ) blows up before or at time t . Then each K t is an H -hull with hcap( K t ) = 2 t and g K t = g t . We call g t and K t the chordal Loewner maps and hulls driven by λ . Chordal SLE κ is defined by taking λ t = √ κ B t , where κ > 0 and B t is a standard Brownian motion. Proposition [LSW01] ( K t ) are chordal Loewner hulls driven by some continuous function iff it is a normalized H -Loewner chain. Moreover, when the above holds, the driving function λ is given by � 0 <ε< T − t K t + ε / K t = { λ t } , 0 ≤ t < T . Dapeng Zhan Complex Analysis in Backward SLE 8 / 32
Preliminary Conformal transformation Applications Sketch proof The backward chordal Loewner equation driven by λ is − 2 ∂ t f t ( z ) = , f 0 ( z ) = z . f t ( z ) − λ t For every t , f t is well defined on H , and maps H conformally onto H \ L t for some H -hull L t . We have hcap( L t ) = 2 t and f t = f L t . But ( L t ) may not be an increasing family. Instead, it satisfies that L t 1 ≺ L t 2 if t 1 ≤ t 2 . To describe other properties of ( L t ), we need the notation of quotient Loewner chain. Dapeng Zhan Complex Analysis in Backward SLE 9 / 32
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