Complex Generalized Integral Means Spectrum of Whole-Plane SLE - PowerPoint PPT Presentation

Complex Generalized Integral Means Spectrum of Whole-Plane SLE ⋆ Bertrand Duplantier † , Xuan Hieu Ho ‡ Thanh Binh Le ∗ , Michel Zinsmeister ‡ ⋆⋆ Dmitry Beliaev ∗∗ , B. D. , M. Z. † Paris-Saclay/ ‡ Orl´ eans/ ∗ Quy Nhon/ ∗∗ Oxford ⋆ Commun. Math. Phys. 359 823-868 (2018) ⋆⋆ Commun. Math. Phys. 353 119-133 (2017) Random Conformal Geometry and Related Fields KIAS, Seoul, Korea June 18 – 22, 2018

Whole-Plane Schramm-Loewner Evolution f t t � ( ) t � ( )= f t ( ) � ( ) t 0 1 � ( )= 0 f 1 ( ) z t = f � 1 0 t ( ) 8 f t 0 ( )= 0 ◮ ◮ λ ( t ) − z , λ ( t ) = exp( i √ κ B t ) ∂ t f t ( z ) = z ∂ ∂ ∂ z f t ( z ) λ ( t ) + z z ∈ D , e t z f t ( e − t z ) → z , t → + ∞ ; κ = 0 , f t ( z ) = ( Koebe ) (1 − z ) 2 ◮ f [ − 1] ( z ) := 1 / f (1 / z ) is the bounded exterior version from C \ D to the slit plane [Beliaev & Smirnov, Lawler].

Integral Means Spectrum ◮ Consider an injective Riemann map Φ ∈ S , i.e., Φ : D → C , Φ(0) = 0 , Φ ′ (0) = 1 . ◮ The integral means of Φ are � 2 π | Φ ′ ( re i θ ) | p d θ, 0 < r < 1 , p ∈ R ; I ( r , p , Φ) := 0 ◮ Φ random : � 2 π | Φ ′ ( re i θ ) | p � � Expectation: E I ( r , p , Φ) := E d θ. 0 ◮ One then defines log( I ( r , p , Φ)) β Φ ( p ) := lim sup ; 1 log( 1 − r ) r → 1 − ◮ If the limit exists, 1 r → 1 − I ( r , p , Φ) ≍ (1 − r ) β Φ ( p ) .

Integral means spectrum & harmonic measure ◮ The integral means spectrum is related to the multifractal spectrum of the harmonic measure ω on the boundary of the image domain. ◮ Define, for α ≥ 1 / 2, E α as being the set of points z on the boundary where ω ( B ( z , r )) ∼ r α , as r → 0. ◮ The multifractal spectrum of ω is the function f ( α ) = D Hausdorff ( E α ). ◮ One goes from the integral means spectrum β to f by a Legendre transform , 1 � β ( p ) − p + 1 + 1 � α f ( α ) = inf , α p p � 1 � β ( p ) = sup α ( f ( α ) − p ) + p − 1 . α

Universal Integral Means Spectrum ◮ B ( p ) = sup { β Φ ( p ) , Φ ∈ S} . ◮ B bd ( p ) = sup { β Φ ( p ) , Φ ∈ S , Φ bounded } . ◮ Theorem (Makarov): B ( p ) = max { B bd ( p ) , 3 p − 1 } .

Generalized Integral Means Spectrum ◮ Consider a ( random ) injective Riemann map Φ ∈ S , i.e., Φ : D → C , Φ(0) = 0 , Φ ′ (0) = 1 . ◮ For ( p , q ) ∈ R 2 , define the generalized integral means � 2 π | Φ ′ ( re i θ ) | p I ( r , p , q , Φ) := | Φ( re i θ ) | q d θ, 0 < r < 1; 0 � 2 π E | Φ ′ ( re i θ ) | p ◮ Expected: E I ( r , p , q , Φ) := | Φ( re i θ ) | q d θ, 0 < r < 1 . 0 ◮ Define log( I ( r , p , q , Φ)) β Φ ( p , q ) := lim sup ; 1 log( 1 − r ) r → 1 − ◮ If the limit exists, 1 r → 1 − I ( r , p , q , Φ) ≍ (1 − r ) β Φ ( p , q ) .

Generalized Integral Means Spectrum ◮ Unified treatment of the bounded and the unbounded cases. ◮ Φ ∈ S ⇒ Ψ = 1 Φ is bounded, ◮ | Ψ ′ | p = | Φ ′ | p | Φ | 2 p . ◮ m -fold transform of f ∈ S : f [ m ] ( z ) := � f ( z m ) , m ∈ Z + , m holomorphic branch with derivative 1 at 0. For m ∈ Z − and z ∈ D − := C \ D , f [ m ] ( z ) := 1 / f [ − m ] (1 / z ). For m < 0, f [ m ] ( D − ) has bounded boundary. For m = − 1, f [ − 1] ( z ) = 1 / f (1 / z ) , is the bounded exterior whole-plane of Beliaev & Smirnov. ◮ | f ′ ( z m ) | p | ( f [ m ] ) ′ ( z ) | p = | z | p ( m − 1) m ) . | f ( z m ) | p (1 − 1

Generalized Integral Means Spectrum ◮ One finds various standard spectra in the ( p , q ) plane: ◮ The standard integral means spectrum on the line q = 0, ◮ The bounded one on the line q = 2 p , 1 ◮ The spectrum for the m - fold f [ m ] ( z ) = ( f ( z m )) m , m ∈ Z + , β [ m ] ( p ) = β [1] ( p , q m ) , q m := p (1 − 1 / m ); ◮ The standard spectrum for the m -fold for m ∈ Z − .

Universal Generalized Integral Means Spectrum ◮ One can similarly define a universal generalized integral means spectrum. ◮ Theorem (Astala, D., Zinsmeister): B ( p , q ) = max { B bd ( p ) , 3 p − 2 q − 1 } . q p � 1 p � � 1 2 p /4 p p q 3 � 2 � 1

Beliaev-Smirnov Generalized PDE ◮ Let f be a whole-plane (inner) SLE κ , z ∈ D , ( p . q ) ∈ R 2 � z � q � � q � � � � z 2 p f ′ ( z ) | f ′ ( z ) | p � � F ( z ) := E , G ( z , ¯ z ) := E . 2 � � f ( z ) f ( z ) � � ◮ Using the SLE equation and Itˆ o calculus, one derives a differential equation satisfied by F , � − κ 2( z ∂ z ) 2 − 1 + z P ( ∂ )[ F ( z )] = 1 − z z ∂ z � p q − (1 − z ) 2 + 1 − z + p − q F ( z ) = 0 , ◮ and a partial differential equation satisfied by G , � − κ z ) 2 − 1 + z 1 − z z ∂ z − 1 + ¯ z P ( D )[ G ( z , ¯ z )] = 2( z ∂ z − ¯ z ∂ ¯ z ¯ z ∂ ¯ z 1 − ¯ � p p q q − (1 − z ) 2 − z ) 2 + 1 − z + z + 2( p − q ) G ( z , ¯ z ) = 0 . (1 − ¯ 1 − ¯

Integrable Probability ◮ Let f be a time 0 whole-plane (inner) SLE κ , and ( p , q ) ∈ R 2 , � z � q � � q � � � � z 2 p f ′ ( z ) | f ′ ( z ) | p � � F ( z ) := E , G ( z , ¯ z ) := E . 2 � � f ( z ) f ( z ) � � ◮ Integrable parabola with parameterization, p ( γ ) := (2 + κ 2) γ − κ 2 γ 2 , γ ∈ R , q ( γ ) := (3 + κ 2) γ − κγ 2 . ◮ Theorem [DHLZ ’18]: If p = p ( γ ) and q = q ( γ ), then z 2 ) = (1 − z 1 ) γ (1 − ¯ z 2 ) γ F ( z ) = (1 − z ) γ , G ( z 1 , ¯ . z 2 ) κγ 2 / 2 (1 − z 1 ¯

q 2 p -0.5 0.5 1 1.5 2 p ( ) -2 � -4 -6 -8 Integrable parabola for κ ∈ { 2 , 4 , 6 } Other integrable parabolae.

Generalized Integral Means Spectrum of Whole-Plane SLE ◮ The generalized spectrum is [D., Ho, Le & Zinsmeister ’18], β 1 ( p , q ; κ ) := 3 p − 2 q − 1 2 − 1 � 1 + 2 κ ( p − q ) . 2 ◮ Phase transition lines: green parabola & blue quartic q p ( ) � D’ D lin 0 0 III D 1 p � 0 ( ) P 0 p II p � tip ( ) IV � 1 p q , ( ) I Q 0

SLE Standard Integral Means Spectrum ◮ As predicted in Lawler & Werner ’99, D. ’99, (BM), D.’00, and Hastings ’02, and proven in Beliaev & Smirnov ’05, and Beliaev, D. & Zinsmeister ’17, the average spectrum of SLE κ involves 3 phases: β tip ( p , κ ) = − p − 1 + 1 � � � (4 + κ ) 2 − 8 κ p 4 + κ − , 4 β 0 ( p , κ ) = − p + (4 + κ ) 2 − (4 + κ ) � (4 + κ ) 2 − 8 κ p , 4 κ 4 κ β lin ( p , κ ) = p − (4 + κ ) 2 . 16 κ ◮ a.s. β tip [Johansson Viklund & Lawler ’12] ◮ a.s. β 0 [Gwynne, Miller & Sun ’18] ◮ a.s. boundary spectrum [Alberts, Binder & Viklund ’16] [Schoug ’18]

8 , p 3 = 3(4 + κ ) 2 p 2 = − 1 − 3 κ 32 κ Average integral means spectrum for bounded whole-plane SLE.

Unbounded Whole-plane SLE ◮ In this case, [D., Nguyen, Nguyen & Zinsmeister ’14] (see also [Loutsenko & Yermolayeva ’13]) have shown the existence of a phase transition at p 0 := (4+ κ ) 2 − 4 − 2 √ 2(4+ κ ) 2 +4 to 16 κ β 1 ( p , 0; κ ) := 3 p − 1 2 − 1 � 1 + 2 κ p . 2 (Related to SLE derivative exponents [Lawler, Schramm, Werner ’01] and ‘tip’ quantum gravity ones [D.’03].)

Remarks ◮ This β 1 spectrum for the unbounded interior case is proven in a finite p -interval above the transition point p 0 . ◮ In the bounded exterior case, the original Beliaev-Smirnov proof has a gap for negative enough p , namely when p ≤ p 1 := − (4 + κ ) 2 (8 + κ ) , 128 a sub-/super solution to the PDE being no longer positive . ◮ This corresponds to a phase transition to a ‘second tip’ spectrum , that requires a new proof [Beliaev, D. & Zinsmeister ’17].

Phase Diagram q p ( ) � D’ D lin 0 0 III D 1 p � 0 ( ) P 0 p II � tip p IV ( ) � 1 p q ( ) , I Q 0 ◮ ◮ Phase transition lines: green parabola & blue quartic ◮ β 1 ( p , q ; κ ) := 3 p − 2 q − 1 2 − 1 � 1 + 2 κ ( p − q ) . 2

Bounded whole-plane SLE ◮ The Beliaev-Smirnov line q = 2 p does not intersect the green parabola part, and is asymptotically parallel to the blue quartic .

Subjacent β 1 spectrum ◮ Zooming below Q 0 The bounded SLE line intersects the continuation of the green parabola at p 1 . For p < p 1 , the β 1 spectrum dominates the bulk one, β 0 , but not the tip one, β tip .

β 1 ( p , 2 p ; κ ) := − p − 1 2 − 1 � 1 − 2 κ p . 2 ‘Second tip’ spectrum [Beliaev, D. & Zinsmeister ’17]

Domain of Proof ◮ Domain where the form of the generalized integral means spectrum has been established: q D D’ 0 0 D 1 P 0 p P 2 P 3 D 2 Q 0