Connectivity properties of the adjacency graph of SLE bubbles for - PowerPoint PPT Presentation

Connectivity properties of the adjacency graph of SLE κ bubbles for κ ∈ (4 , 8) Random Conformal Geometry and Related Fields, KIAS, Seoul Joshua Pfeffer Joint work with Ewain Gwynne MIT June 20, 2018 1 / 36

Introduction 1 Review of LQG 2 Proof of our main result 3 Defining an ( L , R )-Markovian path to infinity Reducing to a single bubble Estimate for a single bubble Converse and Open Problems 4 3 / 36

Introduction Consider a chordal SLE κ curve η from 0 to ∞ in ❍ for κ ∈ (4 , 8). A bubble of η is a connected component of ❍ \ η . 4 / 36

Introduction Today’s talk is about the following question, originally posed by Duplantier, Miller and Sheffield (2014): Question Is the adjacency graph of bubbles connected? I.e., is there a finite path in the adjacency graph between any pair of bubbles? The analogous question for Brownian motion is a well-known open problem. 5 / 36

Introduction Is the adjacency graph of bubbles even connected for any κ ? The answer is not obvious. The set of points on the curve which do not lie on the boundary of any bubble has full Hausdorff dimension. There could exist pairs of macroscopic bubbles separated by an infinite “cloud” of small bubbles. Figure: An SLE 6 in a square domain. Simulation by Jason Miller. 6 / 36

Main Result Theorem (Gwynne, P. (2018)) For each fixed κ ∈ (4 , κ 0 ] , the adjacency graph of bubbles of a chordal SLE κ curve is almost surely connected, where κ 0 ≈ 5 . 6158 is the unique solution of the equation π cot( πκ/ 4) + ψ (2 − κ/ 4) − ψ (1) = 0 on the interval (4 , 8) . Here, ψ ( x ) = Γ ′ ( x ) Γ( x ) denotes the digamma function. Corollary (Gwynne, P. (2018)) For κ ∈ (4 , κ 0 ] , the set of points on a chordal SLE κ curve which do not lie on the boundary of any bubble is almost surely totally disconnected. 7 / 36

Our Approach By theory of LQG, the left and right boundaries of an SLE κ curve (with a particular parametrization) are a pair ( L , R ) of independent κ/ 4-stable Levy processes. η ( t ) R t = ν h (blue) L t = ν h (green) − ν h (purple) − ν h (orange) 0 8 / 36

Our Approach η ( t ) R t = ν h (blue) L t = ν h (green) − ν h (purple) − ν h (orange) 0 We use ( L , R ) to define a stronger “Markovian” connectivity condition for the graph of bubbles. We will show that this condition holds for κ ∈ (4 , κ 0 ] and fails for κ sufficiently close to 8. (Reminder: κ 0 ≈ 5 . 6158 is the unique solution of the equation π cot( πκ/ 4) + ψ (2 − κ/ 4) − ψ (1) = 0 on the interval (4 , 8).) 9 / 36

Introduction 1 Review of LQG 2 Proof of our main result 3 Defining an ( L , R )-Markovian path to infinity Reducing to a single bubble Estimate for a single bubble Converse and Open Problems 4 10 / 36

Review of LQG To define ( L , R ) precisely, we need some definitions from LQG. 11 / 36

Review of LQG D ⊂ ❈ open h a GFF-type distribution on D “Definition” The γ -LQG surface associated with h is the random Riemannian surface with Riemann metric tensor e γ h ( z ) ( dx 2 + dy 2 ), where dx 2 + dy 2 is the Euclidean metric tensor. This definition does not make literal sense since h is a distribution, not a pointwise-defined function. 12 / 36

Review of LQG However, certain objects associated with γ -LQG surfaces can be defined rigorously using regularization procedures: ◮ a γ -LQG area measure on D (defined as limit of regularized versions of e γ h ( z ) dz ) ◮ a γ -LQG length measure on certain curves in D , such as ∂ D and SLE κ -type curves for κ = γ 2 (or, equivalently, the outer boundaries of SLE 16 /κ curves by SLE duality) 13 / 36

Review of LQG We parametrize our SLE κ curve by quantum natural time. Roughly speaking, this is the same as parametrizing by “quantum Minkowski content” It is the quantum analogue of the so-called natural parametrization of SLE. 14 / 36

Definition of ( L , R ) Sample an SLE κ curve independently on a 4 γ − γ 2 - quantum wedge with γ = 4 / √ κ . γ − γ To construct the 4 2 - quantum wedge: � � Take the distribution ˜ γ − γ log | · | on ❍ , where ˜ 4 h − h a free boundary 2 GFF on ❍ . “Zoom in near the origin” Rescale so the γ -LQG mass of ❉ ∩ ❍ remains of constant order. 15 / 36

Definition of ( L , R ) γ − γ Sample an SLE κ curve independently on a 4 2 - quantum wedge with γ = 4 / √ κ . η ( t ) R t = ν h (blue) L t = ν h (green) − ν h (purple) − ν h (orange) 0 Theorem (Duplantier, Miller and Sheffield (2014)) The processes L t and R t are i.i.d. totally asymmetric κ 4 -stable Levy processes with only negative jumps. 16 / 36

SLE-Levy process dictionary We can use ( L , R ) to describe some geometric features of an SLE κ curve. Points on boundary of bubble Points of the boundary of two different bubbles ⇔ (local) cut points ⇔ edges of adjacency graph of bubbles 0 ρ x 17 / 36

Introduction 1 Review of LQG 2 Proof of our main result 3 Defining an ( L , R )-Markovian path to infinity Reducing to a single bubble Estimate for a single bubble Converse and Open Problems 4 18 / 36

Proof of our main result 1 We define an “( L , R )-Markovian path to infinity” ◮ If it exists, the graph must be connected. 2 We reduce the task of proving existence of this path to an estimate for a single bubble. 3 We outline the proof of the estimate for a single bubble. After the proof, we will remark on our converse result for large κ and discuss open problems. 19 / 36

Table of Contents Introduction 1 Review of LQG 2 Proof of our main result 3 Defining an ( L , R )-Markovian path to infinity Reducing to a single bubble Estimate for a single bubble Converse and Open Problems 4 20 / 36

Step 1: ( L , R )-Markovian path to infinity Definition For κ ∈ (4 , 8), an ( L , R ) -Markovian path to infinity in the adjacency graph of bubbles of η is an infinite increasing sequence of stopping times τ 1 < τ 2 < τ 3 < · · · for ( L , R ) such that almost surely τ k → ∞ , η forms a bubble b k at each time τ k (equivalently, either L or R has a downward jump at time τ k ), and b k and b k +1 are connected in the adjacency graph (i.e., ∂ b k ∩ ∂ b k +1 � = ∅ ) for each k . Note: this is a random path defined for almost every realization of the SLE κ curve. 21 / 36

Step 1: ( L , R )-Markovian path to infinity There is an ( L , R )-Markovian path to infinity started at b 2 η ( τ 2 ) every stopping time at which η forms a bubble almost surely. b 1 η ( τ 1 ) ⇓ Adjacency graph of bubbles is connected almost surely. 0 0 22 / 36

Table of Contents Introduction 1 Review of LQG 2 Proof of our main result 3 Defining an ( L , R )-Markovian path to infinity Reducing to a single bubble Estimate for a single bubble Converse and Open Problems 4 23 / 36

Step 2: Reducing to a single bubble There is an ( L , R )-Markovian ⇒ path to infinity started at ❊ log( L τ − L σ ) ≥ 0 every stopping time at which η forms a bubble almost surely. η ( σ ) ⇓ 0 η ( τ ) ρ 1 Adjacency graph of bubbles is connected almost surely. 24 / 36

Step 2: Reducing to a single bubble Using ❊ log( L τ − L σ ) ≥ 0 to construct ( L , R ) Markovian path to infinity: η ( σ 2 ) b 2 η ( σ 1 ) η ( τ 2 ) b 1 0 η ( τ 1 ) X 1 = L τ 1 − L σ 1 i . i . d . X 2 = R τ 2 − R σ 2 X k +1 / X k ≡ L τ − L σ etc. 25 / 36

Step 2: Reducing to a single bubble There is an ( L , R )-Markovian ⇒ path to infinity started at ❊ log( L τ − L σ ) ≥ 0 every stopping time at which η forms a bubble almost surely. η ( σ ) ⇓ 0 η ( τ ) ρ 1 Adjacency graph of bubbles is connected almost surely. 26 / 36

Table of Contents Introduction 1 Review of LQG 2 Proof of our main result 3 Defining an ( L , R )-Markovian path to infinity Reducing to a single bubble Estimate for a single bubble Converse and Open Problems 4 27 / 36

Step 3: Estimate for a single bubble We have reduced the theorem to proving ❊ log( L τ − L σ ) ≥ 0. η ( σ ) L t ( τ, L τ ) t 0 η ( τ ) ρ 1 ( σ, L σ ) L τ − L σ stochastically dominates R τ − − R σ . Why is this estimate tricky? The laws of τ and σ are not known explicitly! 28 / 36

Step 3: Estimate for a single bubble L t ( τ, L τ ) t η ( σ ) ( σ, L σ ) R t ( τ, R τ − ) 0 η ( τ ) ρ 1 t ( σ, R σ ) − 1 ( τ, R τ ) 29 / 36

Step 3: Estimate for a single bubble L t ( τ, L τ ) t η ( σ ) ( σ, L σ ) R t ( τ, R τ − ) 0 η ( τ ) ρ 1 L τ − L σ stochastically dominates t R τ − − R σ . ( σ, R σ ) − 1 ( τ, R τ ) 29 / 36

Step 3: Estimate for a single bubble L t ( τ, L τ ) t η ( σ ) ( σ, L σ ) R t ( τ, R τ − ) ρ 1 0 η ( ξ ) η ( τ ) L τ − L σ stochastically dominates t R τ − − R σ . ( σ, R σ ) ( ξ, R ξ ) − 1 ( τ, R τ ) 30 / 36

Step 3: Estimate for a single bubble ( L , R ) run backwards from τ to ξ ( L , R ) run backward L = conditional on { R τ − − R ξ = r } until R hits − r For fixed r , we know the law of the last simultaneous running infimum of ( L , R ) run backward until R hits − r . ◮ The set of simultaneous running infima is a subordinator with known index (by result of Brownian motions), so can use arcsine law for subordinators. Now the only thing we need to know is the law of R τ − − R ξ . Fortunately, this is known! 31 / 36