T ree Embedding via the Generalized Loewner Equation Vivian - PowerPoint PPT Presentation

T ree Embedding via the Generalized Loewner Equation Vivian Olsiewski Healey University of Chicago Joint work with Govind Menon KIAS, June 19, 2018

Preview: Main Idea Just heard from Peter Lin: Embedding the CRT as a limit of embeddings of finite trees. This talk: ✤ Originally motivated by same question (embedding CRT) ✤ Different approach: embed trees as growth processes ✤ Use Loewner equation ✤ Approach adds geometric difficulty ✤ Benefits: • Links embedding problem to SLE • Geometric and analytic properties of independent interest • Hope: useful for scaling limit of discrete processes?

Preview: Main Idea Galton-Watson trees Chordal Loewner equation Describe the genealogy of birth- death processes. Growing hulls ⇐ ⇒ Evolving real measures Continuum Random Tree scaling limit of GW trees, conditioned to be large

Preview: Main Idea Question 1: Can we use the Loewner equation to construct embeddings of Galton-Watson trees in the upper half plane (as growth processes)? Question 2: Can we construct an embedding of the CRT as a limit of these embeddings of finite Galton-Watson trees?

Preview: Main Idea Answer to Q1: Let μ t be the evolving real measure: • supp( μ t ) is a particle system on the real line • branching determined by a tree T - “birth” in T : particle duplicates - “death” in T : particle disappears • repulsion ~ ( x i − x j ) − 1 “Theorem”: Let K s = hull generated by the Loewner equation with driving measure μ above. ✤ K s is a graph embedding of the subtree of T s ✤ K s ⊂ K s’ if s < s’.

Contents ✤ Background • Plane trees • Loewner equation and SLE ✤ A specific tree embedding ✤ Finding the scaling limit: tightness and an SPDE

The Continuum Random Tree Definition: The continuum random tree (CRT) is the random metric tree coded by the normalized Brownian excursion . − → Uniform distribution on Dyck paths (length 2n ) . − → Uniform distribution on plane trees ( n edges) CRT. − → 1 � � (d) C n ( 2 nt ) n →∞ ( t ) 0 ≤ t ≤ 1 √ − − − → 2 n 0 ≤ t ≤ 1

The Continuum Random Tree Note: many different contour functions code the same metric tree. CRT •Limit of metric trees distributed according to the uniform distribution on Dyck paths. •A random metric space. Not directly a limit of random planar maps. Goal: Take the scaling limit of embedded plane trees to get an embedding of the CRT.

Plane trees as growth processes • Our approach: think of trees as growth processes • Graph distance from the root = time parameter: branching processes plane trees. ⇐ ⇒ time →

) The Loewner Equation & SLE Let γ : (0, T ] → ℍ be a simple curve with γ (0) ∈ ℝ . g t γ ( t ) U ( t ) Loewner (1920s): g t satisfies the initial value problem ˙ b ( t ) g t ( z ) = g 0 ( z ) = z . ˙ g t ( z ) − U ( t ) ,

The Loewner Equation & SLE Generalized version: Let g t ( z ) denote the solution to the initial value problem μ t ( du ) � g t ( z ) = ˙ g 0 ( z ) = z . g t ( z ) − u , R Let H t = { z ∈ ℍ } for which g t ( z ) ∈ ℍ is well defined. Then g t is the unique conformal map from H t onto ℍ with the hydrodynamic normalization. Hull generated by μ t : K t = ℍ ∖ H t . Idea: The measure is supported on points that are escaping ℍ . Geometry of H t ? Need to know fine properties of μ t .

The Loewner Equation & SLE We consider the generalized Loewner equation for discrete driving measures μ t : μ t ( du ) � g t ( z ) = ˙ g 0 ( z ) = z . g t ( z ) − u , R Examples: 1) μ t = δ U ( t ) μ t = δ √ κB t 2) generates SLE κ . N � μ t = δ U i ( t ) 3) produces the multislit equation: N i = 1 1 � g t ( z ) = ˙ g t ( z ) − U i ( t ) . i = 1 Goal: Piece together simple curves in multislit equation. ( N is varying.)

Tree Embedding: ( α , β )-approach Setup: ✤ U 1 , . . . , U n continuous functions U i : [0, T ] → ℝ ✤ U j (0) = U j+1 (0) ✤ mutually nonintersecting: U i ( t ) < U i+1 (t), i = 1, . . . , n, t ∈ [0, T ] n � μ t = c δ U i ( t ) i = 1 Local behavior: want hulls ρ K t to converge in the Hausdorff metric to V α , β inside the disc D R centered at 0. (Call this property ( α , β )-approach.) (Motivated by Schleissinger ’12.)

Tree Embedding: ( α , β )-approach Theorem (H, Menon, 2017): In the setting above, if U j ( t ) − U j ( 0 ) = φ 1 ( α , β ) − φ 2 ( α , β ) lim √ t t � 0 U j + 1 ( t ) − U j + 1 ( 0 ) = φ 1 ( α , β ) + φ 2 ( α , β ) , lim √ t t � 0 for φ 1 ( α , β ) and φ 2 ( α , β ) given below, then the hulls K t approach ℝ in ( α , β )-direction at U j (0). ( 1 + x − 3 a − 3 bx ) φ 1 ( α , β ) = √ c � a ( 1 − a ) − 2 abx + b ( 1 − b ) x 2 � ( 1 − a ) 2 + 2 x ( a + b + ab − 1 ) + x 2 ( 1 − b ) 2 φ 2 ( α , β ) = √ c , a ( 1 − a ) − 2 abx + b ( 1 − b ) x 2 where α = a π , β =b π , and x = x ( a,b ) is the unique negative root of − a + a 3 + 3 ax − 3 a 2 x − 3 abx + 3 a 2 bx + 3 bx 2 − 3 abx 2 − 3 b 2 x 2 + 3 ab 2 x 2 − bx 3 + b 3 x 3 .

Tree Embedding: ( α , β )-approach Balanced case: If 0 < α = β < π /2, then φ 1 and φ 2 simplify to � π − 2 α √ φ 1 ( α , α ) = 0 and φ 2 ( α , α ) = 2 c . α Intuitively: Loewner scaling •If μ t generates hulls K t , then ρμ t/ ρ 2 generates the hulls ρ K t . •So, expect to see √ t whenever a hull is preserved under dilation. Advantage of explicit expressions for φ 1 and φ 2 : • Gives the precise angles.

Tree Embedding: ( α , β )-approach Proof idea: Use estimates on conformal radius (comparable to Euclidean distance) rad( w, H \ ρ K t ) = 2 = ( g ρ t ( w )) � � � ( g ρ t ) 0 ( w ) � � ∂ � � ∂ z ( h t ( z )) − g ρ t ( z )) � . Need to uniformly bound � � (Show contribution of other driving points is negligible.) w w K t ρ K t

A Specific Tree Embedding Let T = { ν , h ( ν )} be a marked plane tree. ✤ Think: h ( ν ) = time of death of ν � Let μ t be supported on elements of T alive at t: μ t = c δ U ν ( t ) . ν ∈ Δ t T On time intervals without branching, how should the U ν evolve? ✤ Dyson Brownian motion? We’ll come back to this at the end.

A Specific Tree Embedding Let T = { ν , h ( ν )} be a marked plane tree. ✤ Think: h ( ν ) = time of death of ν � Let μ t be supported on elements of T alive at t: μ t = c δ U ν ( t ) . ν ∈ Δ t T On time intervals without branching: c 1 ˙ � U ν ( t ) = U ν ( t ) − U η ( t ) . ν � = η � Δ t T Theorem (H, Menon, 2017) : Let T be a binary tree with h ν ≠ h η . Let { K s } be the hulls generated by the Loewner equation driven by μ . Then each K s is a graph embedding of the subtree T s = { ν ∈ T : h ( p ( ν )) < s } in ℍ , with the image of the root on the real line, and K s ⊂ K s’ if s < s’ .

A Specific Tree Embedding Proof (idea): The proof relies on analyzing the interacting particle system c 1 ˙ � U ν ( t ) = U ν ( t ) − U η ( t ) . ν � = η � Δ t T • Extend the solution backward to the initial condition U j (0) = U j+ 1 (0). • Show that the solution gives simple curves away from t = 0. (Use Marshall & Rohde ’05, Lind ’05, Schleissinger ’13) • Show that the generated hull approaches ℝ in ( α , α )-direction for π α = . 2 + c 1 2 c □

Application: Galton-Watson Trees Resulting embedding of a sample of a binary Galton-Watson tree with exponential lifetimes. (Simulation courtesy of Brent Werness.)

Scaling Limit? Question 2 (geometric scaling limit): • Let { T k } be a sequence of random trees that (when appropriately rescaled) converges in distribution to the CRT when T k is conditioned on having k edges. • Does the law of the generated hulls converge to a scaling limit? Question 2a (first step): • Find the scaling limit of the corresponding sequence of measure- valued processes.

Choosing a Sequence of Measures Let { T k } be a sequence of random trees. Let and be two sequences in ℝ + . For each k, define { c k } { c k 1 } μ k t = c k � δ U ν ( t ) , ν ∈ Δ t T k where the U ν ( t ) evolve according to c k ˙ � 1 U ν ( t ) = U ν ( t ) − U η ( t ) . ν � = η � Δ t T k • Same setting as tree embedding theorem. { c k } { c k • Remains to choose random trees { T k } and constants and . 1 }

The Scaling Limit Choose: T k distributed as a critical binary Galton-Watson tree with 1 exponential lifetimes of mean , conditioned to have k edges. √ 2 k Theorem (Aldous ’91): T k converges in distribution to the CRT as k → ∞ .

The Scaling Limit Theorem (H, Menon ’17): For each k , let T k be distributed as a critical 1 binary Galton-Watson tree with exponential lifetimes of mean , √ 2 k conditioned to have k edges, and let { μ k } be the corresponding sequence of measures. If the scaling constants are 1 = 1 c k = c k √ k then the sequence { μ k } is tight in D M f (ˆ R ) [ 0 , ∞ ) . c k = c k c k 1 / c k • Choose , since the ratio determines the branching angle. 1 √ c k = 1 / k • is the rescaling for which the total population process of L t T k converges to , the local time at level t of the normalized Brownian excursion.