The Cut tree of the Brownian Continuum Random Tree and the Reverse - PowerPoint PPT Presentation

The Cut tree of the Brownian Continuum Random Tree and the Reverse Problem Minmin Wang Joint work with Nicolas Broutin Universit´ e de Pierre et Marie Curie, Paris, France 24 June 2014

Motivation Introduction to the Brownian CRT ◮ Let T n be a uniform tree of n vertices.

Motivation Introduction to the Brownian CRT ◮ Let T n be a uniform tree of n vertices. ◮ Let each edge have length 1 / √ n � metric space ◮ Put mass 1 / n at each vertex � uniform distribution ◮ Denote by 1 √ n T n the obtained metric measure space.

Motivation Introduction to the Brownian CRT ◮ Let T n be a uniform tree of n vertices. ◮ Let each edge have length 1 / √ n � metric space ◮ Put mass 1 / n at each vertex � uniform distribution ◮ Denote by 1 √ n T n the obtained metric measure space. ◮ Aldous (’91): 1 √ nT n = ⇒ T , n → ∞ , where T is the Brownian CRT ( Continuum Random Tree ).

Motivation Brownian CRT seen from Brownian excursion Let B e be the normalized Brownian excursion. Then T is encoded by 2 B e . 2 B e leaf b a branching point 0 1

Motivation Brownian CRT T is ◮ a (random) compact metric space such that ∀ u , v ∈ T , ∃ unique geodesic � u , v � between u and v ; ◮ equipped with a probability measure µ (mass measure), concentrated on the leaves; ◮ equipped with a σ -finite measure ℓ (length measure) such that ℓ ( � u , v � ) = distance between u and v .

Motivation Aldous–Pitman’s fragmentation process Let P be a Poisson point process on [0 , ∞ ) × T of intensity dt ⊗ ℓ ( dx ). ◮ P t := { x ∈ T : ∃ s ≤ t such that ( s , x ) ∈ P} . ◮ If v ∈ T , let T v ( t ) be the connected component of T \ P t containing v .

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V 2 V 3 V 1 V 4

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k 0 V 2 t 1 V 3 x 1 V 1 V 4 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k 0 V 2 x 1 t 1 V 3 x 1 V 2 V 1 V 4 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k 0 V 2 x 1 t 1 V 3 x 1 V 2 t 2 V 1 V 4 x 2 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k 0 V 2 x 1 t 1 V 3 x 1 V 2 x 2 t 2 V 1 V 1 V 4 x 2 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k 0 V 2 x 1 t 1 V 3 x 1 V 2 x 2 t 2 V 1 V 1 V 4 x 2 x 3 x 3 t 3 V 4 V 3 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . S k subtree of T spanned by V 1 , · · · , V k 0 V 2 x 1 t 1 V 3 x 1 V 2 x 2 t 2 V 1 V 1 V 4 x 2 x 3 x 3 t 3 V 4 V 3 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V k +1 0 V 2 V 3 V 5 V 1 V 4 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V k +1 0 V 2 x 1 t 1 V 3 x 1 V 5 V 2 V 1 V 4 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V k +1 0 V 2 x 1 t 1 V 3 x 1 V 5 V 2 t 2 x 2 V 1 V 4 x 2 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V k +1 0 V 2 x 1 t 1 V 3 x 1 V 5 V 2 t 2 x 2 V 1 x ′ t ′ V 4 x ′ x 2 V 1 V 5 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V k +1 S k +1 0 V 2 x 1 t 1 V 3 x 1 V 5 V 2 t 2 x 2 V 1 x ′ t ′ V 4 x ′ x 2 x 3 V 1 V 5 x 3 t 3 V 4 V 3 t

Motivation Genealogy of Aldous-Pitman’s fragmentation Let V 1 , V 2 , · · · be independent leaves picked from µ . subtree of T spanned by V 1 , · · · , V k V k +1 S k ⊂ S k +1 0 V 2 x 1 t 1 V 3 x 1 V 5 V 2 t 2 x 2 V 1 x ′ t ′ V 4 x ′ x 2 x 3 V 1 V 5 x 3 t 3 V 4 V 3 t

Motivation Cut tree of the Brownian CRT S k 0 root � t 1 0 µ i ( s ) ds i = 1 , 2 , 3 , 4 x 1 t 1 Equip S k with a distance d such that � ∞ t 1 µ 2 ( t ) dt � ∞ V 2 d (root , V i ) = µ i ( t ) dt := L i , x 2 t 2 0 with µ i ( t ) := µ ( T V i ( t )). V 1 x 3 t 3 V 4 V 3 t

Motivation Cut tree of the Brownian CRT Note that S k ⊂ S k +1 (as metric space). Let cut( T ) = ∪ S k .

Motivation Cut tree of the Brownian CRT Note that S k ⊂ S k +1 (as metric space). Let cut( T ) = ∪ S k . Bertoin & Miermont, 2012 cut( T ) d = T .

Motivation Cut tree of the Brownian CRT Note that S k ⊂ S k +1 (as metric space). Let cut( T ) = ∪ S k . Bertoin & Miermont, 2012 cut( T ) d = T . Question: given cut( T ), can we recover T ?

Motivation Cut tree of the Brownian CRT Note that S k ⊂ S k +1 (as metric space). Let cut( T ) = ∪ S k . Bertoin & Miermont, 2012 cut( T ) d = T . Question: given cut( T ), can we recover T ? Not completely.

Motivation Cut tree of the Brownian CRT Note that S k ⊂ S k +1 (as metric space). Let cut( T ) = ∪ S k . Bertoin & Miermont, 2012 cut( T ) d = T . Question: given cut( T ), can we recover T ? Not completely. Theorem (Broutin & W., 2014) Let H be the Brownian CRT. Almost surely, there exist shuff( H ) such that (shuff( H ) , H ) d = ( T , cut( T )) .

Related discrete model Cutting down uniform tree A uniform tree T n V 2 V 3 V 1 B 1 B 2

Related discrete model Cutting down uniform tree A uniform tree T n V 2 V 3 V 1 B 1 B 2

Related discrete model Cutting down uniform tree A uniform tree T n V 2 V 3 V 1 B 1 B 2

Related discrete model Cutting down uniform tree A uniform tree T n B 1 V 2 V 3 V 2 V 3 V 1 B 1 B 2

Related discrete model Cutting down uniform tree A uniform tree T n B 1 V 2 V 1 V 3 V 2 V 3 V 1 B 1 B 2 B 2

Related discrete model Cutting down uniform tree A uniform tree T n B 1 V 2 V 1 V 3 V 2 V 3 V 1 B 1 B 2 cut( T n ) B 2

Related discrete model Cut tree of T n B 1 For v ∈ T n , 1 2 let L n ( v ) := nb. of picks affecting the V 1 3 size of the connected component containing v . V 2 V 3 Then, L n ( v ) = nb. of vertices between the root and v in cut( T n ). B 2 L n � distance on T n cut( T n )

Related discrete model Convergence of cut trees ◮ Meir & Moon, Panholzer, etc if V n is uniform on T n , then L n ( V n ) / √ n = ⇒ Rayleigh distribution (of density xe − x 2 / 2 )

Related discrete model Convergence of cut trees ◮ Meir & Moon, Panholzer, etc if V n is uniform on T n , then L n ( V n ) / √ n = ⇒ Rayleigh distribution (of density xe − x 2 / 2 ) cut( T n ) d ◮ Broutin & W., 2013 = T n (Eq 1)

Related discrete model Convergence of cut trees ◮ Meir & Moon, Panholzer, etc if V n is uniform on T n , then L n ( V n ) / √ n = ⇒ Rayleigh distribution (of density xe − x 2 / 2 ) cut( T n ) d ◮ Broutin & W., 2013 = T n (Eq 1) ◮ Broutin & W., 2013 � 1 √ nT n , 1 � � � √ n cut( T n ) = ⇒ T , cut( T ) , n → ∞ . (Eq 2)

Related discrete model Convergence of cut trees ◮ Meir & Moon, Panholzer, etc if V n is uniform on T n , then L n ( V n ) / √ n = ⇒ Rayleigh distribution (of density xe − x 2 / 2 ) cut( T n ) d ◮ Broutin & W., 2013 = T n (Eq 1) ◮ Broutin & W., 2013 � 1 √ nT n , 1 � � � √ n cut( T n ) = ⇒ T , cut( T ) , n → ∞ . (Eq 2) cut( T ) d ◮ (Eq 1) and (Eq 2) entail that = T (Eq 3)

Related discrete model Convergence of cut trees ◮ Meir & Moon, Panholzer, etc if V n is uniform on T n , then L n ( V n ) / √ n = ⇒ Rayleigh distribution (of density xe − x 2 / 2 ) cut( T n ) d ◮ Broutin & W., 2013 = T n (Eq 1) ◮ Broutin & W., 2013 � 1 √ nT n , 1 � � � √ n cut( T n ) = ⇒ T , cut( T ) , n → ∞ . (Eq 2) cut( T ) d ◮ (Eq 1) and (Eq 2) entail that = T (Eq 3) ◮ 1 (Eq 2) ⇒ L ( V ) d √ nL n ( V n ) = = d T (root , V ) , by Eq (3)

Related discrete model Reverse transformation From cut( T n ) to T n A uniform tree T n B 1 V 2 V 3 V 1 V 2 V 3 V 1 B 1 B 2 cut( T n ) B 2

Related discrete model Reverse transformation From cut( T n ) to T n : destroy all the edges in cut( T n ) A uniform tree T n B 1 V 2 V 3 V 1 V 2 V 3 V 1 B 1 B 2 cut( T n ) B 2

Related discrete model Reverse transformation From cut( T n ) to T n : replace them with the edges in T n A uniform tree T n A uniform tree T n B 1 B 1 V 2 V 2 V 3 V 3 V 1 V 1 V 2 V 2 V 3 V 3 V 1 V 1 B 1 B 1 B 2 B 2 cut( T n ) cut( T n ) B 2 B 2