The Cut tree of the Brownian ContinuumRandom Tree and the Reverse Problem

Minmin Wang

Joint work with Nicolas Broutin

Universite de Pierre et Marie Curie, Paris, France

24 June 2014

MotivationIntroduction to the Brownian CRT

I Let Tn be a uniform tree of n vertices.

I Let each edge have length 1/√n metric space

I Put mass 1/n at each vertex uniform distributionI Denote by 1√

nTn the obtained metric measure space.

I Aldous (’91):1√nTn =⇒ T , n→∞,

where T is the Brownian CRT (Continuum Random Tree).

MotivationIntroduction to the Brownian CRT

I Let Tn be a uniform tree of n vertices.

I Let each edge have length 1/√n metric space

I Put mass 1/n at each vertex uniform distributionI Denote by 1√

nTn the obtained metric measure space.

I Aldous (’91):1√nTn =⇒ T , n→∞,

where T is the Brownian CRT (Continuum Random Tree).

MotivationIntroduction to the Brownian CRT

I Let Tn be a uniform tree of n vertices.

I Let each edge have length 1/√n metric space

I Put mass 1/n at each vertex uniform distributionI Denote by 1√

nTn the obtained metric measure space.

I Aldous (’91):1√nTn =⇒ T , n→∞,

where T is the Brownian CRT (Continuum Random Tree).

MotivationBrownian CRT seen from Brownian excursion

Let Be be the normalized Brownian excursion. Then T is encodedby 2Be .

0 1

2Be

leaf

branching pointa b

MotivationBrownian CRT

T is

I a (random) compact metric space such that ∀u, v ∈ T , ∃unique geodesic Ju, vK between u and v ;

I equipped with a probability measure µ (mass measure),concentrated on the leaves;

I equipped with a σ-finite measure ` (length measure) such that`(Ju, vK) = distance between u and v .

MotivationAldous–Pitman’s fragmentation process

Let P be a Poisson point process on [0,∞)× T of intensitydt ⊗ `(dx).

I Pt := {x ∈ T : ∃ s ≤ t such that (s, x) ∈ P}.I If v ∈ T , let Tv (t) be the connected component of T \ Pt

containing v .

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

V2x1

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2V2

x1

x2

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

x1

x2V2

V1

x1

x2

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

t3

x1

x2

x3

V2

V1

V3 V4

x1

x2x3

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

t3

x1

x2

x3

Sk

V2

V1

V3 V4

x1

x2x3

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

V5

Vk+1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

V5

Vk+1

V2x1

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2V5

Vk+1

x2V2

x1

x2

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2V5

Vk+1

t′ x′

V1 V5

x2V2

x1

x2x′

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

t3 x3

V5

Vk+1

t′ x′

V1 V5

Sk+1

V3 V4

x2V2

x1

x2x3

x′

x1

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

t3 x3

V5

Vk+1

t′ x′

V1 V5

Sk+1

V3 V4

x2V2

Sk ⊂

x1

x2x3

x′

x1

MotivationCut tree of the Brownian CRT

Equip Sk with a distance d such that

d(root,Vi ) =

∫ ∞

0µi (t)dt := Li ,

with µi (t) := µ(TVi(t)).

0

t

t1

t2

t3

V2

V1

V3 V4

Sk

∫ t10 µi(s)ds

i = 1, 2, 3, 4

∫∞t1µ2(t)dt

x1

x2

x3

root

MotivationCut tree of the Brownian CRT

Note that Sk ⊂ Sk+1 (as metric space). Let cut(T ) = ∪Sk .

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 )).

MotivationCut tree of the Brownian CRT

Note that Sk ⊂ Sk+1 (as metric space). Let cut(T ) = ∪Sk .

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 )).

MotivationCut tree of the Brownian CRT

Note that Sk ⊂ Sk+1 (as metric space). Let cut(T ) = ∪Sk .

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 )).

MotivationCut tree of the Brownian CRT

Note that Sk ⊂ Sk+1 (as metric space). Let cut(T ) = ∪Sk .

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 )).

MotivationCut tree of the Brownian CRT

Note that Sk ⊂ Sk+1 (as metric space). Let cut(T ) = ∪Sk .

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 modelCutting down uniform tree

V1

V2

V3

A uniform tree Tn

B1

B2

Related discrete modelCutting down uniform tree

V1

V2

V3

A uniform tree Tn

B1

B2

Related discrete modelCutting down uniform tree

V1

V2

V3

A uniform tree Tn

B1

B2

Related discrete modelCutting down uniform tree

V1

V2

V3

A uniform tree Tn

B1

B2

B1

V2 V3

Related discrete modelCutting down uniform tree

V1

V2

V3

A uniform tree Tn

B1

B2

B1

V2 V3

V1

B2

Related discrete modelCutting down uniform tree

V1

V2

V3

A uniform tree Tn

B1

B2

B1

cut(Tn)

V2 V3

V1

B2

Related discrete modelCut tree of Tn

For v ∈ Tn,

let Ln(v) := nb. of picks affecting thesize of the connected componentcontaining v .

Then, Ln(v) = nb. of vertices betweenthe root and v in cut(Tn).

Ln distance on Tn

B1

cut(Tn)

V2 V3

V1

B2

123

Related discrete modelConvergence of cut trees

I Meir & Moon, Panholzer, etc if Vn is uniform on Tn, then

Ln(Vn)/√n =⇒ Rayleigh distribution (of density xe−x

2/2)

I Broutin & W., 2013 cut(Tn)d= Tn (Eq 1)

I Broutin & W., 2013

( 1√nTn,

1√n

cut(Tn))

=⇒(T , cut(T )

), n→∞. (Eq 2)

I (Eq 1) and (Eq 2) entail that cut(T )d= T (Eq 3)

I1√nLn(Vn)

(Eq 2)=⇒ L(V )

d= dT (root,V ), by Eq (3)

Related discrete modelConvergence of cut trees

I Meir & Moon, Panholzer, etc if Vn is uniform on Tn, then

Ln(Vn)/√n =⇒ Rayleigh distribution (of density xe−x

2/2)

I Broutin & W., 2013 cut(Tn)d= Tn (Eq 1)

I Broutin & W., 2013

( 1√nTn,

1√n

cut(Tn))

=⇒(T , cut(T )

), n→∞. (Eq 2)

I (Eq 1) and (Eq 2) entail that cut(T )d= T (Eq 3)

I1√nLn(Vn)

(Eq 2)=⇒ L(V )

d= dT (root,V ), by Eq (3)

Related discrete modelConvergence of cut trees

I Meir & Moon, Panholzer, etc if Vn is uniform on Tn, then

Ln(Vn)/√n =⇒ Rayleigh distribution (of density xe−x

2/2)

I Broutin & W., 2013 cut(Tn)d= Tn (Eq 1)

I Broutin & W., 2013

( 1√nTn,

1√n

cut(Tn))

=⇒(T , cut(T )

), n→∞. (Eq 2)

I (Eq 1) and (Eq 2) entail that cut(T )d= T (Eq 3)

I1√nLn(Vn)

(Eq 2)=⇒ L(V )

d= dT (root,V ), by Eq (3)

Related discrete modelConvergence of cut trees

I Meir & Moon, Panholzer, etc if Vn is uniform on Tn, then

Ln(Vn)/√n =⇒ Rayleigh distribution (of density xe−x

2/2)

I Broutin & W., 2013 cut(Tn)d= Tn (Eq 1)

I Broutin & W., 2013

( 1√nTn,

1√n

cut(Tn))

=⇒(T , cut(T )

), n→∞. (Eq 2)

I (Eq 1) and (Eq 2) entail that cut(T )d= T (Eq 3)

I1√nLn(Vn)

(Eq 2)=⇒ L(V )

d= dT (root,V ), by Eq (3)

Related discrete modelConvergence of cut trees

I Meir & Moon, Panholzer, etc if Vn is uniform on Tn, then

Ln(Vn)/√n =⇒ Rayleigh distribution (of density xe−x

2/2)

I Broutin & W., 2013 cut(Tn)d= Tn (Eq 1)

I Broutin & W., 2013

( 1√nTn,

1√n

cut(Tn))

=⇒(T , cut(T )

), n→∞. (Eq 2)

I (Eq 1) and (Eq 2) entail that cut(T )d= T (Eq 3)

I1√nLn(Vn)

(Eq 2)=⇒ L(V )

d= dT (root,V ), by Eq (3)

Related discrete modelReverse transformation

From cut(Tn) to Tn

V1

V2

V3

A uniform tree Tn

B1

B2

B1

cut(Tn)

V2 V3

V1

B2

Related discrete modelReverse transformation

From cut(Tn) to Tn: destroy all the edges in cut(Tn)

B1

cut(Tn)

V2 V3

V1

B2

V1

V2

V3

A uniform tree Tn

B1

B2

Related discrete modelReverse transformation

From cut(Tn) to Tn: replace them with the edges in Tn

V1

V2

V3

A uniform tree Tn

B1

B2

B1

cut(Tn)

V2 V3

V1

B2

B1

cut(Tn)

V2 V3

V1

B2

V1

V2

V3

A uniform tree Tn

B1

B2

Related discrete modelReverse transformation

From cut(Tn) to Tn: or equivalently...

V1

V2

V3

A uniform tree Tn

B1

B2

B1

cut(Tn)

V2 V3

V1

B2

B1

cut(Tn)

V2 V3

V1

B2

V1

V2

V3

A uniform tree Tn

B1

B2

Construction of shuff(H)

Br(H) = {x1, x2, x3, · · · }. For each n ≥ 1, sample An below xnaccording to the mass measure µ.

H = Brownian CRT

xn

Hr

Construction of shuff(H)

Br(H) = {x1, x2, x3, · · · }. For each n ≥ 1, sample An below xnaccording to the mass measure µ.

H = Brownian CRT

xn

An

Hr

Construction of shuff(H)

Br(H) = {x1, x2, x3, · · · }. For each n ≥ 1, sample An below xnaccording to the mass measure µ.

H = Brownian CRT

xn

An

Hr

Construction of shuff(H)

Br(H) = {x1, x2, x3, · · · }. For each n ≥ 1, sample An below xnaccording to the mass measure µ.

H = Brownian CRT

xn

An

Hr

Construction of shuff(H)

I Almost surely, the transformation converges as n→∞.Denote by shuff(H) the limit tree.

I shuff(H) does not depend on the order of the sequence (xn)

I It satisfies(shuff(H),H)

d= (T , cut(T )).

Thank you!

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

t3

x1

x2

x3

Sk

V2

V1

V3 V4

x1

x2x3

MotivationGenealogy of Aldous-Pitman’s fragmentation

Let V1,V2, · · · be independent leaves picked from µ.

subtree of T spanned by V1, · · · , Vk

V1

V2

V3

V4

t

0

t1

t2

t3

x1

x2

x3

Sk

V2

V1

V3 V4

x1

x2x3