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 Tn be a uniform tree of n vertices.

Motivation

Introduction to the Brownian CRT

◮ Let Tn 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 √nTn the obtained metric measure space.

Motivation

Introduction to the Brownian CRT

◮ Let Tn 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 √nTn the obtained metric measure space.

◮ Aldous (’91):

1 √nTn = ⇒ T , n → ∞, where T is the Brownian CRT (Continuum Random Tree).

Motivation

Brownian CRT seen from Brownian excursion

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

1

2Be

leaf branching point a b

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).

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

containing v.

Motivation

Genealogy 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

Motivation

Genealogy 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 t1 x1

Motivation

Genealogy 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 t1

V2

x1 x1

Motivation

Genealogy 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 t1 t2

V2

x1 x2 x1

Motivation

Genealogy 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 t1 t2 x1 x2

V2 V1

x1 x2

Motivation

Genealogy 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 t1 t2 t3 x1 x2 x3

V2 V1 V3 V4

x1 x2 x3

Motivation

Genealogy 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 t1 t2 t3 x1 x2 x3

Sk V2 V1 V3 V4

x1 x2 x3

Motivation

Genealogy 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

V5 Vk+1

Motivation

Genealogy 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 t1

V5 Vk+1 V2

x1 x1

Motivation

Genealogy 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 t1 t2

V5 Vk+1

x2

V2

x1 x2 x1

Motivation

Genealogy 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 t1 t2

V5 Vk+1

t′ x′

V1 V5

x2

V2

x1 x2 x′ x1

Motivation

Genealogy 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 t1 t2 t3 x3

V5 Vk+1

t′ x′

V1 V5 Sk+1 V3 V4

x2

V2

x1 x2 x3 x′ x1

Motivation

Genealogy 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 t1 t2 t3 x3

V5 Vk+1

t′ x′

V1 V5 Sk+1 V3 V4

x2

V2 Sk ⊂

x1 x2 x3 x′ x1

Motivation

Cut tree of the Brownian CRT

Equip Sk with a distance d such that d(root, Vi) = ∞ µi(t)dt := Li, with µi(t) := µ(TVi(t)). t t1 t2

t3 V2 V1 V3 V4

Sk

t1 0 µi(s)ds

i = 1, 2, 3, 4

∞ t1 µ2(t)dt

x1 x2 x3

root

Motivation

Cut tree of the Brownian CRT

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

Motivation

Cut tree of the Brownian CRT

Note that Sk ⊂ Sk+1 (as metric space). Let cut(T ) = ∪Sk. Bertoin & Miermont, 2012 cut(T ) d = T .

Motivation

Cut 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 ?

Motivation

Cut 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.

Motivation

Cut 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 model

Cutting down uniform tree

V1 V2 V3

A uniform tree Tn

B1 B2

Related discrete model

Cutting down uniform tree

V1 V2 V3

A uniform tree Tn

B1 B2

Related discrete model

Cutting down uniform tree

V1 V2 V3

A uniform tree Tn

B1 B2

Related discrete model

Cutting down uniform tree

V1 V2 V3

A uniform tree Tn

B1 B2 B1 V2 V3

Related discrete model

Cutting down uniform tree

V1 V2 V3

A uniform tree Tn

B1 B2 B1 V2 V3 V1 B2

Related discrete model

Cutting down uniform tree

V1 V2 V3

A uniform tree Tn

B1 B2 B1

cut(Tn)

V2 V3 V1 B2

Related discrete model

Cut tree of Tn

For v ∈ Tn, let Ln(v) := nb. of picks affecting the size of the connected component containing v. Then, Ln(v) = nb. of vertices between the root and v in cut(Tn). Ln distance on Tn B1

cut(Tn)

V2 V3 V1 B2 1 2 3

Related discrete model

Convergence of cut trees

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

Ln(Vn)/√n = ⇒ Rayleigh distribution (of density xe−x2/2)

Related discrete model

Convergence of cut trees

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

Ln(Vn)/√n = ⇒ Rayleigh distribution (of density xe−x2/2)

◮ Broutin & W., 2013

cut(Tn) d = Tn (Eq 1)

Related discrete model

Convergence of cut trees

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

Ln(Vn)/√n = ⇒ Rayleigh distribution (of density xe−x2/2)

◮ Broutin & W., 2013

cut(Tn) d = Tn (Eq 1)

◮ Broutin & W., 2013

1 √nTn, 1 √n cut(Tn)

• =

⇒

• T , cut(T )
• ,

n → ∞. (Eq 2)

Related discrete model

Convergence of cut trees

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

Ln(Vn)/√n = ⇒ Rayleigh distribution (of density xe−x2/2)

◮ Broutin & W., 2013

cut(Tn) d = Tn (Eq 1)

◮ Broutin & W., 2013

1 √nTn, 1 √n cut(Tn)

• =

⇒

• T , cut(T )
• ,

n → ∞. (Eq 2)

◮ (Eq 1) and (Eq 2) entail that

cut(T ) d = T (Eq 3)

Related discrete model

Convergence of cut trees

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

Ln(Vn)/√n = ⇒ Rayleigh distribution (of density xe−x2/2)

◮ Broutin & W., 2013

cut(Tn) d = Tn (Eq 1)

◮ Broutin & W., 2013

1 √nTn, 1 √n cut(Tn)

• =

⇒

• T , cut(T )
• ,

n → ∞. (Eq 2)

◮ (Eq 1) and (Eq 2) entail that

cut(T ) d = T (Eq 3)

◮

1 √nLn(Vn)

(Eq 2)

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

Related discrete model

Reverse transformation

From cut(Tn) to Tn

V1 V2 V3

A uniform tree Tn

B1 B2 B1

cut(Tn)

V2 V3 V1 B2

Related discrete model

Reverse 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 model

Reverse 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 model

Reverse 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 xn according to the mass measure µ.

H = Brownian CRT xn H r

Construction of shuff(H)

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

H = Brownian CRT xn An H r

Construction of shuff(H)

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

H = Brownian CRT xn An H r

Construction of shuff(H)

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

H = Brownian CRT xn An H r

Construction of shuff(H)

◮ Almost surely, the transformation converges as n → ∞.

Denote by shuff(H) the limit tree.

◮ shuff(H) does not depend on the order of the sequence (xn) ◮ It satisfies

(shuff(H), H) d = (T , cut(T )).

Thank you!

Motivation

Genealogy 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 t1 t2 t3 x1 x2 x3

Sk V2 V1 V3 V4

x1 x2 x3

Motivation

Genealogy 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 t1 t2 t3 x1 x2 x3

Sk V2 V1 V3 V4

x1 x2 x3