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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
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