The Cut tree of the Brownian Continuum Random Tree and the ... -...
Transcript of The Cut tree of the Brownian Continuum Random Tree and the ... -...
![Page 1: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/1.jpg)
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
![Page 2: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/2.jpg)
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).
![Page 3: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/3.jpg)
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).
![Page 4: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/4.jpg)
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).
![Page 5: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/5.jpg)
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
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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 .
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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 .
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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
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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
![Page 10: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/10.jpg)
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
![Page 11: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/11.jpg)
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
![Page 12: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/12.jpg)
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
![Page 13: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/13.jpg)
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
![Page 14: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/14.jpg)
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
![Page 15: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/15.jpg)
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
![Page 16: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/16.jpg)
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
![Page 17: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/17.jpg)
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
![Page 18: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/18.jpg)
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
![Page 19: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/19.jpg)
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
![Page 20: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/20.jpg)
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
![Page 21: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/21.jpg)
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
![Page 22: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/22.jpg)
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 )).
![Page 23: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/23.jpg)
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 )).
![Page 24: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/24.jpg)
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 )).
![Page 25: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/25.jpg)
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 )).
![Page 26: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/26.jpg)
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 )).
![Page 27: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/27.jpg)
Related discrete modelCutting down uniform tree
V1
V2
V3
A uniform tree Tn
B1
B2
![Page 28: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/28.jpg)
Related discrete modelCutting down uniform tree
V1
V2
V3
A uniform tree Tn
B1
B2
![Page 29: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/29.jpg)
Related discrete modelCutting down uniform tree
V1
V2
V3
A uniform tree Tn
B1
B2
![Page 30: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/30.jpg)
Related discrete modelCutting down uniform tree
V1
V2
V3
A uniform tree Tn
B1
B2
B1
V2 V3
![Page 31: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/31.jpg)
Related discrete modelCutting down uniform tree
V1
V2
V3
A uniform tree Tn
B1
B2
B1
V2 V3
V1
B2
![Page 32: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/32.jpg)
Related discrete modelCutting down uniform tree
V1
V2
V3
A uniform tree Tn
B1
B2
B1
cut(Tn)
V2 V3
V1
B2
![Page 33: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/33.jpg)
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
![Page 34: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/34.jpg)
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)
![Page 35: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/35.jpg)
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)
![Page 36: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/36.jpg)
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)
![Page 37: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/37.jpg)
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)
![Page 38: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/38.jpg)
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)
![Page 39: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/39.jpg)
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
![Page 40: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/40.jpg)
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
![Page 41: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/41.jpg)
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
![Page 42: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/42.jpg)
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
![Page 43: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/43.jpg)
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
![Page 44: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/44.jpg)
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
![Page 45: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/45.jpg)
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
![Page 46: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/46.jpg)
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
![Page 47: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/47.jpg)
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 )).
![Page 48: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/48.jpg)
Thank you!
![Page 49: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/49.jpg)
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
![Page 50: The Cut tree of the Brownian Continuum Random Tree and the ... - …people.bath.ac.uk/ados20/ris2014/slides/minmin_wang.pdf · 2014. 6. 24. · 24 June 2014. Motivation Introduction](https://reader035.fdocuments.in/reader035/viewer/2022071214/6042b68fb44b8935580582bb/html5/thumbnails/50.jpg)
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