a universal limit for random plane non-crossing congurations

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Discrete non-crossing models The continuous model Proving the convergence Application to uniform dissections

The Brownian triangulation:

a universal limit for random plane non-crossing congurations

(joint work with Nicolas Curien)

Igor Kortchemski (Université Paris-Sud, Orsay, France)

MIT Probability Seminar, April 2nd 2012

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Motivations

Let(Xn)_n>1 be a sequence of discrete objects converging towards a continuous objectX:

Xn −→

n→∞ X.

Several consequences:

- From the discrete to the continuous world: if a propertyPis satised by all theXn and passes to the limit, thenXsatisesP.

- From the continuous world to the discrete world: if a propertyPis satised byXand passes to the limit,Xn satises approximately Pfornlarge. - Universality: if(Yn)_n_>₁ is another sequence of objects converging towards

X, thenXn andYn share approximately the same properties forn large. What is the sense of the convergence when the objects arerandom?

→Convergence in distribution

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Motivations

Xn −→

n→∞ X.

- From the continuous world to the discrete world: if a propertyPis satised byXand passes to the limit,Xn satises approximately Pfornlarge. - Universality: if(Yn)_n_>₁ is another sequence of objects converging towards

X, thenXn andYn share approximately the same properties forn large. What is the sense of the convergence when the objects arerandom?

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Motivations

Xn −→

n→∞ X.

- From the continuous world to the discrete world: if a propertyPis satised byXand passes to the limit,Xn satises approximately Pfornlarge.

- Universality: if(Yn)_n_>₁ is another sequence of objects converging towards X, thenXn andYn share approximately the same properties forn large. What is the sense of the convergence when the objects arerandom?

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Motivations

Xn −→

n→∞ X.

- Universality: if(Yn)_n_>₁ is another sequence of objects converging towards X, thenXn andYn share approximately the same properties forn large.

What is the sense of the convergence when the objects arerandom?

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Motivations

Xn −→

n→∞ X.

- Universality: if(Yn)_n_>₁ is another sequence of objects converging towards X, thenXn andYn share approximately the same properties forn large.

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Convergence in distribution

Let(Xn)_n_>₁andXbe random variables with values in a metric space(E,d). Xn

converges in distribution towardsXif

E[F(Xn)] −→

n→∞ E[F(X)] for every bounded continuous functionF:E→R. We write:

Xn

−→(d)

n→∞ X

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Convergence in distribution

converges in distribution towardsXif E[F(Xn)] −→

n→∞ E[F(X)]

for every bounded continuous functionF:E→R.

We write:

Xn

−→(d)

n→∞ X

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Convergence in distribution

converges in distribution towardsXif E[F(Xn)] −→

n→∞ E[F(X)]

for every bounded continuous functionF:E→R.

We write:

Xn

−→(d)

n→∞ X

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Outline

I. The discrete object

II. The limiting continuous object III. Proving the convergence

IV. Application to the study of uniform dissections

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I. The discrete objects

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LetPn be the polygon whose vertices aree^2iπjⁿ (j=0,1,. . .,n−1).

Framework: choose a random non-crossingconguration obtained from the vertices ofPn, that is a collection of non-intersecting diagonals.

What happens forn large?

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What happens fornlarge?

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Case of dissections ofPn.

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Dissections

A dissection ofPn is the union of the sides of Pn and of a collection of diagonals that may intersect only at their endpoints.

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Dissections

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Dissections

A dissection ofP is the union of the sides of P and of a collection of

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Dissections

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Dissections

A dissection ofP is the union of the sides of P and of a collection of

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Dissections

LetDn be a random dissection, chosenuniformlyat random among all dissections ofPn. What doesDn look like whennis large?

Samples ofD18 andD15000.

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Dissections

LetDn be a random dissection, chosenuniformlyat random among all dissections ofPn. What doesDn look like whennis large?

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Case ofnon-crossingtrees ofPn.

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Non-crossing trees

Example of anon-crossing tree ofP10:

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Non-crossing trees

LetTn be a randomnon-crossingtree, chosenuniformlyat random among all those ofPn. What does Tn look like for largen ?

Samples ofT500andT1000.

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Non-crossing trees

LetTn be a randomnon-crossingtree, chosenuniformlyat random among all those ofPn. What does Tn look like for largen ?

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Case of non-crossingpair-partitions ofP2n .

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Non-crossing pair partitions

Example of anon-crossing pair-partition ofP20:

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Non-crossing pair partitions

LetQn be a randomnon-crossing pair-partitition ofP_2n, chosenuniformly among all those ofP_2n. What doesQn look like fornlarge ?

Samples ofQ250andQ1000.

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Non-crossing pair partitions

LetQn be a randomnon-crossing pair-partitition ofP_2n, chosenuniformly among all those ofP_2n. What doesQn look like fornlarge ?

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History of non-crossing congurations of P

_n

Combinatorical point of view:

I Counting and bijections for non-crossing trees: Dulucq & Penaud (1993), Noy (1998), ...

I Counting of various non-crossing congurations: Flajolet & Noy (1999)

Probabilistical combinatorics point of view:

I Uniform triangulations (maximal degree): Devroye, Flajolet, Hurtado, Noy

& Steiger (1999) et Gao & Wormald (2000)

I Non-crossing trees (total length, maximal degree): Deutsch & Noy (2002), Marckert & Panholzer (2002)

I Uniform dissections (degrees, maximal degree): Bernasconi, Panagiotou & Steger (2010)

Geometrical point of view:

I Aldous (1994): large uniform triangulations

I K' (2011): dissections with large faces (non uniform)

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History of non-crossing congurations of P

_n

I Counting of various non-crossing congurations: Flajolet & Noy (1999) Probabilistical combinatorics point of view:

I Uniform dissections (degrees, maximal degree): Bernasconi, Panagiotou &

Steger (2010)

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History of non-crossing congurations of P

_n

I Counting of various non-crossing congurations: Flajolet & Noy (1999) Probabilistical combinatorics point of view:

I Uniform dissections (degrees, maximal degree): Bernasconi, Panagiotou &

Steger (2010)

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II. Construction of the continuous limiting object:

theBrownian triangulation(Aldous, '94)

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Interlude: Brownian motion and the Brownian excursion

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Brownian motion

Theorem (Donsker)

Let(Xn)_n>1 be a sequence of i.i.d random variables withE[X1] =0 and σ²=E

X₁²

<∞.

SetSn=X₁+X₂+· · ·+Xn. Then: Snt

σ√ n,t>0

(d)

n→∞−→ (Wt,t>0),

where (Wt,t>0)is a continuous random function called Brownian motion (which does not depend onσ).

Snt

σ√

n,06t61

forn=100.000:

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Brownian motion

Theorem (Donsker)

X₁²

<∞. SetSn=X₁+X₂+· · ·+Xn.

Then: Snt

σ√ n,t>0

(d)

n→∞−→ (Wt,t>0),

Snt

σ√

n,06t61

forn=100.000:

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Brownian motion

Theorem (Donsker)

X₁²

<∞. SetSn=X₁+X₂+· · ·+Xn. Then:

Snt

σ√ n,t>0

(d) n→∞−→

(Wt,t>0),

Snt

σ√

n,06t61

forn=100.000:

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Brownian motion

Theorem (Donsker)

X₁²

Snt

σ√ n,t>0

(d)

n→∞−→ (Wt,t>0),

Snt

σ√

n,06t61

forn=100.000:

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Brownian motion

Theorem (Donsker)

X₁²

Snt

σ√ n,t>0

(d)

n→∞−→ (Wt,t>0),

Snt

σ√

n,06t61

forn=100.000:

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Brownian motion

Theorem (Donsker)

Let(Xn)_n_>₁ be a sequence of i.i.d random variables withE[X1] =0 and σ²=E

X12

<∞. SetSn=X1+X2+· · ·+Xn. Then:

Snt

σ√ n,t>0

(d)

n−→→∞ (Wt,t>0),

Snt

σ√

n,06t61

forn=100:

Snt

σ√

n,06t61

forn=100.000:

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Brownian motion

Theorem (Donsker)

X₁²

Snt

σ√ n,t>0

(d)

n→∞−→ (Wt,t>0),

S√nt ,06t61

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Theorem (Donsker, conditioned version)

Let(Xn)_n>1 be a sequence of i.i.d. random variables withE[X1] =0and σ²=E

X₁²

<∞.

SetSn=X₁+X₂+· · ·+Xn. Then: Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d)

n→∞−→ (et,t>0), where (et,t>0)is a continuous random function called the Brownian excursion.

The Brownian excursion can be seen as Brownian motion (Wt,06t61) conditioned onW₁=0 andWt>0 fort∈(0,1).

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Theorem (Donsker, conditioned version)

X₁²

<∞. SetSn=X₁+X₂+· · ·+Xn.

Then: Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d)

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Theorem (Donsker, conditioned version)

X₁²

Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d) n→∞−→

(et,t>0), where (et,t>0)is a continuous random function called the Brownian excursion.

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Theorem (Donsker, conditioned version)

X₁²

Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d)

n→∞−→ (et,t>0),

where (et,t>0)is a continuous random function called the Brownian excursion.

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Theorem (Donsker, conditioned version)

X₁²

Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d)

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Theorem (Donsker, conditioned version)

X₁²

Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d)

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Theorem (Donsker, conditioned version)

X₁²

Snt

σ√ n,t>0

Sn=0,S_i>0for i<n

(d)

The Brownian excursion can be seen as Brownian motion (Wt,06t61) conditioned onW =0 andW >0 fort∈(0,1).

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Construction of the limiting object

We start from the Brownian excursion e:

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0. t

Lettbe a local minimum time.

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0. t

Let be a local minimum time. Set sup{s t; e e}and

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.gt t dt

Lettbe a local minimum time. Set gt=sup{s<t; es =et}and dt=inf{s>t; es =et}.

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.gt t dt

Lettbe a local minimum time. Set gt=sup{s<t; es =et}and inf{s t; e e}. Then draw the chords _−2iπg _−2iπt,

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.gt t dt

e^−2iπg^t

e

⁻^2iπd^t

e

⁻^2iπt

Lettbe a local minimum time. Set gt=sup{s<t; es =et}and dt=inf{s>t; es =et}. Then draw the chords

e^−2iπg^t,e^−2iπt , e^−2iπt,e^−2iπd^t

and

e^−2iπg^t,e^−2iπd^t .

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.gt t dt

e^−2iπg^t

e

⁻^2iπd^t

e

⁻^2iπt

Lettbe a local minimum time. Set gt=sup{s<t; es =et}and d =inf{s>t; e =e}. Then draw the chords

e^−2iπg^t,e^−2iπt,

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.

e^−2iπg^t,e^−2iπt , e^−2iπt,e^−2iπd^t and

e^−2iπg^t,e^−2iπd^t.

Repeat this operation for all local minimum times.

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Construction of the limiting object

0.2 0.4 0.6 0.8 1

0.

e^−2iπg^t,e^−2iπt,

−2iπt −2iπd and ^−2iπg ^−2iπd .

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Theorem (Curien & K. '12)

Forn>3, letχ_n be auniformly distributed dissection ofPn, or a uniformly distributed non-crossingtree ofPn or auniformlydistributednon-crossing pair-partition of P2n.

Then:

χ_n −−−→^(d)

n→∞ L(e),

where the convergence holds in distribution for the Hausdor distance on compact subsets of the unit disk.

Applications:

I The length of the longest diagonal ofχ_n converges in distribution towards the probability measure with density:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

This stems from a small calculation when χ_n is a triangulation (Aldous '94)!

I The area of the largest face ofχ_nconverges in distribution towards the area of the largest triangle ofL(e).

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Theorem (Curien & K. '12)

Forn>3, letχ_n be auniformly distributed dissection ofPn, or a uniformly distributed non-crossingtree ofPn or auniformlydistributednon-crossing pair-partition of P2n. Then:

χ_n −−−→^(d)

n→∞ L(e),

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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Theorem (Curien & K. '12)

χ_n −−−→^(d)

n→∞ L(e),

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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Theorem (Curien & K. '12)

Forn>3, letχ_n be auniformly distributed dissection ofPn, or a uniformly distributed non-crossingtree ofPn or auniformlydistributednon-crossing pair-partition of P_2n. Then:

χ_n −−−→^(d)

n→∞ L(e),

Remarks:

I Aldous '94: this holds whenχ_n is auniformlydistributed triangulation of Pn.

I There exists a stable analog ofL(e)with big holes (K. '11). Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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Theorem (Curien & K. '12)

Forn>3, letχ_n be auniformly distributed dissection ofPn, or a uniformly distributed non-crossingtree ofPn or auniformlydistributednon-crossing pair-partition of P_2n. Then:

χ_n −−−→^(d)

n→∞ L(e),

Remarks:

I Aldous '94: this holds whenχ_n is auniformlydistributed triangulation of Pn.

I There exists a stable analog ofL(e)with big holes (K. '11).

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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Theorem (Curien & K. '12)

χ_n −−−→^(d)

n→∞ L(e),

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

I The area of the largest face ofχ_n converges in distribution towards the area of the largest triangle ofL(e).

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Theorem (Curien & K. '12)

χ_n −−−→^(d)

n→∞ L(e),

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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Theorem (Curien & K. '12)

χ_n −−−→^(d)

n→∞ L(e),

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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Theorem (Curien & K. '12)

χ_n −−−→^(d)

n→∞ L(e),

Applications:

1 π

3x−1 x²(1−x)²√

1−2x1{¹36x6¹₂}dx.

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III. How does one establish the convergence of all these non-crossing uniformly distributed models towards the

Brownian triangulation?

Key point: Each one of the previous models can be coded by aconditioned Galton-Watson tree.

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III. How does one establish the convergence of all these non-crossing uniformly distributed models towards the

Brownian triangulation?

Key point: Each one of the previous models can be coded by aconditioned Galton-Watson tree.

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Coding trees

Denition (of the contour function)

A platypus explores the tree at unit speed. For06t62(ζ(τ) −1),Ct(τ)is dened as the distance from the root at the position of the beast at time t.

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Coding trees

Denition (of the contour function)

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Coding trees

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Coding trees

Denition (of the contour function)

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Coding trees

0 10 20 30 40 50

0 1 2 3 4 5 6 7

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0.2 0.4 0.6 0.8 1

0.

Scaled contour function of a large conditioned Galton-Watson tree.

Strategy to prove the convergence towards the Brownian triangulation:

I Each one of thenon-crossing uniformly distributedmodels can be coded by aconditionedGalton-Watsontree.

I The scaled contour functions ofconditionedGalton-Watsontrees converge towards the Brownian excursion.

I The Brownian excursion codes the Brownian triangulationL(e).

It follows that thenon-crossing uniformly distributedmodels converge towards L(e).

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0.2 0.4 0.6 0.8 1

0.

I Each one of thenon-crossinguniformly distributed models can be coded by aconditionedGalton-Watsontree.

I The scaled contour functions ofconditionedGalton-Watsontrees converge towards theBrownian excursion.

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0.2 0.4 0.6 0.8 1

0.

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0.2 0.4 0.6 0.8 1

0.

I The scaled contour functions ofconditionedGalton-Watsontrees converge

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0.2 0.4 0.6 0.8 1

0.

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0.2 0.4 0.6 0.8 1

0.

I The scaled contour functions ofconditionedGalton-Watsontrees converge

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Brief recap on Galton-Watson trees

We consider rooted plane (oriented) trees.

Letρ be a probability measure onNwith mean61 s.t. ρ(1)<1. The law of aGalton-Watson tree with ospring distributionρis the unique probability distributionPρ on the set of all trees such that:

1. k_∅ is distributed according toρ, where k_∅ is the number of children ofthe root.

2. for everyj>1 withρ(j)>0, conditionally onPρ(·|k_∅=j), thejsubtrees of thejchildren ofthe rootare independent with law Pρ.

Here,k_∅=2.

The probability of getting this tree isρ(2)²ρ(0)³.

Here,ζ(τ) =5andλ(τ) =3.

ζ(τ)is the total number of vertices and λ(τ)is the total number of leaves.

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Brief recap on Galton-Watson trees

Letρ be a probability measure onNwith mean61 s.t. ρ(1)<1.

The law of aGalton-Watson tree with ospring distributionρis the unique probability distributionPρ on the set of all trees such that:

Here,k_∅=2.

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Brief recap on Galton-Watson trees

Here,k_∅=2.

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Brief recap on Galton-Watson trees

Here,k_∅=2.

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Brief recap on Galton-Watson trees

Here,k_∅=2.

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Brief recap on Galton-Watson trees

Here,k_∅=2.