Random stable looptrees and percolation on random maps

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Random stable looptrees and percolation on random maps

Igor Kortchemski (joint work with Nicolas Curien)

Universität Zürich

Seminar on Stochastic Processes – Zürich – October 2014

October 7, 2014

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Galton–Watson trees Looptrees Looptrees and preferential attachment Looptrees and percolation

Outline

0. Motivation

I. Galton–Watson trees and their scaling limits II. Looptrees

III. Looptrees and preferential attachment

IV. Looptrees and percolation on random triangulations

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Motivations

Let (X_n)_n_>₁ be “discrete” objects converging towards a “continuous” object X:

X_n _n !

!1 X.

Several consequences:

- From the discrete to the continuous world: if a property P is satisfied by all the X_n and passes to the limit, then X satisfies P.

- From the world to the discrete world: if a property P is satisfied by X and passes to the limit, X_n satisfies “approximately” P for n large.

- Universality: if (Y_n)_n_>₁ is another sequence of objects converging towards X, then X_n and Y_n share approximately the same properties for n large.

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Motivations

X_n _n !

!1 X.

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Motivations

X_n _n !

!1 X.

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Motivations

X_n _n !

!1 X.

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Motivations

X_n _n !

!1 X.

What is the sense of the convergence when the objects are random?

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Motivations

X_n _n !

!1 X.

What is the sense of the convergence when the objects are random?

y Convergence in distribution

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Motivations

X_n _n !

!1 X.

What discrete objets will we consider ?

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Motivations

X_n _n !

!1 X.

What discrete objets will we consider ?

y Finite graphs, seen as compact metric spaces

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Motivations

X_n _n !

!1 X.

What will be the notion of convergence?

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Motivations

X_n _n !

!1 X.

What will be the notion of convergence?

y Convergence of compact metric spaces for the Gromov–Hausdorﬀ topology

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The Gromov–Hausdorff distance

Let X, Y be two compact metric spaces.

The Gromov–Hausdorﬀ distance between X and Y is the minimal Hausdorﬀ distance between all possible embeddings of X and Y into a common metric space Z.

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The Gromov–Hausdorff distance

Let X, Y be two compact metric spaces.

The Gromov–Hausdorﬀ distance between X and Y is the minimal Hausdorﬀ distance between all possible embeddings of X and Y into a common metric space Z.

X Z Y

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Motivations

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0. Motivations

I. Galton–Watson trees and their scaling limits

II. Looptrees

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Reminder on Galton–Watson trees

Let ⇢ be a probability measure on {0, 1, 2, . . .} such that P

i i⇢(i) 6 1 and

⇢(1) < 1.

A Galton–Watson tree with oﬀspring distribution ⇢ is a random plane tree such that:

the number of children of the root is distributed according to ⇢,

Conditionally on the fact that the root has j children, the number of children of these j children are independent of law ⇢, and so on.

Here, the root has 2 children.

Here, ⌧ has 8 vertices.

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Reminder on Galton–Watson trees

i i⇢(i) 6 1 and

⇢(1) < 1. A Galton–Watson tree with oﬀspring distribution ⇢ is a random plane tree such that:

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Reminder on Galton–Watson trees

i i⇢(i) 6 1 and

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Reminder on Galton–Watson trees

i i⇢(i) 6 1 and

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Reminder on Galton–Watson trees

i i⇢(i) 6 1 and

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Reminder on Galton–Watson trees

i i⇢(i) 6 1 and

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Scaling limits: finite variance case

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Scaling limits: finite variance

Let µ be an oﬀspring distribution such that X

i>0

iµ_i = 1 (µ is critical) X

i>0

i²µ_i < 1 (µ has finite variance)

Let t_n be a GWµ tree conditioned on having n vertices.

View at t_n as a compact metric space (the vertices of t_n are endowed with the graph distance).

What does t_n look like for large n?

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Scaling limits: finite variance

i>0

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Scaling limits: finite variance

i>0

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Scaling limits: finite variance

i>0

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A simulation of a large random critical GW tree

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Scaling limits: finite variance case

Let µ be a critical oﬀspring distribution with finite variance.

Let t_n be a GWµ

tree conditioned on having n vertices.

View t_n as a compact metric space (the vertices of t_n are endowed with the graph distance).

Theorem (Aldous ’93)

There exists a random compact metric space T such that: 2p

n · t_n ^(d)!

n!1 T,

where the convergence holds in distribution for the Gromov-Hausdorﬀ distance on compact metric spaces.

T is called the Brownian Continuum Random Tree.

Remarks

y All the branchpoints of T are binary.

y T is coded by the normalized Brownian excursion.

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Scaling limits: finite variance case

Let µ be a critical oﬀspring distribution with finite variance. Let t_n be a GWµ

Theorem (Aldous ’93)

n · t_n ^(d)!

n!1 T,

Remarks

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Scaling limits: finite variance case

Theorem (Aldous ’93)

n · t_n ^(d)!

n!1 T,

Remarks

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Scaling limits: finite variance case

Theorem (Aldous ’93)

There exists a random compact metric space T such that:

2p

n · t_n ^(d)!

n!1 T,

Remarks

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Scaling limits: finite variance case

Theorem (Aldous ’93)

2p

n · t_n ^(d)!

n!1 T,

Remarks

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Scaling limits: finite variance case

Theorem (Aldous ’93)

2p

n · t_n ^(d)!

n!1 T,

Remarks

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Scaling limits: finite variance case

Theorem (Aldous ’93)

2p

n · t_n ^(d)!

n!1 T,

Remarks

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Scaling limits: finite variance case

Theorem (Aldous ’93)

2p

n · t_n ^(d)!

n!1 T,

Remarks

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What is the Brownian Continuum Random Tree?

First define the contour function of a tree:

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What is the Brownian Continuum Random Tree?

Knowing the contour function, it is easy to recover the tree by gluing:

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What is the Brownian Continuum Random Tree?

The Brownian tree T is obtained by gluing from the Brownian excursion e.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Figure: A simulation of e.

p

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A simulation of the Brownian CRT

Figure: A non isometric plane embedding of a realization of T .

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Scaling limits: infinite variance case

p

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Scaling limits: domain of attraction of a stable law

Fix ↵ 2 (1, 2). Let µ be an oﬀspring distribution such that X

i>0

iµ_i = 1 (µ is critical)

µ_i ⇠

i!1

c

i^1+↵ (µ has a heavy tail)

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Scaling limits: domain of attraction of a stable law

i>0

µ_i ⇠

i!1

c

p

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Scaling limits: domain of attraction of a stable law

i>0

µ_i ⇠

i!1

c

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Scaling limits: domain of attraction of a stable law

i>0

µ_i ⇠

i!1

c

p

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Figure: A large ↵ = 1.1 – stable tree

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p

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Scaling limits: domain of attraction of a stable law

Fix ↵ 2 (1, 2). Let µ be a critical oﬀspring distribution such that µ_i ⇠ c/i^1+↵. Let t_n be a GWµ tree conditioned on having n vertices.

Theorem (Duquesne ’03)

There exists a random compact metric space T_↵ such that:

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

Remarks

y All the branchpoints of T_↵ are of infinite degree.

y The maximal degree of t_n is of order n^1/↵.

y T_↵ is coded by the normalized excursion of a spectrally positive stable Lévy process of index ↵.

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Scaling limits: domain of attraction of a stable law

Theorem (Duquesne ’03)

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

Remarks

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Scaling limits: domain of attraction of a stable law

Theorem (Duquesne ’03)

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

Remarks

p

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Scaling limits: domain of attraction of a stable law

Theorem (Duquesne ’03)

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

The tree T_↵ is called the stable tree of index ↵ (introduced by Le Gall & Le Jan).

Remarks

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Scaling limits: domain of attraction of a stable law

Theorem (Duquesne ’03)

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

Remarks

p

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Scaling limits: domain of attraction of a stable law

Theorem (Duquesne ’03)

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

Remarks

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Scaling limits: domain of attraction of a stable law

Theorem (Duquesne ’03)

(c| (1 - ↵)|)^-1/^↵

n^1-1/^↵ · t_n ^(d)!

n!1 T_↵,

Remarks

p

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What happens if µ is not critical?

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Scaling limits: nongeneric case

Fix ↵ > 1. Let µ be a subcritical oﬀspring distribution such that µ_i ⇠ c/i^1+↵.

Let t_n be a GWµ tree conditioned on having n vertices. The tree t_n is said to be nongeneric (Jonsson & Stefánsson).

Theorem (Jonsson & Stefánsson ’11)

A condensation phenomenon occurs: there exists a unique vertex of t_n of degree of order n, and all the other degrees are o(n).

The height of t_n is of order ln(n). Theorem (K.).

p

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Scaling limits: nongeneric case

Fix ↵ > 1. Let µ be a subcritical oﬀspring distribution such that µ_i ⇠ c/i^1+↵. Let t_n be a GWµ tree conditioned on having n vertices.

The tree t_n is said to be nongeneric (Jonsson & Stefánsson).

Theorem (Jonsson & Stefánsson ’11)

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Scaling limits: nongeneric case

Fix ↵ > 1. Let µ be a subcritical oﬀspring distribution such that µ_i ⇠ c/i^1+↵. Let t_n be a GWµ tree conditioned on having n vertices. The tree t_n is said to be nongeneric (Jonsson & Stefánsson).

Theorem (Jonsson & Stefánsson ’11)

p

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Scaling limits: nongeneric case

Figure: A large nongeneric tree

Theorem (Jonsson & Stefánsson ’11)

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Scaling limits: nongeneric case

Theorem (Jonsson & Stefánsson ’11)

p

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Scaling limits: nongeneric case

Theorem (Jonsson & Stefánsson ’11)

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0. Motivations

II. Looptrees

p

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Discrete looptrees

Given a plane tree ⌧, define Loop(⌧) as the graph obtained from ⌧

by replacing each vertex u by a loop with deg(u) vertices,

then by gluing the loops together according to the tree structure of ⌧.

Figure: A plane tree ⌧ and its associated discrete looptree Loop(⌧).

We view Loop(⌧) as a compact metric space.

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Discrete looptrees

Given a plane tree ⌧, define Loop(⌧) as the graph obtained from ⌧ by replacing each vertex u by a loop with deg(u) vertices,

p

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Discrete looptrees

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Discrete looptrees

p

a

c

d(a,b)=2 d(b,c)=3 d(a,c)=4

b

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Discrete looptrees

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0. Motivations

II. Looptrees

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Trees built by preferential attachment

Let (T_n)_n_>₁ be a sequence of random plane trees grown recursively at random:

I T₁ is just one vertex,

I For every n > 1, T_n+1 is obtained from T_n by adding an edge into a corner of T_n chosen uniformly at random.

This is the preferential attachement model (Szymánski ’87; Albert & Barabási

’99; Bollobás, Riordan, Spencer & Tusnády ’01).

y Does the sequence (T_n) admit scaling limits? It is known that the

diameter of T_n is of order log(n): Does _log(n)¹ · T_n converge towards a limiting compact metric space?

y Answer: no.

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Trees built by preferential attachment

y Answer: no.

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Trees built by preferential attachment

y Answer: no.

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Trees built by preferential attachment

y Answer: no.

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Trees built by preferential attachment

y Does the sequence (T_n) admit scaling limits?

It is known that the

y Answer: no.

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Trees built by preferential attachment

y Does the sequence (T_n) admit scaling limits? It is known that the diameter of T_n is of order log(n):

Does _log(n)¹ · T_n converge towards a limiting compact metric space?

y Answer: no.

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Trees built by preferential attachment

y Answer: no.

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Trees built by preferential attachment

y Answer: no.

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Scaling limits of trees built by preferential attachment

There exists a random compact metric space L such that:

n^-1/2 · Loop(T_n) _n^a.s.!

!1 L,

where the convergence holds almost surely for the Gromov–Hausdorﬀ topology.

Theorem (Curien, Duquesne, K., Manolescu).

Figure: The looptree of a large tree built by preferential attachement.

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Scaling limits of trees built by preferential attachment

There exists a random compact metric space L such that:

n^-1/2 · Loop(T_n) _n^a.s.!

!1 L,

where the convergence holds almost surely for the Gromov–Hausdorﬀ topology.

Theorem (Curien, Duquesne, K., Manolescu).

Figure: The looptree of a large tree built by preferential attachement.Igor Kortchemski Random stable looptrees and random maps 28 / 20

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0. Motivations

I. Galton–Watson trees and their scaling limits II. Looptrees

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Random stable looptrees