Random planar maps & growth-fragmentations

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Igor Kortchemski

(joint work with J. Bertoin and N. Curien)

Random planar maps & growth-fragmentations

CNRS & École polytechnique

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Motivation Boltzmann triangulations Peeling explorations Growth-fragmentations

Motivation

What does a “typical” random surfacelook like?

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y Idea: construct a (two-dimensional) random surfaceas a limit of random discrete surfaces.

Consider ntriangles, and glue them uniformly at random in such a way to get a surface homeomorphic to a sphere.

Figure: A largerandom triangulation(simulation by Nicolas Curien)

Igor Kortchemski Random planar maps & growth-fragmentations 2 / 672

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The Brownian map

Problem(Schrammat ICM ’06): LetTn be a random uniform triangulation of the sphere withntriangles.

ViewTn as a compact metric space, by equipping its vertices with the graph distance. Show that n^−1/4·Tn converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorfftopology.

Solved byLe Gallin 2011.

Since, many different models of discrete surfaces have been shown to converge to the Brownian map (Miermont,Beltran&Le Gall,Addario-Berry&Albenque, Bettinelli& Jacob&Miermont,Abraham)

, using various techniques (in particular bijective codings by labelled trees).

(seeLe Gall’s proceeding at ICM ’14 for more information and references)

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The Brownian map

Problem(Schrammat ICM ’06): LetTn be a random uniform triangulation of the sphere withntriangles. ViewTn as a compact metric space, by equipping its vertices with the graph distance.

Show thatn^−1/4·Tn converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorfftopology.

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The Brownian map

Problem(Schrammat ICM ’06): LetTn be a random uniform triangulation of the sphere withntriangles. ViewTn as a compact metric space, by equipping its vertices with the graph distance. Show that n^−1/4·Tn converges towards a random compact metric space (the Brownian map)

, in distribution for the Gromov–Hausdorfftopology.

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The Brownian map

Problem(Schrammat ICM ’06): LetTn be a random uniform triangulation of the sphere withntriangles. ViewTn as a compact metric space, by equipping its vertices with the graph distance. Show that n^−1/4·Tn converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorfftopology.

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y Other motivations:

– links with two dimensional Liouville Quantum Gravity (David,Duplantier, Garban,Kupianen, Maillard,Miller,Rhodes,Sheffield,Vargas,Zeitouni) c.f. the talks ofJason Miller,Scott SheffieldandVincent Vargas.

– study of random planar maps decorated with statistical physics models (Angel, Berestycki,Borot,Bouttier,Guitter,Chen,Curien,Gwynne,K.,Laslier,Mao, Ray,Sheffield, Sun,Wilson), c.f. the talk byGourab Ray.

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Outline

I. Boltzmann triangulations with a boundary II. Peeling explorations

III. Cycles & growth-fragmentations

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I. Boltzmann triangulations with a boundary

II. Peeling explorations

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Triangulations

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Definitions

Amapis a finite connected graph properly embedded in the sphere (up to orientation preserving continuous deformations).

A map is atriangulation when all the faces are triangles. A map isrootedwhen an oriented edge is distinguished.

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Definitions

Amapis a finite connected graph properly embedded in the sphere (up to orientation preserving continuous deformations).

A map is atriangulation when all the faces are triangles. A map isrootedwhen an oriented edge is distinguished.

Figure: Two identical triangulations.

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Definitions

Amapis a finite connected graph properly embedded in the sphere (up to orientation preserving continuous deformations). A map is atriangulation when all the faces are triangles.

A map isrootedwhen an oriented edge is distinguished.

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Definitions

Amapis a finite connected graph properly embedded in the sphere (up to orientation preserving continuous deformations). A map is atriangulation when all the faces are triangles. A map isrootedwhen an oriented edge is distinguished.

Figure: Two identical triangulations.

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Definitions

Amapis a finite connected graph properly embedded in the sphere (up to orientation preserving continuous deformations). A map is atriangulation when all the faces are triangles. A map isrootedwhen an oriented edge is distinguished.

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Triangulations with a boundary

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Definitions

Atriangulation with a boundaryis a map where all faces are triangles, except possibly the one to the right of the root edge, called the external face.

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Another example of a triangulation with a boundary

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Definitions

Atriangulation of thep-gon is a triangulation with a simple boundary of lengthp.

A triangulation of thep-gon chosen at random proportionally to (12√

3)−#(internal vertices)

is called a (critical)Boltzmann triangulation of the p-gon.

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Definitions

Atriangulation of thep-gon is a triangulation with a simple boundary of lengthp.

A triangulation of thep-gon chosen at random proportionally to (12√

3)−#(internal vertices)

is called a (critical)Boltzmann triangulation of the p-gon.

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Cycles at heights

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The goal

LetT^(p)be a randomBoltzmann triangulationof thep-gon

,B_r(T^(p))its ball of radiusr, and

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

be the lengths (or perimeters) of the cycles ofBr(T^(p))ranked in decreasing order.

t Br(t)

r

y Goal: obtain a functional invariance principle for(L^(p)(r);r>0). In this spirit, a “breadth-first search” description of theBrownian mapis given by Miller

& Sheffield’15.

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The goal

LetT^(p)be a randomBoltzmann triangulationof thep-gon,B_r(T^(p))its ball of radiusr

, and

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

t Br(t)

r

& Sheffield’15.

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The goal

LetT^(p)be a randomBoltzmann triangulationof thep-gon,B_r(T^(p))its ball of radiusr

, and

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

t Br(t)

r

& Sheffield’15.

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The goal

LetT^(p)be a randomBoltzmann triangulationof thep-gon,B_r(T^(p))its ball of radiusr, and

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

t Br(t)

r

& Sheffield’15.

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The goal

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

t Br(t)

r

y Goal: obtain a functional invariance principle for(L^(p)(r);r>0).

In this spirit, a “breadth-first search” description of theBrownian mapis given by Miller

& Sheffield’15.

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The goal

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

t Br(t)

r

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I. Boltzmann triangulations with a boundary

II. Peeling explorations

Igor Kortchemski Random planar maps & growth-fragmentations 15 /ℵ0

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Geometry of random maps

Several techniques to studyrandom maps:

– bijective techniques, following the work ofSchaeffer’98.

– peeling process, which is an algorithmic procedure that explores a map step-by-step in a Markovian way (Watabiki’95, Angel’03).

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Geometry of random maps

– peeling process, which is an algorithmic procedure that explores a map step-by-step in a Markovian way (Watabiki’95, Angel’03).

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Geometry of random maps

– peeling process, which is an algorithmic procedure that explores a map step-by-step in a Markovian way (Watabiki’95,Angel ’03).

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Branching peeling explorations

Intuitively speaking, thebranching peeling processof atriangulationtis a way to iteratively exploretstarting from its boundary and by discovering at each step a new triangle bypeeling an edgedetermined by a peeling algorithmA.

The goal

LetT^(p)be a randomBoltzmann triangulationof thep-gon,Br(T^(p))its ball of radiusr, and

L^(p)(r) :=

L^(p)₁ (r),L^(p)₂ (r), . . . .

t Br(t)

r

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Following the locally largest cycle

y Idea: follow the locally largest cycle at each peeling step and consider its lengtheL^(p)(r)afterr peeling steps.

eL⁽⁴⁾(0) =4

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Following the locally largest cycle

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Following the locally largest cycle

eL⁽⁴⁾(0) =4,eL⁽⁴⁾(1) =5

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Following the locally largest cycle

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Following the locally largest cycle

eL⁽⁴⁾(0) =4,eL⁽⁴⁾(1) =5,eL⁽⁴⁾(2) =3

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Following the locally largest cycle

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Following the locally largest cycle

eL⁽⁴⁾(0) =4,eL⁽⁴⁾(1) =5,eL⁽⁴⁾(2) =3,eL⁽⁴⁾(3) =3

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Following the locally largest cycle

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Following the locally largest cycle

eL⁽⁴⁾(0) =4,eL⁽⁴⁾(1) =5,eL⁽⁴⁾(2) =3,eL⁽⁴⁾(3) =3,eL⁽⁴⁾(4) =2

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Following the locally largest cycle

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Following the locally largest cycle

eL⁽⁴⁾(0) =4,eL⁽⁴⁾(1) =5,eL⁽⁴⁾(2) =3,eL⁽⁴⁾(3) =3,eL⁽⁴⁾(4) =2,eL⁽⁴⁾(5) =0.

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Scaling limit for the locally largest cycle

Recall thateL^(p)(r)the length of the locally largest cycle afterr peeling steps.

y Key fact: (eL^(p)(r);r>0)is a Markov chain on the nonnegative integers, started at p, absorbed at0 and with explicit transitions. In addition, the triangulations filling-in the unexplored holes areBoltzmann triangulations. IfL^(p)(r)the length of the locally largest cycle at heightr, with the help of Bertoin & K. ’14 and Curien & Le Gall ’14, we get that:

We have 1

pL^(p)(b√

p·tc);t>0

(d) p→−→∞

X

3

2√ π·t

;t>0

, Proposition(Bertoin, Curien & K. ’15).

where Xis a càdlàg self-similar process withX(0)=1and absorbed at 0.

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Scaling limit for the locally largest cycle

y Key fact: (eL^(p)(r);r>0)is a Markov chain on the nonnegative integers, started at p, absorbed at0 and with explicit transitions.

In addition, the triangulations filling-in the unexplored holes areBoltzmann triangulations. IfL^(p)(r)the length of the locally largest cycle at heightr, with the help of Bertoin & K. ’14 and Curien & Le Gall ’14, we get that:

We have 1

pL^(p)(b√

p·tc);t>0

(d) p→−→∞

X

3

2√ π·t

;t>0

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Scaling limit for the locally largest cycle

y Key fact: (eL^(p)(r);r>0)is a Markov chain on the nonnegative integers, started at p, absorbed at0 and with explicit transitions. In addition, the triangulations filling-in the unexplored holes areBoltzmann triangulations.

IfL^(p)(r)the length of the locally largest cycle at heightr, with the help of Bertoin & K. ’14 and Curien & Le Gall ’14, we get that:

We have 1

pL^(p)(b√

p·tc);t>0

(d) p→−→∞

X

3

2√ π·t

;t>0

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Scaling limit for the locally largest cycle

IfL^(p)(r)the length of the locally largest cycle at heightr

, with the help of Bertoin & K. ’14 and Curien & Le Gall ’14, we get that:

We have 1

pL^(p)(b√

p·tc);t>0

(d) p→−→∞

X

3

2√ π·t

;t>0

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Scaling limit for the locally largest cycle

IfL^(p)(r)the length of the locally largest cycle at heightr, with the help of Bertoin & K. ’14 and Curien & Le Gall ’14, we get that:

We have 1

pL^(p)(b√

p·tc);t>0

(d) p→−→∞

X

3

2√ π·t

;t>0

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The self-similar process X

20 000 40 000 60 000 80 000

500 1000 1500 2000 2500

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The self-similar process X

Letξ be the spectrally negative Lévy process with Laplace exponent

Ψ(q) = −8 3q+

Z1 1/2

(x^q−1+q(1−x)) x(1−x)−5/2

dx, so thatE[exp(qξ(t))] =exp(tΨ(q))for everyt>0 andq>0.

Then set

τ(t) =inf

u>0; Zu

0

^ξ(s)/2ds > t

, t>0 with the convention thatinf∅=∞, i.e.τ(t) =∞whenevert>R_∞

0 ^ξ(s)/2ds. Finally, set

X(t)=exp(ξ(τ(t))), t>0

(with the conventionexp(ξ(∞)) =0), which is a self-similar Markov process (Lamperti transformation).

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We use Xto define a self-similar growth-fragmentation process with binary dislocations.

We viewX(t)as the size of a typical particle or cell at aget, and: – Start at time 0from a single cell with size1, and suppose that its size evolves according toX. We interpret each (negative) jump ofXas a division event for the cell, in the sense that whenever∆X(t)=X(t)−X(t−)= −y <0, the cell divides at timetinto a mother cell and a daughter.

y After the splitting event, the mother cell has sizeX(t)and the daughter cell has sizeyand the evolution of the daughter cell is then governed by the law of the same self-similar Markov processX(starting of course fromy), and is independent of the processes of all the other daughter particles.

And so on for the granddaughters, then great-granddaughters, ...

ByBertoin’15, for everyt>0, the family of the sizes of cells which are present in the system at time tis cube-summable, and can therefore be ranked in non-increasing order. This yields a random variable with values in `^↓₃ which we denote byX(t)= (X1(t),X2(t), . . .).

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Defining the growth-fragmentation

We use Xto define a self-similar growth-fragmentation process with binary dislocations. We viewX(t)as the size of a typical particle or cell at aget, and:

– Start at time 0from a single cell with size1, and suppose that its size evolves according toX. We interpret each (negative) jump ofXas a division event for the cell, in the sense that whenever∆X(t)=X(t)−X(t−)= −y <0, the cell divides at timetinto a mother cell and a daughter.

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Defining the growth-fragmentation

– Start at time 0from a single cell with size1, and suppose that its size evolves according toX.

We interpret each (negative) jump ofXas a division event for the cell, in the sense that whenever∆X(t)=X(t)−X(t−)= −y <0, the cell divides at timetinto a mother cell and a daughter.

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Defining the growth-fragmentation

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Defining the growth-fragmentation

y After the splitting event, the mother cell has sizeX(t)and the daughter cell has sizey

and the evolution of the daughter cell is then governed by the law of the same self-similar Markov processX(starting of course fromy), and is independent of the processes of all the other daughter particles.

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Defining the growth-fragmentation

y After the splitting event, the mother cell has sizeX(t)and the daughter cell has sizeyand the evolution of the daughter cell is then governed by the law of the same self-similar Markov processX(starting of course fromy)

, and is independent of the processes of all the other daughter particles.

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Defining the growth-fragmentation

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Defining the growth-fragmentation

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Defining the growth-fragmentation

, and can therefore be ranked in non-increasing order. This yields a random variable with values in `^↓₃ which we denote byX(t)= (X1(t),X2(t), . . .).

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Defining the growth-fragmentation

ByBertoin’15, for everyt>0, the family of the sizes of cells which are present in the system at timet is cube-summable, and can therefore be ranked in non-increasing order.

This yields a random variable with values in`^↓₃ which we denote byX(t)= (X1(t),X2(t), . . .).

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Defining the growth-fragmentation

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Description of the growth-fragmentation

We can think ofXas a self-similar compensated fragmentation, in the sense that it describes the evolution of particles that grow and divide independently one of the other as time passes:

y Xfulfills the branching property, and is self-similar with index−1/2, in the sense that for everyc >0, the rescaled process(cX(c^−1/2t),t>0)has the same law asXstarted from the sequence(c, 0, 0, . . .).

y The dislocations occurring inXare binary, i.e. they correspond to replacing some massmin the system by two smaller massesm₁andm₂with

m₁+m₂=m. Informally, inX, each massm >0 splits into a pair of smaller masses(xm,(1−x)m)at ratem^−1/2ν(dx), where

ν(dx) = (x(1−x))^−5/2dx.

y We haveR

(1−x)²ν(dx)<∞, butR

(1−x)ν(dx) =∞which underlines the necessity of compensating the dislocations.

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Description of the growth-fragmentation

y Xfulfills the branching property, and is self-similar with index−1/2

, in the sense that for everyc >0, the rescaled process(cX(c^−1/2t),t>0)has the same law asXstarted from the sequence(c, 0, 0, . . .).

ν(dx) = (x(1−x))^−5/2dx.

y We haveR

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Description of the growth-fragmentation

ν(dx) = (x(1−x))^−5/2dx.

y We haveR

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Description of the growth-fragmentation

y The dislocations occurring inXare binary, i.e. they correspond to replacing some massmin the system by two smaller massesm₁andm₂ with

m₁+m₂=m.

Informally, inX, each massm >0 splits into a pair of smaller masses(xm,(1−x)m)at ratem^−1/2ν(dx), where

ν(dx) = (x(1−x))^−5/2dx.

y We haveR