Scaling limits of random dissections

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Scaling limits of random dissections

Igor Kortchemski (work with N. Curien and B. Haas) DMA – École Normale Supérieure

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Motivation Definition: Boltzmann dissections Theorem Applications Proof

Outline

O. Motivation I. Definition II. Theorem III. Application IV. Proof

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O. Motivation

I. Definition II. Theorem III. Application IV. Proof

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Scaling limits of random maps

Goal: choose a random mapM_n of sizenand study its global geometry as n→∞.

If possible, show that a continuous limit exists (Chassaing & Schaeffer

’04).

Problem(Schramm at ICM ’06): LetTn be a random uniform triangulation of the sphere withnvertices. ViewTn as a compact metric space. Show that n^−1/4·Tn converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorff topology.

Solved by Le Gall in 2011.

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

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Scaling limits of random maps

Goal: choose a random mapM_n of sizenand study its global geometry as n→∞. If possible, show that a continuous limit exists (Chassaing & Schaeffer

’04).

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Scaling limits of random maps

’04).

Problem(Schramm at ICM ’06): LetTn be a random uniform triangulation of the sphere withnvertices.

ViewTn as a compact metric space. Show that n^−1/4·Tn converges towards a random compact metric space (the Brownian map), in distribution for the Gromov–Hausdorff topology.

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Scaling limits of random maps

’04).

Problem(Schramm at ICM ’06): LetTn be a random uniform triangulation of the sphere withnvertices. ViewTn as a compact metric space.

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

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Scaling limits of random maps

’04).

Problem(Schramm at ICM ’06): LetTn be a random uniform triangulation of the sphere withnvertices. ViewTn as a compact metric space. Show that n^−1/4·Tn converges towards a random compact metric space (the Brownian map)

, in distribution for the Gromov–Hausdorff topology. Solved by Le Gall in 2011.

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Scaling limits of random maps

’04).

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Scaling limits of random maps

’04).

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Scaling limits of random maps

’04).

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A simulation of the Brownian CRT

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Random maps having the CRT as a scaling limit

I Albenque & Marckert (’07): Uniform stack triangulationswith2nfaces

I Janson & Steffánsson (’12): Boltzmann-type bipartite maps withnedges, having a face of macroscopic degree.

I Bettinelli (’11): Uniform quadrangulationswithnfaces with fixed boundary√

n.

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Random maps having the CRT as a scaling limit

n.

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Random maps having the CRT as a scaling limit

n.

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Random maps having the CRT as a scaling limit

n.

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Random maps having the CRT as a scaling limit

n.

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Random maps having the CRT as a scaling limit

n.

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O. Motivation

I. Definition: Boltzmann dissections

II. Theorem III. Application IV. Proof

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Dissections

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

Adissection ofPn is the union of the sidesPn and of a collection of noncrossing diagonals.

We will viewdissectionsascompact metric spaces.

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Dissections

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Dissections

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Dissections à la Boltzmann

LetDⁿ be the set of alldissectionsofPn.

Fix a sequence(µi)i>2of nonnegative real numbers. Associate a weight π(ω)with everydissectionω∈Dn:

π(ω)= Y

ffaces ofω

µ_deg(f)−1.

Then define a probability measure onDn by normalizing the weights: Zn= X

ω∈Dn

π(ω)

and whenZn 6=0, set for everyω∈Dn: P^µn(ω) = 1

Zn

π(ω).

We call P^µn(ω)aBoltzmann probability distribution.

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Dissections à la Boltzmann

Fix a sequence(µi)i>2of nonnegative real numbers.

Associate a weight π(ω)with everydissectionω∈Dn: π(ω)= Y

ffaces ofω

µ_deg(f)−1.

ω∈Dn

π(ω)

Zn

π(ω).

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Dissections à la Boltzmann

ffaces ofω

µ_deg(f)−1.

ω∈Dn

π(ω)

Zn

π(ω).

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Dissections à la Boltzmann

ffaces ofω

µ_deg(f)−1.

ω∈Dn

π(ω)

Zn

π(ω).

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Dissections à la Boltzmann

ffaces ofω

µ_deg(f)−1.

Then define a probability measure onDn by normalizing the weights:

Z_n = X

ω∈Dn

π(ω)

and whenZn6=0, set for everyω∈Dn:

µ 1

We call P^µⁿ(ω)aBoltzmann probability distribution.

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Dissections à la Boltzmann

ffaces ofω

µ_deg(f)−1.

Then define a probability measure onDn by normalizing the weights:

Z_n = X

ω∈Dn

π(ω)

and whenZn6=0, set for everyω∈Dn: P^µn(ω) = 1

Zn

π(ω).

We call P^µⁿ(ω)aBoltzmann probability distribution.

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Dissections à la Boltzmann

Assume that νi=λⁱ⁻¹µifor everyi>2 withλ >0.

ThenP^νn=P^µn.

Proposition.

Suppose that there existsλ >0such that P

i>2iλⁱ⁻¹µi=1. Set

ν0=1−X

i>2

λⁱ⁻¹µi, ν1=0, νi=λⁱ⁻¹µi (i>2),

Then P^νn=P^µⁿ andνis a critical probability measure on Z+withν₁=0.

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Dissections à la Boltzmann

Assume that νi=λⁱ⁻¹µifor everyi>2 withλ >0. ThenP^νn=P^µn. Proposition.

ν0=1−X

i>2

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Dissections à la Boltzmann

i>2iλⁱ⁻¹µi=1.

Set ν0=1−X

i>2

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Dissections à la Boltzmann

ν0=1−X

i>2

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Dissections à la Boltzmann

ν0=1−X

i>2

ThenP^νn=P^µⁿ andνis a critical probability measure on Z+ withν₁=0.

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Dissections à la Boltzmann: examples

I Uniform p-angulations (p>3). Set µ^(p)₀ =1− 1

p−1, µ^(p)_p−1= 1

p−1, µ^(p)_i =0 (i6=0,i6=p−1).

ThenP^µ

(p)

n is the uniform measure over allp-angulationsofPn.

I Uniform dissections. Set µ0=2−√

2, µ1=0, µi= 2−√ 2 2

!i−1

i>2, then P^µn is the uniform measure on the set of alldissectionsofP_n.

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Dissections à la Boltzmann: examples

p−1, µ^(p)_p−1= 1

p−1, µ^(p)_i =0 (i6=0,i6=p−1).

ThenP^µ

(p)

2, µ1=0, µi= 2−√ 2 2

!i−1

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Dissections à la Boltzmann: examples

p−1, µ^(p)_p−1= 1

p−1, µ^(p)_i =0 (i6=0,i6=p−1).

ThenP^µ

(p)

2, µ1=0, µi= 2−√ 2 2

!i−1

i>2,

then P^µn is the uniform measure on the set of alldissectionsofP_n.

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Dissections à la Boltzmann: examples

p−1, µ^(p)_p−1= 1

p−1, µ^(p)_i =0 (i6=0,i6=p−1).

ThenP^µ

(p)

2, µ1=0, µi= 2−√ 2 2

!i−1

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I. Definition

II. Theorem: scaling limits of random dissections

III. Application IV. Proof

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Scaling limit of random dissections

Letµbe a probability measure over{0, 2, 3, . . .}of mean1 s.t. P

i>0e^λiµi<∞ for some λ >0.

LetD^µn be a randomdissectiondistributed according toP^µⁿ.

What does D^µn look like, fornlarge,as a metric space?

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Scaling limit of random dissections

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Scaling limit of random dissections

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Simulations

Figure :A uniformdissectionofP45.

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Simulations

Figure : A uniformdissectionofP260.

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Simulations

Figure : A uniformdissectionofP387.

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Simulations

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Simulations

Figure :A uniformdissectionofP .

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Scaling limit of random dissections

i>0e^λiµ_i<∞ for some λ >0. LetD^µn be a randomdissectiondistributed according to P^µn.

There exists a constantc(µ)such that:

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

whereTeis a random compact metric space, called theBrownian CRTand the convergence holds in distribution for the Gromov–Hausdorff topology on compact metric spaces.

In addition, c(µ) = 2 σ√

µ₀· 1 4

σ²+ µ0µ2Z+

2µ₂_Z₊−µ₀

,

where µ2Z+ =µ0+µ2+µ4+· · · andσ²∈(0,∞)is the variance ofµ.

Theorem(Curien, Haas & K.).

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Scaling limit of random dissections

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

µ₀· 1 4

σ²+ µ0µ2Z+

2µ₂_Z₊−µ₀

,

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Scaling limit of random dissections

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

whereTeis a random compact metric space, called theBrownian CRT

and the convergence holds in distribution for the Gromov–Hausdorff topology on compact metric spaces.

µ₀· 1 4

σ²+ µ0µ2Z+

2µ₂_Z₊−µ₀

,

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Scaling limit of random dissections

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

µ₀· 1 4

σ²+ µ0µ2Z+

2µ₂_Z₊−µ₀

,

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Scaling limit of random dissections

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

In addition, c(µ) = 2

√ · 1

σ²+ µ0µ2Z+

,

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Scaling limit of random dissections

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

µ₀· 1 4

σ²+ µ0µ2Z+

2µ₂_Z₊−µ₀

,

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What is the Brownian Continuum Random Tree?

First define thecontour functionof a tree:

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What is the Brownian Continuum Random Tree?

Knowing thecontour function, it is easy to recover the tree bygluing:

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What is the Brownian Continuum Random Tree?

The Brownian treeTeis obtained bygluingfrom the Brownian excursione.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

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A simulation of the Brownian CRT

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Recap

√1 n·D^µn

−−−→(d)

n→∞ c(µ)·Te,

In addition, c(µ) = 2

√ · 1

σ²+ µ0µ2Z+

, Theorem(Curien, Haas & K.).

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I. Definition II. Theorem

III. Combinatorial applications

IV. Proof

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Applications

Combinatorial properties of random dissections have been studied by various authors :

I Uniformtriangulations: Devroye, Flajolet, Hurtado, Noy & Steiger (maximal degree,longest diagonal, 1999) and Gao & Wormald (maximal degree, 2000),

I Uniformdissections(andtriangulations) : Bernasconi, Panagiotou & Steger (degrees,maximal degree, 2010) and Drmota, de Mier & Noy (diameter, 2012).

y Whenµi∼c/i^1+α asi→∞, loops remain in the scaling limit, which is thestable looptreeof indexα∈(1, 2)(Curien & K.). Thelooptree of index α=3/2describes the scaling limit of boundaries of critical site percolation on the Uniform Infinite Planar Triangulation (Curien & K.).

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Applications

y Whenµi∼c/i^1+α asi→∞, loops remain in the scaling limit, which is thestable looptreeof indexα∈(1, 2)(Curien & K.).

Thelooptree of index α=3/2describes the scaling limit of boundaries of critical site percolation on the Uniform Infinite Planar Triangulation (Curien & K.).

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Applications

y Whenµi∼c/i^1+α asi→∞, loops remain in the scaling limit, which is thestable looptreeof indexα∈(1, 2)(Curien & K.). Thelooptree of index

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Applications

The diameterDiam(D^µn)is the maximal distance between two points ofD^µn.

We have for everyp >0:

E h

Diam(D^µn)^pi

n→∼∞

c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2

.

In particular, forp=1,E h

Diam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

For uniform dissections, we getE h

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

πn. This strenghtens a result of Drmota, de Mier & Noy who proved that

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞

c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2

.

In particular, forp=1,E

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞

c(µ)^p· Z_∞

0

x^pf_D(x)dx

·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p·

Z_∞

0

x^pf_D(x)dx

·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

where, setting b_k,x= (16πk/x)²,f_D(x)is

√2π 3 times X2⁹

x⁴ 4b⁴_k,x−36b³_k,x+75b²_k,x−30bk,x

+2⁴

x² 4b³_k,x−10b²_k,x

e^−b^k,x.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

where, setting b_k,x= (16πk/x)²,f_D(x)is

√2π 3 times X

k>1

2⁹

x⁴ 4b⁴_k,x−36b³_k,x+75b²_k,x−30bk,x

+2⁴

x² 4b³_k,x−10b²_k,x

e^−b^k,x.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

y Remark: c(µ)is explicit forp-angulationsanduniform dissections.

For uniform dissections, we getE

hDiam(D^µn)i

∼ 1 21(3+

√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

y Remark: c(µ)is explicit forp-angulationsanduniform dissections. For instance, for uniform dissections:

c(µ) = 1 7(3+√

2)2^3/4.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

πn.

This strenghtens a result of Drmota, de Mier & Noy who proved that

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn '0.74√

πn '1.5√

πn.

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Applications

E h

Diam(D^µn)^pi

n→∼∞ c(µ)^p· Z_∞

0

x^pf_D(x)dx·n^p/2.

hDiam(D^µn)i

n→∼∞ c(µ)· 2√ 2π 3 ·√

n.

Corollary.

Diam(D^µn)i

∼ 1 21(3+√

2)2^9/4 √ πn '0.99988√

(3+√ 2)2^1/4 7

√πn6 E h

Diam(D^µn)i

6 2·(3+√ 2)2^1/4 7

√πn

√ √

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I. Definition II. Theorem III. Application

IV. Proof

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Proof

gStep 1. Consider the dual tree ofD^µn:

Figure : A dissection and its dual treeTn^µ.

y Key fact. T^µn is a (planted) Galton–Watson tree with offspring distribution µ conditioned on havingn−1leaves.

It is known (Rizzolo or K.) that:

√1 n·Tn^µ

−→(d) n→∞

2 σ√

µ0

·Te.

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Proof

y Key fact. T^µn is a (planted) Galton–Watson tree with offspring distribution µconditioned on having n−1leaves.

It is known (Rizzolo or K.) that:

√1 n·Tn^µ

−→(d) n→∞

2 σ√

µ0

·Te.

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Proof

y Key fact. T^µn is a (planted) Galton–Watson tree with offspring distribution µconditioned on having n−1leaves. It is known (Rizzolo or K.) that:

√1 n·Tn^µ

−→(d) n→∞

2 σ√

µ0

·Te.

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Proof

gStep 2. We show that:

D^µn ' 1 4

σ²+ µ0µ2Z+

2µ2Z+−µ0

·Tn^µ.

To this end, we compare the length of geodesics inD^µn and inT^µn by using an

“exploration” Markov Chain:

Figure :A geodesic inTn^µ(in light blue) and the associated geodesic inD^µn (in red).

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Proof

gStep 2. We show that:

D^µn ' 1 4

σ²+ µ0µ2Z+

2µ2Z+−µ0

·Tn^µ.

To this end, we compare the length of geodesics inD^µn and inT^µn by using an

“exploration” Markov Chain:

Figure :A geodesic inTn^µ(in light blue) and the associated geodesic inD^µn (in red).

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The Markov Chain

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The Markov Chain

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The Markov Chain

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The Markov Chain

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The Markov Chain

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The Markov Chain

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