triangulations convergence of simplealbenque/slides/albenque_simpletrig.pdf · convergence of...
TRANSCRIPT
Convergence of simpleTriangulations
Journees Cartes, 20th June 2013
Marie Albenque (CNRS, LIX, Ecole Polytechnique)
Louigi Addario-Berry (McGill University Montreal)
A planar map is the embedding of a connected graph in thesphere up to continuous deformations.
Planar Maps – Triangulations.
Triangulation = all faces are triangles.
A planar map is the embedding of a connected graph in thesphere up to continuous deformations.
Planar Maps – Triangulations.
Plane maps are rooted. Face that contains the root = outer face
Triangulation = all faces are triangles.
Distance between two vertices = number of edges between them.Planar map = Metric space
A planar map is the embedding of a connected graph in thesphere up to continuous deformations.
Planar Maps – Triangulations.
Triangulation = all faces are triangles.
Simple map = no loops nor multiple edges
SimpleTriangulation
Model + Motivation
SimpleTriangulation
Euler Formula : v + f = 2 + eTriangulation : 2e = 3f
Mn = {Simple triangulations of size n}= n+ 2 vertices, 2n faces, 3n edges
Mn = Random element of Mn
What is the behavior of Mn when n goes to infinity ?typical distances ? convergence towards a continuous object ?
One motivation : Circle-packing theorem
Mn = Random element of Mn
Model + Motivation
Mn = {Simple triangulations of size n}
What is the behavior of Mn when n goes to infinity ?typical distances ? convergence to a continuous object ?
Each simple triangulation M has a unique (up toMobius transformations and reflections) circlepacking whose tangency graph is M .[Koebe-Andreev-Thurston]
Gives a canonical embedding of simpletriangulations in the sphere and possibly of theirlimit.
Random circle packing
Random circle packing =canonical embedding ofrandom simple triangulation inthe sphere.
Gives a way to define acanonical embedding of theirlimit ?
Team effort : code by Kenneth Stephenson, EricFusy and our own.
Convergence of uniform quadrangulations
[Chassaing, Schaeffer ’04] :
Typical distance is n1/4 + convergence of the profile
Convergence of uniform quadrangulations
[Chassaing, Schaeffer ’04] :
Typical distance is n1/4 + convergence of the profile
[Marckert, Mokkadem ’06] :
1st Def of Brownian map + weak convergence of quadrangulations.
Convergence of uniform quadrangulations
[Chassaing, Schaeffer ’04] :
Typical distance is n1/4 + convergence of the profile
[Marckert, Mokkadem ’06] :
1st Def of Brownian map + weak convergence of quadrangulations.
[Le Gall ’07] :
Hausdorff dimension of the Brownian map is 4.
[Le Gall-Paulin ’08, Miermont ’08] :
Topology of Brownian map = sphere
Convergence of uniform quadrangulations
[Chassaing, Schaeffer ’04] :
Typical distance is n1/4 + convergence of the profile
[Miermont ’12, Le Gall ’12] :
Convergence towards the Brownian map(quadrangulations + 2p-angulations and triangulations)
[Marckert, Mokkadem ’06] :
1st Def of Brownian map + weak convergence of quadrangulations.
[Le Gall ’07] :
Hausdorff dimension of the Brownian map is 4.
[Le Gall-Paulin ’08, Miermont ’08] :
Topology of Brownian map = sphere
Convergence of uniform quadrangulations
[Chassaing, Schaeffer ’04] :
Typical distance is n1/4 + convergence of the profile
The Brownian map is a universal limiting object.All ”reasonable models” of maps (properly rescaled) areexpected to converge towards it.
[Chassaing, Schaeffer ’04] :
[Marckert, Mokkadem ’06] :
1st Def of Brownian map + weak convergence of quadrangulations.
[Miermont ’12, Le Gall ’12] :
Convergence towards the Brownian map(quadrangulations + 2p-angulations and triangulations)
Idea :
Convergence of uniform quadrangulations
[Chassaing, Schaeffer ’04] :
Typical distance is n1/4 + convergence of the profile
The Brownian map is a universal limiting object.All ”reasonable models” of maps (properly rescaled) areexpected to converge towards it.
[Chassaing, Schaeffer ’04] :
[Marckert, Mokkadem ’06] :
1st Def of Brownian map + weak convergence of quadrangulations.
[Miermont ’12, Le Gall ’12] :
Convergence towards the Brownian map(quadrangulations + 2p-angulations and triangulations)
Idea :
Problem : These results relie on nice bijections between maps and labeledtrees [Schaeffer ’98], [Bouttier-Di Francesco-Guitter ’04].
general mapsNOT simple maps
The result
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
The result
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
• same scaling n1/4 as for general maps
The result
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
• same scaling n1/4 as for general maps
• distance between compact spaces.
The result
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
• same scaling n1/4 as for general maps
• distance between compact spaces.
• The Brownian Map
The result
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
• same scaling n1/4 as for general maps
• distance between compact spaces.
• The Brownian Map
Exactly the same kind of result as Le Gall and Miermont’s.
Gromov-Hausdorff distance
Hausdorff distance between X and Y two compact sets of (E, d) :
supx∈X d(x, Y )
supy∈Y d(y,X)
X
Y
dH(X,Y ) = max{supx∈X
d(x, Y ), supy∈Y
d(y,X)}
Gromov-Hausdorff distance
Hausdorff distance between X and Y two compact sets of (E, d) :
supx∈X d(x, Y )
supy∈Y d(y,X)
X
Y
Gromov-Hausdorff distance btw two compact metric spaces E and F :
dH(X,Y ) = max{supx∈X
d(x, Y ), supy∈Y
d(y,X)}
dGH(E,F) = inf dH(φ(E), ψ(F))
φ(X)
ψ(Y )
Infimum taken on : • all the metric spaces M• all the isometric embeddings φ, ψ : E,F →M .
Gromov-Hausdorff distance
Hausdorff distance between X and Y two compact sets of (E, d) :
supx∈X d(x, Y )
supy∈Y d(y,X)
X
Y
Gromov-Hausdorff distance btw two compact metric spaces E and F :
dH(X,Y ) = max{supx∈X
d(x, Y ), supy∈Y
d(y,X)}
dGH(E,F) = inf dH(φ(E), ψ(F))
φ(X)
ψ(Y )
{isometry classes of compact metric spaces with GH distance}= complete and separable (= “Polish” ) space.
The result
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
Idea of proof :• encode the simple triangulations by some trees,• study the limits of trees,• interpret the distance in the maps by some function of the tree.
From blossoming trees to simple triangulations
2-blossoming tree:planted plane tree such that each
vertex carries two leaves
plane tree:plane map that is a tree
rooted plane tree:one corner is distinguished
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
• and repeat !
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
• and repeat !
When finished two vertices have stilltwo leaves and others have one.
Tree balanced = root corner has two leaves
A∗
A
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
• and repeat !
When finished two vertices have stilltwo leaves and others have one.
• label A and A?, the vertices with two leaves ,
Tree balanced = root corner has two leaves
A∗
A
B C
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
• and repeat !
When finished two vertices have stilltwo leaves and others have one.
• label A and A?, the vertices with two leaves ,
• Add two new vertices in the outer face,
Tree balanced = root corner has two leaves
A∗
A
B C
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
• and repeat !
When finished two vertices have stilltwo leaves and others have one.
• label A and A?, the vertices with two leaves ,
• Add two new vertices in the outer face,
• Connect leaves to the vertex on their side,
Tree balanced = root corner has two leaves
A∗
A
B C
From blossoming trees to simple triangulations
Given a planted 2-blossoming tree:
• If a leaf is followed by two internal edges,
• close it to make a triangle.
• and repeat !
When finished two vertices have stilltwo leaves and others have one.
• label A and A?, the vertices with two leaves ,
• Add two new vertices in the outer face,
• Connect leaves to the vertex on their side,
• Connect B and C.
Tree balanced = root corner has two leaves
• out(v) = 3 for v an inner vertexA∗
A
B C
From blossoming trees to simple triangulations
Simple triangulation endowed withits unique orientation such that :
• out(A) = 2, out(B) = 1 andout(C) = 0
• no counterclockwise cycle
• out(v) = 3 for v an inner vertexA∗
A
B C
From blossoming trees to simple triangulations
Simple triangulation endowed withits unique orientation such that :
• out(A) = 2, out(B) = 1 andout(C) = 0
• no counterclockwise cycle
The orientations characterize simpletriangulations [Schnyder]
• out(v) = 3 for v an inner vertexA∗
A
B C
From blossoming trees to simple triangulations
Simple triangulation endowed withits unique orientation such that :
• out(A) = 2, out(B) = 1 andout(C) = 0
• no counterclockwise cycle
Given the orientation the blossomingtree is the leftmost spanning tree ofthe map (after removing B and C).
The orientations characterize simpletriangulations [Schnyder]
A∗
A
B C
From blossoming trees to simple triangulations
Proposition: [Poulalhon, Schaeffer ’07]
The closure operation is a bijectionbetween balanced 2-blossomingtrees and simple triangulations.
2
Same bijection with corner labels
• Start with a planted 2-blossoming tree.• Give the root corner label 2.
2
2
Same bijection with corner labels
• Non-leaf to leaf, label does not change.• Leaf to non-leaf, label increases by 1.• Non-leaf to non-leaf, label decreases by 1.
In contour order, apply the following rules:
• Start with a planted 2-blossoming tree.• Give the root corner label 2.
2
23
Same bijection with corner labels
• Non-leaf to leaf, label does not change.• Leaf to non-leaf, label increases by 1.• Non-leaf to non-leaf, label decreases by 1.
In contour order, apply the following rules:
• Start with a planted 2-blossoming tree.• Give the root corner label 2.
2
2
2
3
Same bijection with corner labels
• Non-leaf to leaf, label does not change.• Leaf to non-leaf, label increases by 1.• Non-leaf to non-leaf, label decreases by 1.
In contour order, apply the following rules:
2
2
22
22
2
22
2
3
3
3
3
3
3
33
3
3
3
3
3
33
33
4
4
4
4 4
44
44
3
5
5
Same bijection with corner labels
• Non-leaf to leaf, label does not change.• Leaf to non-leaf, label increases by 1.• Non-leaf to non-leaf, label decreases by 1.
In contour order, apply the following rules:
4
• Start with a planted 2-blossoming tree.• Give the root corner label 2.
2
2
22
22
2
22
2
3
3
3
3
3
3
33
3
3
3
3
3
33
33
4
4
4
4 4
44
44
3
5
5
Same bijection with corner labels
Aside: Tree is balanced ⇔all labels ≥ 2
+root corner incident to two stemsClosure: Merge each leaf with the firstsubsequent corner with a smaller label.
• Non-leaf to leaf, label does not change.• Leaf to non-leaf, label increases by 1.• Non-leaf to non-leaf, label decreases by 1.
In contour order, apply the following rules:
4
• Start with a planted 2-blossoming tree.• Give the root corner label 2.
C
A
B
2
2
22
22
2
22
2
3
3
3
3
3
3
33
3
3
3
3
3
33
33
4
4
4
4 4
44
44
3
5
5
22
2
2
3
3
21
Same bijection with corner labels
Aside: Tree is balanced ⇔all labels ≥ 2
+root corner incident to two stemsClosure: Merge each leaf with the firstsubsequent corner with a smaller label.
4
C
A
B
2
2
2
2
22
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4 4
4
4
5
5
22
2
2
3
3
11
1
1
222
1
21
Same bijection with corner labels
Aside: Tree is balanced ⇔all labels ≥ 2
+root corner incident to two stemsClosure: Merge each leaf with the firstsubsequent corner with a smaller label.
4
From blossoming trees to labeled trees
2
2
22
22
2
22
2
3
3
3
3
3
3
3
3 3
3
3
3
3
33
33
4
4
4
44
44
44
3
5
5
4
2
2
2
22
3
3label of a vertex =minimum label of its corners
In the following:Labels gives approximatedistances to the root in the map
From blossoming trees to labeled trees
2
2
22
22
2
22
2
3
3
3
3
3
3
3
3 3
3
3
3
3
33
33
4
4
4
44
44
44
3
5
5
4
2
2
2
22
3
3
0
+1
−1
0−1
−1
Generic vertex :
i+1
ii−1
i−1
i i
i+1i+1i−1
From blossoming trees to labeled trees
2
2
2
22
3
3
0
+1
−1
0−1
−1
Generic vertex :
i+1
ii−1
i−1
i i
i+1i+1i−1
• Can retrieve the blossoming treefrom the labeled tree.
• Labeled tree = GW trees +random displacements on edges uniform on
{(−1,−1, . . . ,−1, 0, 0, . . . , 0, 1, 1 . . . , 1)}.
almost the setting of [Janson-Marckert] and [Marckert-Miermont] butr.v are not ”locally centered” ⇒ symmetrization required
Convergence of labeled trees
Contour and label processes of a labeled tree
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
T CTn (or Cn) = contour process
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
If T is a labeled tree, (Cn(i), Zn(i)) = contour and label processes
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
If T is a labeled tree, (Cn(i), Zn(i)) = contour and label processes
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Convergence of labeled trees
Contour and label processes of a labeled tree
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
If T is a labeled tree, (Cn(i), Zn(i)) = contour and label processes
Theorem : [Addario-Berry, A.]For a sequence of simple random triangulations (Mn), the contourand label process of the associated labeled tree satisfie:(
(3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0 1
Brownian snake (et, Zt)0≤t≤1
1st step : the Brownian tree [Aldous]
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
0 1
Brownian snake (et, Zt)0≤t≤1
1st step : the Brownian tree [Aldous]
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
0 1
(et)0≤t≤1= Brownian excursion
0 1
Brownian snake (et, Zt)0≤t≤1
1st step : the Brownian tree [Aldous]
i jT CTn (or Cn) = contour process
i and j = same vertex of T
Cn(i) = Cn(j) = mini≤k≤j
Cn(k)iff
0 1
(et)0≤t≤1= Brownian excursionTe
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
u v
u
ρ = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Brownian snake (et, Zt)0≤t≤1
1st step : the Brownian tree [Aldous]
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Conditional on Te, Z a centered Gaussian process with Zρ = 0 andE[(Zs − Zt)2] = de(s, t)
Z ∼ Brownian motion on the tree
Brownian snake (et, Zt)0≤t≤1
2nd step : Brownian labels
1st step : the Brownian tree [Aldous]
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Conditional on Te, Z a centered Gaussian process with Zρ = 0 andE[(Zs − Zt)2] = de(s, t)
Z ∼ Brownian motion on the tree
Brownian snake (et, Zt)0≤t≤1
2nd step : Brownian labels
1st step : the Brownian tree [Aldous]
Theorem : [Addario-Berry, A.]((3n)−1/2Cbntc, (4n/3)−1/4Zbntc
)0≤t≤1
(d)→n→∞
(et, Zt)0≤t≤1,
0
Idea of proof :
Start with one of “our” tree and apply a random permutation at each vertex
0−1 0
−1
−1
1
0
1
0 0
0
1
−1
−1
01
0
−10
1
1
0 0
0
Idea of proof :
Start with one of “our” tree and apply a random permutation at each vertex
0−1 0
−1
−1
1
0
1
0 0
0
1
−1
−1
01
0
−10
1
1
0 0New tree satisfies the assumptions of[Marckert-Miermont]⇒ convergence result known
But modification tooimportant to derive someproperties of first model.
Idea of proof :
Start with one of “our” tree and apply a random permutation at each vertex
0−1 0
−1
−1
1
0
1
0 0
0
1
New tree satisfies the assumptions of[Marckert-Miermont]⇒ convergence result known
But modification tooimportant to derive someproperties of first model.
0−1 0
−101
−10
1
1
0 0
Gives a coupling between “our” model and the fully permuted model:sufficient control to prove convergence for the true model.
Solution: • consider subtree T 〈k〉 spanned by k random vertices• permute displacements and edges only outside 〈T 〉.• permute only displacements on 〈T 〉.
Distances in simple triangulations
Theorem : [Addario-Berry, A.]Mn= random simple triangulation, then for all ε > 0:
P(
sup0≤t≤1
{∣∣∣Zbntc − Zbntc∣∣∣} ≥ εn1/4)→ 0.
i.e. the label process of the tree gives the distance to theroot in the map.
Mn = simple triangulation
(Cbntc, Zbntc) = contour and label process of the associated tree
Zbntc = distance in the map between vertex ”bntc” and the root.
Distances in simple triangulations
First observation : In the tree, the labels of two adjacent verticesdiffer by at most 1. What can go wrong with closures ?
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Distances in simple triangulations
First observation : In the tree, the labels of two adjacent verticesdiffer by at most 1. What can go wrong with closures ?
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
i−1
i−1
i−1
i−2
i−3i−3
i+1
ii
i
i+1
i+1
i+2
i+2i+2
i
Distances in simple triangulations
First observation : In the tree, the labels of two adjacent verticesdiffer by at most 1. What can go wrong with closures ?
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
i−1
i−1
i−1
i−2
i−3i−3
i+1
ii
i
i+1
i+1
i+2
i+2i+2
i
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
u
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
• LMP are not self-intersecting⇒ they reach the outer face u
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
• LMP are not self-intersecting⇒ they reach the outer face
• On the left of a LMP, corner labelsdecrease exactly by one.
i+1
ii
i
i+1i+1
i+2
i+2i+2
i+1
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
• LMP are not self-intersecting⇒ they reach the outer face
• On the left of a LMP, corner labelsdecrease exactly by one.
i+1
ii
i
i+1i+1
i+2
i+2i+2
i+ 1
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
• LMP are not self-intersecting⇒ they reach the outer face
• On the left of a LMP, corner labelsdecrease exactly by one.
i+1
ii
i
i+1i+1
i+2
i+2i+2
i+ 1 i
i−1
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
• LMP are not self-intersecting⇒ they reach the outer face
• On the left of a LMP, corner labelsdecrease exactly by one.
i+ 3
i+1
ii
i
i+1i+1
i+2
i+2i+2
Distances in simple triangulations
Claim 1: 3dMn(root, ui) ≥ Ln(ui)
Claim 2 : dMn(root, ui) ≤ Ln(ui)
• Consider the Left Most Path from (u, v)to the root face.
• For each inner vertex : 3 LMP
• LMP are not self-intersecting⇒ they reach the outer face
• On the left of a LMP, corner labelsdecrease exactly by one.
i+ 3
i+1
ii
i
i+1i+1
i+2
i+2i+2i+2
i+1
LMP are almost geodesic
u
w
Tq
lq
Leftmost path
Another path: can it be shorter ?
lp
LMP are almost geodesic
Euler Formula :|E(Tq)| = 3|V (Tq)|−3− (`p+`q)
3-orientation + LMP :|E(Tq)| ≥ 3|V (Tq)| − 2`q − 2
=⇒ `q ≥ `p + 1u
w
Tq
lq
Leftmost path
Another path: can it be shorter ?
lp
LMP are almost geodesic
Leftmost path
Another path: can it be shorter ?
A
u
`p`q
`q ≥ `p + 1
LMP are almost geodesic
Leftmost path
Another path: can it be shorter ?
A
u
`p
`q ≥ `p
A
u
`p`q
`q ≥ `p + 3
A
u
`p`q
`q ≥ `p − 2
A
u
`p`q
`q ≥ `p + 1
YES
LMP are almost geodesic
YES ... but not too oftenLeftmost path
Another path: can it be shorter ?
A Bad configuration =too many windings around the LMP
But w.h.p a winding cannot be too short.
=⇒ w.h.p the number of windings is o(n1/4).
LMP are almost geodesic
Proposition:
For ε > 0, let An,ε be the event that there existsu ∈Mn such that Ln(u) ≥ dMn(u, root) + εn1/4.Then under the uniform law on Mn, for all ε > 0:
P (An,ε)→ 0.
YES ... but not too oftenLeftmost path
Another path: can it be shorter ?
A Bad configuration =too many windings around the LMP
But w.h.p a winding cannot be too short.
=⇒ w.h.p the number of windings is o(n1/4).
Distances are tight
u v
Lu Lv
Distances are tight
u v
Lu Lv
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Lu−2
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Lu−2
Lu−3
Modified LMP:at each step, we take the first edgein the tree
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Lu−2
Lu−3
Modified LMP:at each step, we take the first edgein the tree
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Lu,v−1
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Lu,v−1
Distances are tight
u v
Lu Lv
Lu−1
Lu−2
Lu,v
Lu,v = min{Ls, u ≤ s ≤ v}
Lu,v−1
Blue path = path of length Lu + Lv − 2Lu,v + 2
Since (n−1/4Zbntc) converges ⇒ (dn) tight
The result for the last time
Theorem : [Addario-Berry, A.](Mn) = sequence of random simple triangulations, then:(
Mn,
(3
4n
)1/4
dMn
)(d)−−→ (M,D?),
for the distance of Gromov-Hausdorff on the isometry classes ofcompact metric spaces.
The Brownian Map ??
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Conditional on Te, Z a centered Gaussian process with Zρ = 0 andE[(Zs − Zt)2] = de(s, t) Z ∼ Brownian motion on the tree
The Brownian map
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Conditional on Te, Z a centered Gaussian process with Zρ = 0 andE[(Zs − Zt)2] = de(s, t) Z ∼ Brownian motion on the tree
D◦(s, t) = Zs + Zt − 2 max(
infs≤u≤t
Zu, inft≤u≤s
Zu
), s, t ∈ [0, 1] .
The Brownian map
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Conditional on Te, Z a centered Gaussian process with Zρ = 0 andE[(Zs − Zt)2] = de(s, t) Z ∼ Brownian motion on the tree
D◦(s, t) = Zs + Zt − 2 max(
infs≤u≤t
Zu, inft≤u≤s
Zu
), s, t ∈ [0, 1] .
D∗(a, b) = inf{ k−1∑i=1
D◦(ai, ai+1) : k ≥ 1, a = a1, a2, . . . , ak−1, ak = b},
The Brownian map
0 1u v
u
Te = [0, 1]/ ∼eu ∼e v iff de(u, v) = 0
de(u, v) = eu + ev − 2 minu≤s≤v es
Conditional on Te, Z a centered Gaussian process with Zρ = 0 andE[(Zs − Zt)2] = de(s, t) Z ∼ Brownian motion on the tree
D◦(s, t) = Zs + Zt − 2 max(
infs≤u≤t
Zu, inft≤u≤s
Zu
), s, t ∈ [0, 1] .
D∗(a, b) = inf{ k−1∑i=1
D◦(ai, ai+1) : k ≥ 1, a = a1, a2, . . . , ak−1, ak = b},
The Brownian map
Then M = (Te/ ∼D? , D∗) is the Brownian map.
Perspectives
Same approach works also for simple quadrangulations.
Can it be generalized to other families of maps ?
• Generic bijection between blossoming trees and maps [Bernardi, Fusy][A.,Poulalhon].Can we say something about distances ?
Can we say something about the embedding of the Brownian map in thesphere via circle packing ?
• Convergence of Hurwitz maps: bijection also with blossoming trees[Duchi, Poulalhon, Schaeffer].
Perspectives
Same approach works also for simple quadrangulations.
Can it be generalized to other families of maps ?
• Generic bijection between blossoming trees and maps [Bernardi, Fusy][A.,Poulalhon].Can we say something about distances ?
Can we say something about the embedding of the Brownian map in thesphere via circle packing ?
Thank you !
• Convergence of Hurwitz maps: bijection also with blossoming trees[Duchi, Poulalhon, Schaeffer].