On the mixing time of the flip walk on triangulations of the sphere

On the mixing time of the flip walk on triangulations of the sphere

Thomas Budzinski

ENS Paris

Journées de l’ANR GRAAL, Nancy 6 Décembre 2016

Planar maps

Definitions

Aplanar map is a finite, connected graph embedded in the sphere in such a way that no two edges cross (except at a common endpoint), considered up to orientation-preserving homeomorphism.

A planar map is arooted type-I triangulation if all its faces have degree 3 and it has a distinguished oriented edge. It may contain multiple edges and loops.

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Thomas Budzinski Flips on triangulations of the sphere

Planar maps

Definitions

Aplanar map is a finite, connected graph embedded in the sphere in such a way that no two edges cross (except at a common endpoint), considered up to orientation-preserving homeomorphism.

A planar map is arooted type-I triangulation if all its faces have degree 3 and it has a distinguished oriented edge. It may contain multiple edges and loops.

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Random planar maps in a nutshell

LetTn be the set of rooted type-I triangulations of the sphere with n vertices, andTn(∞)be a uniform variable on Tn. What does T_n(∞) look like forn large ?

Exact enumeration results[Tutte], the distances inTn(∞)are of order n^1/4 [Chassaing–Schaeffer],

when the distances are renormalized, Tn(∞) to a continuum random metric space called the Brownian map [Le Gall], if we don’t renormalize the distances and look at a neighbourhood of the root, convergence to an infinite triangulation of the plane called the UIPT [Angel-Schramm], the volume of the ball of radius r in the UIPT grows like r⁴ [Angel, Curien-Le Gall].

Thomas Budzinski Flips on triangulations of the sphere

Random planar maps in a nutshell

LetTn be the set of rooted type-I triangulations of the sphere with n vertices, andTn(∞)be a uniform variable on Tn. What does T_n(∞) look like forn large ?

Exact enumeration results[Tutte],

the distances inTn(∞)are of order n^1/4 [Chassaing–Schaeffer],

when the distances are renormalized, Tn(∞) to a continuum random metric space called the Brownian map [Le Gall], if we don’t renormalize the distances and look at a neighbourhood of the root, convergence to an infinite triangulation of the plane called the UIPT [Angel-Schramm], the volume of the ball of radius r in the UIPT grows like r⁴ [Angel, Curien-Le Gall].

Random planar maps in a nutshell

LetTn be the set of rooted type-I triangulations of the sphere with n vertices, andTn(∞)be a uniform variable on Tn. What does T_n(∞) look like forn large ?

Exact enumeration results[Tutte], the distances inTn(∞)are of order n^1/4 [Chassaing–Schaeffer],

when the distances are renormalized, Tn(∞) to a continuum random metric space called the Brownian map [Le Gall], if we don’t renormalize the distances and look at a neighbourhood of the root, convergence to an infinite triangulation of the plane called the UIPT [Angel-Schramm], the volume of the ball of radius r in the UIPT grows like r⁴ [Angel, Curien-Le Gall].

Thomas Budzinski Flips on triangulations of the sphere

Random planar maps in a nutshell

LetTn be the set of rooted type-I triangulations of the sphere with n vertices, andTn(∞)be a uniform variable on Tn. What does T_n(∞) look like forn large ?

Exact enumeration results[Tutte], the distances inTn(∞)are of order n^1/4 [Chassaing–Schaeffer],

when the distances are renormalized, Tn(∞) to a continuum random metric space called the Brownian map [Le Gall],

if we don’t renormalize the distances and look at a neighbourhood of the root, convergence to an infinite triangulation of the plane called the UIPT [Angel-Schramm], the volume of the ball of radius r in the UIPT grows like r⁴ [Angel, Curien-Le Gall].

Random planar maps in a nutshell

LetTn be the set of rooted type-I triangulations of the sphere with n vertices, andTn(∞)be a uniform variable on Tn. What does T_n(∞) look like forn large ?

Exact enumeration results[Tutte], the distances inTn(∞)are of order n^1/4 [Chassaing–Schaeffer],

when the distances are renormalized, Tn(∞) to a continuum random metric space called the Brownian map [Le Gall], if we don’t renormalize the distances and look at a neighbourhood of the root, convergence to an infinite triangulation of the plane called the UIPT [Angel-Schramm],

the volume of the ball of radius r in the UIPT grows like r⁴ [Angel, Curien-Le Gall].

Thomas Budzinski Flips on triangulations of the sphere

Random planar maps in a nutshell

LetTn be the set of rooted type-I triangulations of the sphere with n vertices, andTn(∞)be a uniform variable on Tn. What does T_n(∞) look like forn large ?

Exact enumeration results[Tutte], the distances inTn(∞)are of order n^1/4 [Chassaing–Schaeffer],

when the distances are renormalized, Tn(∞) to a continuum random metric space called the Brownian map [Le Gall], if we don’t renormalize the distances and look at a neighbourhood of the root, convergence to an infinite triangulation of the plane called the UIPT [Angel-Schramm], the volume of the ball of radius r in the UIPT grows like r⁴

A uniform triangulation of the sphere with 10 000 vertices

Thomas Budzinski Flips on triangulations of the sphere

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁) e₂

???

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

t

flip(t,e₂) =t e1

flip(t,e₁) e₂

???

Thomas Budzinski Flips on triangulations of the sphere

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

t

flip(t,e₂) =t

e1

flip(t,e₁) e₂

???

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁) e₂

???

Thomas Budzinski Flips on triangulations of the sphere

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁)

e₂

???

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

t

flip(t,e₂) =t e1

flip(t,e₁)

e₂

???

Thomas Budzinski Flips on triangulations of the sphere

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁) e₂

???

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

t

flip(t,e₂) =t

e1

flip(t,e₁)

e₂

???

Thomas Budzinski Flips on triangulations of the sphere

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁) e₂

???

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁) e₂

???

Thomas Budzinski Flips on triangulations of the sphere

How to sample a large uniform triangulation ?

"Modern" tools : bijections with trees, peeling process.

Back in the 80’s : Monte Carlo methods : we look for a Markov chain on Tn for which the uniform measure is stationary.

A simple local operation on triangulations : flips.

flip(t,te₂) =t e1

flip(t,e₁) e₂

???

A Markov chain on T

We fixt0 ∈Tnand take Tn(0) =t0.

Conditionally on (Tn(k))_0≤i≤k, let ek be a uniform edge of Tn(k)andTn(k+1) =flip(Tn(k),e_k).

The uniform measure on Tn is reversible forTn, thus stationary.

The chainTnis irreducible (the flip graph is connected [Wagner 36]) and aperiodic (non flippable edges), so it converges to the uniform measure.

Question : how quick is the convergence ?

Thomas Budzinski Flips on triangulations of the sphere

A Markov chain on T

We fixt0 ∈Tnand take Tn(0) =t0.

Conditionally on (Tn(k))_0≤i≤k, let ek be a uniform edge of Tn(k)andTn(k+1) =flip(Tn(k),e_k).

The uniform measure on Tn is reversible forTn, thus stationary.

The chainTnis irreducible (the flip graph is connected [Wagner 36]) and aperiodic (non flippable edges), so it converges to the uniform measure.

Question : how quick is the convergence ?

A Markov chain on T

We fixt0 ∈Tnand take Tn(0) =t0.

Conditionally on (Tn(k))_0≤i≤k, let ek be a uniform edge of Tn(k)andTn(k+1) =flip(Tn(k),e_k).

The uniform measure on Tn is reversible forTn, thus stationary.

The chainTn is irreducible (the flip graph is connected [Wagner 36]) and aperiodic (non flippable edges), so it converges to the uniform measure.

Question : how quick is the convergence ?

Thomas Budzinski Flips on triangulations of the sphere

A Markov chain on T

We fixt0 ∈Tnand take Tn(0) =t0.

Conditionally on (Tn(k))_0≤i≤k, let ek be a uniform edge of Tn(k)andTn(k+1) =flip(Tn(k),e_k).

The uniform measure on Tn is reversible forTn, thus stationary.

The chainTn is irreducible (the flip graph is connected [Wagner 36]) and aperiodic (non flippable edges), so it converges to the uniform measure.

Question : how quick is the convergence ?

Mixing time of T

For n≥3 and 0< ε <1 we define the mixing timet_mix(ε,n) as the smallest k such that

tmax0∈Tn

A⊂maxTn

|P(T_n(k)∈A)−P(T_n(∞)∈A)| ≤ε, where we recall that T_n(∞) is uniform onTn.

Theorem (B., 2016)

For all0< ε <1, there is a constant c >0 such that t_mix(ε,n)≥cn^5/4.

Thomas Budzinski Flips on triangulations of the sphere

Mixing time of T

For n≥3 and 0< ε <1 we define the mixing timet_mix(ε,n) as the smallest k such that

tmax0∈Tn

A⊂maxTn

|P(T_n(k)∈A)−P(T_n(∞)∈A)| ≤ε,

where we recall that T_n(∞) is uniform onTn. Theorem (B., 2016)

For all0< ε <1, there is a constant c >0 such that t_mix(ε,n)≥cn^5/4.

Sketch of proof

We will be interested in the existence of small separating cycles.

Theorem (≈Le Gall–Paulin, 2008)

Let`<sub>n</sub>=o(n<sup>1/4</sup>). Then, with probability going to1 asn →+∞, there is no cycle inTn(∞) of length at most`n that separates T_n(∞) in two parts, each of which contains at least ⁿ₄ vertices. LetT_n¹(0) andT_n²(0) be two independent uniform triangulations of a 1-gon with ⁿ₂ inner vertices each, andTn(0) the gluing ofT_n¹(0) andT_n²(0) along their boundary. It is enough to prove

Proposition

Letkn=o(n^5/4). There is a cycle γ in Tn(kn)of length o(n^1/4) in probability that separatesTn(kn) in two parts, each of which contains at least ⁿ₄ vertices.

Thomas Budzinski Flips on triangulations of the sphere

Sketch of proof

We will be interested in the existence of small separating cycles.

Theorem (≈Le Gall–Paulin, 2008)

Let`<sub>n</sub>=o(n<sup>1/4</sup>). Then, with probability going to1 asn →+∞, there is no cycle inTn(∞) of length at most`n that separates T_n(∞) in two parts, each of which contains at least ⁿ₄ vertices.

LetT_n¹(0) andT_n²(0) be two independent uniform triangulations of a 1-gon with ⁿ₂ inner vertices each, andTn(0) the gluing ofT_n¹(0) andT_n²(0) along their boundary. It is enough to prove

Proposition

Letkn=o(n^5/4). There is a cycle γ in Tn(kn)of length o(n^1/4) in probability that separatesTn(kn) in two parts, each of which contains at least ⁿ₄ vertices.

Sketch of proof

We will be interested in the existence of small separating cycles.

Theorem (≈Le Gall–Paulin, 2008)

Let`<sub>n</sub>=o(n<sup>1/4</sup>). Then, with probability going to1 asn →+∞, there is no cycle inTn(∞) of length at most`n that separates T_n(∞) in two parts, each of which contains at least ⁿ₄ vertices.

LetT_n¹(0) andT_n²(0) be two independent uniform triangulations of a 1-gon with ⁿ₂ inner vertices each, andTn(0) the gluing ofT_n¹(0) andT_n²(0) along their boundary.

It is enough to prove Proposition

Letkn=o(n^5/4). There is a cycle γ in Tn(kn)of length o(n^1/4) in probability that separatesTn(kn) in two parts, each of which contains at least ⁿ₄ vertices.

Thomas Budzinski Flips on triangulations of the sphere

Sketch of proof

We will be interested in the existence of small separating cycles.

Theorem (≈Le Gall–Paulin, 2008)

Let`<sub>n</sub>=o(n<sup>1/4</sup>). Then, with probability going to1 asn →+∞, there is no cycle inTn(∞) of length at most`n that separates T_n(∞) in two parts, each of which contains at least ⁿ₄ vertices.

LetT_n¹(0) andT_n²(0) be two independent uniform triangulations of a 1-gon with ⁿ₂ inner vertices each, andTn(0) the gluing ofT_n¹(0) andT_n²(0) along their boundary. It is enough to prove

Proposition

Letkn=o(n^5/4). There is a cycle γ in Tn(kn)of length o(n^1/4) in

Spatial Markov property

We writeTn,p for the set of triangulations of ap-gon withn inner vertices. IfT is uniform inTn,p, we consider an edge e∈∂T and the facef of T adjacent to e.

fe n−1

with probability

#T^n−1,p+1

#Tn,p

f e

m i n−m

with probability

#T^m,i+1#T^n−m,p−i

#Tn,p

Moreover, in every case, the (one or two) connected components of T\f are independent and uniform given their volume and perimeter.

Thomas Budzinski Flips on triangulations of the sphere

Spatial Markov property

We writeTn,p for the set of triangulations of ap-gon withn inner vertices. IfT is uniform inTn,p, we consider an edge e∈∂T and the facef of T adjacent to e.

fe n−1

with probability

#T^n−1,p+1

#Tn,p

f e

m i n−m

with probability

#Tm,i+1#T^n−m,p−i

#Tn,p

Moreover, in every case, the (one or two) connected components of T\f are independent and uniform given their volume and perimeter.

Spatial Markov property

We writeTn,p for the set of triangulations of ap-gon withn inner vertices. IfT is uniform inTn,p, we consider an edge e∈∂T and the facef of T adjacent to e.

fe n−1

with probability

#T^n−1,p+1

#Tn,p

f e

m i n−m

with probability

#Tm,i+1#T^n−m,p−i

#Tn,p

Moreover, in every case, the (one or two) connected components of T\f are independent and uniform given their volume and perimeter.

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2 τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13

T_n¹(0)

T_n¹(1) T_n¹(2) T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0)

T_n²(1) T_n²(2) T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pe_n(0) =1

Pee_n(1) =1 Pe_n(2) =2 Pe_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3

Ve_n(0) =1

Vee_n(1) =1 Ve_n(2) =2 Ve_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2 τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13

T_n¹(0)

T_n¹(1) T_n¹(2) T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0)

T_n²(1) T_n²(2) T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pe_n(0) =1

Pee_n(1) =1 Pe_n(2) =2 Pe_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3

Ve_n(0) =1

Vee_n(1) =1 Ve_n(2) =2 Ve_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2 τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1)

T_n¹(2) T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0)

T_n²(1)

T_n²(2) T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pe_n(0) =1

Pe_n(1) =1

Pee_n(2) =2 Pe_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Ve_n(0) =1

Ve_n(1) =1

Vee_n(2) =2 Ve_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2 τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1)

T_n¹(2) T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0)

T_n²(1)

T_n²(2) T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pe_n(0) =1

Pe_n(1) =1

Pee_n(2) =2 Pe_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Ve_n(0) =1

Ve_n(1) =1

Vee_n(2) =2 Ve_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2 τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1)

T_n¹(2) T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0)

T_n²(1)

T_n²(2) T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pe_n(0) =1

Pe_n(1) =1

Pee_n(2) =2 Pe_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Ve_n(0) =1

Ve_n(1) =1

Vee_n(2) =2 Ve_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2

τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1)

T_n¹(2)

T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1)

T_n²(2)

T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 P_n(1) =1

Pe_n(2) =2

Pee_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 V_n(1) =1

Ve_n(2) =2

Vee_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2

τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1)

T_n¹(2)

T_n¹(3) T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1)

T_n²(2)

T_n²(3) T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 P_n(1) =1

Pe_n(2) =2

Pee_n(3) =2 Pe_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 V_n(1) =1

Ve_n(2) =2

Vee_n(3) =2 Ve_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2

τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2)

T_n¹(3)

T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2)

T_n²(3)

T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 P_n(2) =2

Pe_n(3) =2

Pee_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 V_n(2) =2

Ve_n(3) =2

Vee_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2

τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2)

T_n¹(3)

T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2)

T_n²(3)

T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 P_n(2) =2

Pe_n(3) =2

Pee_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 V_n(2) =2

Ve_n(3) =2

Vee_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0

τ1 =2

τ2 =4 τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2)

T_n¹(3)

T_n¹(4) T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2)

T_n²(3)

T_n²(4) T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 P_n(2) =2

Pe_n(3) =2

Pee_n(4) =3 Pe_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 V_n(2) =2

Ve_n(3) =2

Vee_n(4) =3 Ve_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0 τ1 =2

τ2 =4

τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2) T_n¹(3)

T_n¹(4)

T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2) T_n²(3)

T_n²(4)

T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 Pe_n(2) =2 P_n(3) =2

Pe_n(4) =3

Pee_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 Ve_n(2) =2 V_n(3) =2

Ve_n(4) =3

Vee_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0 τ1 =2

τ2 =4

τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2) T_n¹(3)

T_n¹(4)

T_n¹(5) T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2) T_n²(3)

T_n²(4)

T_n²(5) T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 Pe_n(2) =2 P_n(3) =2

Pe_n(4) =3

Pee_n(5) =3 Pe_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 Ve_n(2) =2 V_n(3) =2

Ve_n(4) =3

Vee_n(5) =3 Ve_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0 τ1 =2

τ2 =4

τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2) T_n¹(3) T_n¹(4)

T_n¹(5)

T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2) T_n²(3) T_n²(4)

T_n²(5)

T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 Pe_n(2) =2 Pe_n(3) =2 P_n(4) =3

Pe_n(5) =3

Pee_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 Ve_n(2) =2 Ve_n(3) =2 V_n(4) =3

Ve_n(5) =3

Vee_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Exploration of T

(k )

e0

e1

e2

e3

e4

e5

e6

e7

e8

e9 e₁₀ e11

e12

τ0 =0 τ1 =2

τ2 =4

τ3 =6 τ4 =8 τ5 =9 τ6 =12 τ7 =13 T_n¹(0)

T_n¹(1) T_n¹(2) T_n¹(3) T_n¹(4)

T_n¹(5)

T_n¹(6) T_n¹(7) T_n¹(8) T_n¹(9) T_n¹(10) T_n¹(11) T_n¹(12) T_n¹(13)

T_n²(0) T_n²(1) T_n²(2) T_n²(3) T_n²(4)

T_n²(5)

T_n²(6) T_n²(7) T_n²(8) T_n²(9) T_n²(10) T_n²(11) T_n²(12)

T_n²(13)

Pee_n(0) =1 Pe_n(1) =1 Pe_n(2) =2 Pe_n(3) =2 P_n(4) =3

Pe_n(5) =3

Pee_n(6) =4 Pe_n(7) =4 Pe_n(8) =5 P_n(9) =4 Pee_n(10) =4 Pe_n(11) =4 Pe_n(12) =5 P_n(13) =3 Vee_n(0) =1 Ve_n(1) =1 Ve_n(2) =2 Ve_n(3) =2 V_n(4) =3

Ve_n(5) =3

Vee_n(6) =4 Ve_n(7) =4 Ve_n(8) =5 V_n(9) =6 Vee_n(10) =6 Ve_n(11) =6 Ve_n(12) =7 V_n(13) =7

Thomas Budzinski Flips on triangulations of the sphere