Arbres, cartes et nombres de Hurwitz CNRS & Gilles Schaeffer - - PowerPoint PPT Presentation

Arbres, cartes et nombres de Hurwitz

Gilles Schaeffer

CNRS & ´ Ecole Polytechnique

ERC Research Starting Grant 208471 ”ExploreMaps” Colloquium du LAREMA, Angers, juin 2013

Plan de l’expos´ e Revˆ etements ramifi´ es et cartes Cartes et arbres ´ Enum´ eration d’arbres et formule d’Hurwitz Revˆ etements et cartes al´ eatoires

Plan de l’expos´ e

Plan de l’expos´ e

Revˆ etements ramifi´ es et cartes Cartes et arbres ´ Enum´ eration d’arbres et formule d’Hurwitz Revˆ etements et cartes al´ eatoires

Ramified coverings of the sphere by itself

A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. By continuity, the number n = |φ−1(x)| of sheets of a covering φ does not depend on x: for instance n = k for φk. φ3 A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. By continuity, the number n = |φ−1(x)| of sheets of a covering φ does not depend on x: for instance n = k for φk. φ3 The number n of sheets is called the degree of the covering. A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Let Ar be the annulus {z | r < |z| < 1} ⊂ C. Consider φk : Ar → Ark with φk(z) = zk. By continuity, the number n = |φ−1(x)| of sheets of a covering φ does not depend on x: for instance n = k for φk. φ3 What is we try to extend from Ar to B? The number n of sheets is called the degree of the covering. A mapping φ : D → I is a covering if, for all x in I there exists n ≥ 1 and a neighborhood V of x such that φ−1(V ) ∼ B × {1, . . . , n}, and the restriction of φ to each sheet Bi (connected component of the preimage) is an homeomorphism φ|Bi : Bi

∼

→ B. Example: x D I Let B = {z | |z| < 1} ⊂ C and let ∼ denote equivalence up to homeomorphisms

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? The mapping φk : B∗ → B∗ is a covering, but not φk : B → B.

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? The mapping φk : B∗ → B∗ is a covering, What happens at x = 0? but not φk : B → B.

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? The mapping φk : B∗ → B∗ is a covering, What happens at x = 0? The mapping φk : B → B has a connected ramification of degree k at x = 0. but not φk : B → B.

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? A mapping φ is ramified at x = 0 if The mapping φk : B∗ → B∗ is a covering, What happens at x = 0? The mapping φk : B → B has a connected ramification of degree k at x = 0.

• the restriction of φ to each component of φ−1(V ) is

homeomorphic to φk for some k.

• there is a neighborhood V of the origin such that

φ−1(V ) ∼ B × [1, . . . , p] and, but not φk : B → B.

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? A mapping φ is ramified at x = 0 if The mapping φk : B∗ → B∗ is a covering, What happens at x = 0? The mapping φk : B → B has a connected ramification of degree k at x = 0.

• the restriction of φ to each component of φ−1(V ) is

homeomorphic to φk for some k.

• there is a neighborhood V of the origin such that

φ−1(V ) ∼ B × [1, . . . , p] and, but not φk : B → B.

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? A mapping φ is ramified at x = 0 if The mapping φk : B∗ → B∗ is a covering, What happens at x = 0? The mapping φk : B → B has a connected ramification of degree k at x = 0.

• the restriction of φ to each component of φ−1(V ) is

homeomorphic to φk for some k.

• there is a neighborhood V of the origin such that

φ−1(V ) ∼ B × [1, . . . , p] and, Regular (aka unramified) value = ramified with φ1 on each component. but not φk : B → B.

Ramified coverings of the sphere by itself

Recall φk : Ar → Ark with φk(z) = zk. φ3 Extend from Ar to B? A mapping φ is ramified at x = 0 if The mapping φk : B∗ → B∗ is a covering, What happens at x = 0? The mapping φk : B → B has a connected ramification of degree k at x = 0.

• the restriction of φ to each component of φ−1(V ) is

homeomorphic to φk for some k.

• there is a neighborhood V of the origin such that

φ−1(V ) ∼ B × [1, . . . , p] and, Regular (aka unramified) value = ramified with φ1 on each component. ramified regular but not φk : B → B.

Ramified coverings of the sphere by itself (Cont’d)

A mapping φ is a ramified covering of S by S if there exists a finite subset X = {x1, . . . , xp} such that:

regular value critical value critical value

D = S I = S

• φS\φ−1(X) is a covering, and
• φ is ramified over each xi

Ramified coverings of the sphere by itself (Cont’d)

A mapping φ is a ramified covering of S by S if there exists a finite subset X = {x1, . . . , xp} such that:

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 regular value critical value critical value

D = S I = S

• φS\φ−1(X) is a covering, and
• φ is ramified over each xi

On each component Vj of φ−1(V (xi)), φ ∼ φλ(i)

j

for some integer λ(i)

j .

Ramified coverings of the sphere by itself (Cont’d)

A mapping φ is a ramified covering of S by S if there exists a finite subset X = {x1, . . . , xp} such that:

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 regular value critical value critical value

D = S I = S

• φS\φ−1(X) is a covering, and
• φ is ramified over each xi

generically n sheets id id id id id φ2 φ2 φ3 φ2 id

On each component Vj of φ−1(V (xi)), φ ∼ φλ(i)

j

for some integer λ(i)

j .

Ramified coverings of the sphere by itself (Cont’d)

A mapping φ is a ramified covering of S by S if there exists a finite subset X = {x1, . . . , xp} such that:

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

D = S I = S

• φS\φ−1(X) is a covering, and
• φ is ramified over each xi

generically n sheets id id id id id φ2 φ2 φ3 φ2 id

The ramification type over a critical value xi is the partition λ(i) On each component Vj of φ−1(V (xi)), φ ∼ φλ(i)

j

for some integer λ(i)

j .

The passport of a ramified covering is the list Λ = (λ(1), . . . , λ(p))

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value φ2 φ2 φ3 φ2 id id id id id id generically n sheets

D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value φ2 φ2 φ3 φ2 id id id id id id generically n sheets

To understand the ”shape” of the covering, draw paths on I and study its preimages. D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages. D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points

• a contractible loop on I

D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points

• a contractible loop on I

yields n contractible loops on D D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points

• a contractible loop on I

yields n contractible loops on D D = S I = S

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points

• a contractible loop on I

yields n contractible loops on D D = S I = S but if we wind around critical points

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points some sheets may get permuted

• a contractible loop on I

yields n contractible loops on D D = S I = S but if we wind around critical points

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points some sheets may get permuted

• visiting critical points create

multiple values or ”vertices”

• a contractible loop on I

yields n contractible loops on D D = S I = S but if we wind around critical points

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points some sheets may get permuted

• visiting critical points create

multiple values or ”vertices”

• a contractible loop on I

yields n contractible loops on D D = S I = S but if we wind around critical points

Ramified coverings of the sphere by itself (Cont’d)

λ(1) = 15 λ(2) = 1, 22 λ(2) = 2, 3 the passport Λ = (λ(1), . . . , λ(p)) of a ramified covering regular value critical value critical value

To understand the ”shape” of the covering, draw paths on I and study its preimages.

• n independant preimages as long

as we stay away from critical points some sheets may get permuted

• visiting critical points create

multiple values or ”vertices” ⇒ The partitions λ(i) are partitions of n, degree of the covering.

• a contractible loop on I

yields n contractible loops on D D = S I = S but if we wind around critical points

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation

1 2 3 4 5

Let us label {1, . . . , n} the preimages of a regular point. D = S I = S

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation

1 2 3 4 5

The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation Contractible loop in S \ X ⇒ identity permutation

1 2 3 4 5

The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation Contractible loop in S \ X ⇒ identity permutation Concatenation of two loops ⇒ product of the permutations

1 2 3 4 5

Example: (1)(2, 3, 4, 5) · (1, 2)(3, 4)(5) The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation Contractible loop in S \ X ⇒ identity permutation Concatenation of two loops ⇒ product of the permutations

1 2 3 4 5

Example: (1)(2, 3, 4, 5) · (1, 2)(3, 4)(5) The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation Contractible loop in S \ X ⇒ identity permutation Concatenation of two loops ⇒ product of the permutations

1 2 3 4 5

Example: (1)(2, 3, 4, 5) · (1, 2)(3, 4)(5) The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation Contractible loop in S \ X ⇒ identity permutation Concatenation of two loops ⇒ product of the permutations

1 2 3 4 5

Example: (1)(2, 3, 4, 5) · (1, 2)(3, 4)(5) The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S ⇒ Equivalence classes of ramified coverings ≡ factorizations of permutations

Monodromy, and permutations

Loop ⇒ permutation of sheet labels Example: (1, 2)(3, 4)(5) in cyclic notation Contractible loop in S \ X ⇒ identity permutation Concatenation of two loops ⇒ product of the permutations

1 2 3 4 5

Example: (1)(2, 3, 4, 5) · (1, 2)(3, 4)(5) The permutation is invariant under continuous deformation of the loop provided it stays in S \ {X} Let us label {1, . . . , n} the preimages of a regular point. D = S I = S but geometric intuition is lost ⇒ Equivalence classes of ramified coverings ≡ factorizations of permutations

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322

D = S I = S

1 2 1 2 1 1 2 2 1 λ = 62 1

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

On I, draw an edge between • and ◦ via the basepoint

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

On I, draw an edge between • and ◦ via the basepoint We get a planar map: that is, a graph embedded on the sphere with simply connected faces

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

On I, draw an edge between • and ◦ via the basepoint We get a planar map: that is, a graph embedded on the sphere with simply connected faces

• Proof. Faces are simply connected

because a loop around the edge in I can be deformed to a loop around

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

On I, draw an edge between • and ◦ via the basepoint We get a planar map: that is, a graph embedded on the sphere with simply connected faces

• Proof. Faces are simply connected

because a loop around the edge in I can be deformed to a loop around

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

On I, draw an edge between • and ◦ via the basepoint We get a planar map: that is, a graph embedded on the sphere with simply connected faces

• Proof. Faces are simply connected

because a loop around the edge in I can be deformed to a loop around

coverings with 3 critical values and bipartite maps

3 critical values

λ• = 2312 λ◦ = 322 1 3 4 5 2 6 7

D = S I = S

1 2 1 2 1 1 2

1 regular value with labeled preimages

2 1 λ = 62 1 8

On I, draw an edge between • and ◦ via the basepoint We get a planar map: that is, a graph embedded on the sphere with simply connected faces

• Proof. Faces are simply connected

because a loop around the edge in I can be deformed to a loop around

• Proposition. This is a bijection

between bipartite planar maps and ramified coverings of S by S with 3 critical values.

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ σ• = (1)(2, 6)(3, 5)(4, 7)(8) with cyclic type λ• Cycle types ⇔ degree distributions 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ σ• = (1)(2, 6)(3, 5)(4, 7)(8) with cyclic type λ• Cycle types ⇔ degree distributions 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ σ• = (1)(2, 6)(3, 5)(4, 7)(8) with cyclic type λ• Cycle types ⇔ degree distributions What about σ and λ ? 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ σ• = (1)(2, 6)(3, 5)(4, 7)(8) with cyclic type λ• Cycle types ⇔ degree distributions What about σ and λ ? σ = (2, 3)(1, 5, 7, 8, 4, 6) loops around = faces 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ σ• = (1)(2, 6)(3, 5)(4, 7)(8) with cyclic type λ• Cycle types ⇔ degree distributions What about σ and λ ? But loop around = concatenate loop around ◦ and • σ = (2, 3)(1, 5, 7, 8, 4, 6) loops around = faces 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

3 critical values, bipartite maps and permutations

1 3 4 5 2 6 7 1 2 1 2 1 1 2 2 1 1 8

A loop around a critical value yields a permutation σ◦ = (1, 3, 6)(2, 5, 4)(7, 8) with cyclic type λ◦ σ• = (1)(2, 6)(3, 5)(4, 7)(8) with cyclic type λ• Cycle types ⇔ degree distributions What about σ and λ ? But loop around = concatenate loop around ◦ and • σ = (2, 3)(1, 5, 7, 8, 4, 6) loops around = faces Proposition: σ◦σ• = σ. 3 critical values

λ• = 2312 λ◦ = 322

D = S I = S 1 regular value with labeled preimages

λ = 62

m + 1 critical values, m-constellations, permutations

1 3 4 2 1 3 2 1 2 3 1 3 2 1 2 4 4 4 4 4

m + 1 critical values 1 regular value with labeled preimages

1 2 3 4

m + 1 critical values, m-constellations, permutations

1 3 4 2 1 3 2 1 2 3 1 3 2 1 2

The preimage of the m-star is called a star-constellation.

• Proposition. Planar star-constellations

with: – n labelled m-stars, – λ

j faces of degree j,

– λ(i)

j

color i vertices of degree j are in bijection with minimal transitive factorizations σ1 · · · σm = σ with σi of cyclic type λ(i).

4 4 4 4 4

m + 1 critical values 1 regular value with labeled preimages

1 2 3 4

Monodromy, permutations, constellations: a summary

• Theorem. There is a bijection between
• Labelled ramified covering of S of type Λ = (λ0, . . . , λm)
• Factorizations (σ1 · · · σm = σ0) of type Λ
• labelled m-star-constellations of type Λ.

D = S ⇔ minimality ⇔ planarity.

Monodromy, permutations, constellations: a summary

• Theorem. There is a bijection between
• Labelled ramified covering of S of type Λ = (λ0, . . . , λm)
• Factorizations (σ1 · · · σm = σ0) of type Λ
• labelled m-star-constellations of type Λ.

Specializations. — m = 2: bipartite maps with n edges D = S ⇔ minimality ⇔ planarity. — m = 2, λ0 = 4n, all faces have degree 4: quadrangulations ⇒ Jean-Fran¸ cois Le Gall’s last year talk at this seminar

Monodromy, permutations, constellations: a summary

• Theorem. There is a bijection between
• Labelled ramified covering of S of type Λ = (λ0, . . . , λm)
• Factorizations (σ1 · · · σm = σ0) of type Λ
• labelled m-star-constellations of type Λ.

Specializations. — m = 2: bipartite maps with n edges D = S ⇔ minimality ⇔ planarity. — m = 2, λ0 = 4n, all faces have degree 4: quadrangulations — for all i ≥ 1, λ(i) = 21n−2: factorizations in transpositions. coverings with only simple branch points ⇒ Jean-Fran¸ cois Le Gall’s last year talk at this seminar

Monodromy, permutations, constellations: a summary

• Theorem. There is a bijection between
• Labelled ramified covering of S of type Λ = (λ0, . . . , λm)
• Factorizations (σ1 · · · σm = σ0) of type Λ
• labelled m-star-constellations of type Λ.

Specializations. — m = 2: bipartite maps with n edges D = S ⇔ minimality ⇔ planarity. — m = 2, λ0 = 4n, all faces have degree 4: quadrangulations — for all i ≥ 1, λ(i) = 21n−2: factorizations in transpositions. coverings with only simple branch points ⇒ Jean-Fran¸ cois Le Gall’s last year talk at this seminar

Today’s topic

Simple ramified covers, increasing quadrangulations

1 3 2 1 2 3 4

Then each face of degree 2 on the image has n − 2 preimages that are faces of degree 2, and 1 that is a quadrangle. A ramified cover is simple if its m ramifications have type 21n−2.

6 5 4 5 6 1 3 4 2

Simple ramified covers, increasing quadrangulations

1 3 2 1 2 3 4

Then each face of degree 2 on the image has n − 2 preimages that are faces of degree 2, and 1 that is a quadrangle. A ramified cover is simple if its m ramifications have type 21n−2.

6 5 4 5 6 1 3 4 2

Upon contracting multiple edges,

• nly quadrangle remains.

Simple ramified covers, increasing quadrangulations

1 3 2 1 2 3 4

Then the faces of the preimage have distinct labels 1, . . . , m that are increasing in ccw direction around black vertices and in cw direction around white vertices. Then each face of degree 2 on the image has n − 2 preimages that are faces of degree 2, and 1 that is a quadrangle. A ramified cover is simple if its m ramifications have type 21n−2.

6 5 4 5 6 1 3 4 2

Upon contracting multiple edges,

• nly quadrangle remains.

Simple ramified covers, increasing quadrangulations

1 3 2 1 2 3 4

Then the faces of the preimage have distinct labels 1, . . . , m that are increasing in ccw direction around black vertices and in cw direction around white vertices. Such a map is called an increasing labelled quadrangulation. Then each face of degree 2 on the image has n − 2 preimages that are faces of degree 2, and 1 that is a quadrangle. A ramified cover is simple if its m ramifications have type 21n−2.

6 5 4 5 6 1 3 4 2

Upon contracting multiple edges,

• nly quadrangle remains.

Simple ramified covers, increasing quadrangulations

1 3 2 1 2 3

• Theorem. Simple ramified covers of S by itself with m ramifications points

are in bijection with increasing labelled quadrangulations with m faces.

4

Then the faces of the preimage have distinct labels 1, . . . , m that are increasing in ccw direction around black vertices and in cw direction around white vertices. Such a map is called an increasing labelled quadrangulation. Then each face of degree 2 on the image has n − 2 preimages that are faces of degree 2, and 1 that is a quadrangle. A ramified cover is simple if its m ramifications have type 21n−2.

6 5 4 5 6 1 3 4 2

Upon contracting multiple edges,

• nly quadrangle remains.

R´ esum´ e du 1er ´ episode

Compter des classes d’´ equivalence de revˆ etements ramifi´ es compter certaines plongements de graphes ⇔

Plan de l’expos´ e Revˆ etements ramifi´ es et cartes Cartes et arbres ´ Enum´ eration d’arbres et formule d’Hurwitz Revˆ etements et cartes al´ eatoires

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces. A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces. A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. The dual map of a map is the map of incidence between faces. A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. The dual map of a map is the map of incidence between faces. Proof ? A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. The dual map of a map is the map of incidence between faces. Euler’s relation: (#vertices-1)+(#faces-1) = #edges Proof ? A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Planar maps, spanning trees and duality

From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. The dual map of a map is the map of incidence between faces. Euler’s relation: (#vertices-1)+(#faces-1) = #edges Proof ? Proof? A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms).

Encoding and counting tree-rooted maps

Starting at a root corner, turn around the tree

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Starting at a root corner, turn around the tree uduuduuddd

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Non visited edges ≡ balanced parenthesis word Starting at a root corner, turn around the tree uduuduuddd

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Non visited edges ≡ balanced parenthesis word Starting at a root corner, turn around the tree uduuduuddd uuuduuddddud

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Non visited edges ≡ balanced parenthesis word Code of the tree-rooted map = tree decorated by a balanced parenthesis word Starting at a root corner, turn around the tree uduuduuddd uuuduuddddud

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Writing the two codes during the walk: Non visited edges ≡ balanced parenthesis word Code of the tree-rooted map = tree decorated by a balanced parenthesis word uuuududuuudududddddudd Starting at a root corner, turn around the tree uduuduuddd uuuduuddddud

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Writing the two codes during the walk: Non visited edges ≡ balanced parenthesis word Code of the tree-rooted map = tree decorated by a balanced parenthesis word = shuffle of two balanced parenthesis words uuuududuuudududddddudd Starting at a root corner, turn around the tree uduuduuddd uuuduuddddud

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Writing the two codes during the walk: Non visited edges ≡ balanced parenthesis word Code of the tree-rooted map = tree decorated by a balanced parenthesis word = shuffle of two balanced parenthesis words The number of tree rooted planar maps with n edges is Pn

i=0

`2n

i

´ CiCn−i where Cn =

1 n+1

`2n

n

´ denotes Catalan numbers, counting balanced parenthesis words. uuuududuuudududddddudd Starting at a root corner, turn around the tree uduuduuddd uuuduuddddud

Encoding and counting tree-rooted maps

Rooted tree ≡ balanced parenthesis word Writing the two codes during the walk: Non visited edges ≡ balanced parenthesis word Code of the tree-rooted map = tree decorated by a balanced parenthesis word = shuffle of two balanced parenthesis words The number of tree rooted planar maps with n edges is Pn

i=0

`2n

i

´ CiCn−i where Cn =

1 n+1

`2n

n

´ denotes Catalan numbers, counting balanced parenthesis words. uuuududuuudududddddudd Starting at a root corner, turn around the tree

Observe that closure edges turn clockwise around the tree.

uduuduuddd uuuduuddddud

but we want rooted (not tree-rooted) maps

Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree

but we want rooted (not tree-rooted) maps

Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree Then write the code of the primal tree on the chosen canonical tree

but we want rooted (not tree-rooted) maps

Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree Then write the code of the primal tree on the chosen canonical tree The map is recovered from the code by closure.

but we want rooted (not tree-rooted) maps

Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree Then write the code of the primal tree on the chosen canonical tree Our code of the map will be a canonical decorated tree Question is How do we choose the canonical spanning tree The map is recovered from the code by closure. so that the resulting decorated trees can be described and counted ?

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root Orient the other edges couterclockwise around the tree

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root Orient the other edges couterclockwise around the tree The resulting orientation has no clockwise circuit.

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root Orient the other edges couterclockwise around the tree It is called a minimal orientation (for the order induced by circuit reversal). The resulting orientation has no clockwise circuit.

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root Orient the other edges couterclockwise around the tree It is called a minimal orientation (for the order induced by circuit reversal). The resulting orientation has no clockwise circuit. A oriented map is accessible if every vertex can be reach by an oriented path from the root.

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root Orient the other edges couterclockwise around the tree It is called a minimal orientation (for the order induced by circuit reversal). The resulting orientation has no clockwise circuit. Theorem (Bernardi 2005) This is a bijection between tree-rooted maps with n edges and minimum accessible maps with n edges A oriented map is accessible if every vertex can be reach by an oriented path from the root.

From tree-rooted maps to minimal accessible maps

Orient the tree edges away from the root Orient the other edges couterclockwise around the tree The tree is recovered by reconstructing its contour . It is called a minimal orientation (for the order induced by circuit reversal). The resulting orientation has no clockwise circuit. Theorem (Bernardi 2005) This is a bijection between tree-rooted maps with n edges and minimum accessible maps with n edges A oriented map is accessible if every vertex can be reach by an oriented path from the root.

Minimal orientations and canonical spanning trees

Idea: Choose a minimal accessible orientation to get a spanning tree Our pb becomes: How to choose a canonical accessible minimal orientation?

Minimal orientations and canonical spanning trees

Idea: A function α : V → N is feasible on a plane map M if there exists an

• rientation of M such that for each vertex v the outdegree of v is f(v).

Choose a minimal accessible orientation to get a spanning tree Our pb becomes: How to choose a canonical accessible minimal orientation?

Minimal orientations and canonical spanning trees

Idea: Theorem (Felsner 2004). Let α be a feasible function on a plane map M. Then the map M has a unique minimal α-orientation. A function α : V → N is feasible on a plane map M if there exists an

• rientation of M such that for each vertex v the outdegree of v is f(v).

Choose a minimal accessible orientation to get a spanning tree Our pb becomes: How to choose a canonical accessible minimal orientation?

Minimal orientations and canonical spanning trees

Idea: Theorem (Felsner 2004). Let α be a feasible function on a plane map M. Then the map M has a unique minimal α-orientation. A function α : V → N is feasible on a plane map M if there exists an

• rientation of M such that for each vertex v the outdegree of v is f(v).

Choose a minimal accessible orientation to get a spanning tree Our pb becomes: How to choose a canonical accessible minimal orientation? Our pb becomes: How to choose a canonical α? (and check accessibility)

Minimal orientations and canonical spanning trees

Idea: Theorem (Felsner 2004). Let α be a feasible function on a plane map M. Then the map M has a unique minimal α-orientation. A function α : V → N is feasible on a plane map M if there exists an

• rientation of M such that for each vertex v the outdegree of v is f(v).

Choose a minimal accessible orientation to get a spanning tree Our pb becomes: How to choose a canonical accessible minimal orientation? Our pb becomes: How to choose a canonical α? (and check accessibility) Fact: For many subclasses F of planar maps, there exists an αF s.t.: A planar map is in F if and only if it admits an αF-orientation.

α-orientations for increasing quadrangulations

Recall increasing quadrangulations are planar maps with faces of degree 4 such that: – faces have labels in {1, . . . , 2n − 2} – around labeled vertices, face labels increase in ccw order – around white vertices, face labels increase in cw order

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

α-orientations for increasing quadrangulations

Recall increasing quadrangulations are planar maps with faces of degree 4 such that: – faces have labels in {1, . . . , 2n − 2} – around labeled vertices, face labels increase in ccw order – around white vertices, face labels increase in cw order Orient each edge so that the minimum incident label is on the left

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

α-orientations for increasing quadrangulations

Recall increasing quadrangulations are planar maps with faces of degree 4 such that: – faces have labels in {1, . . . , 2n − 2} – around labeled vertices, face labels increase in ccw order – around white vertices, face labels increase in cw order Orient each edge so that the minimum incident label is on the left This orientation is accessible, in fact strongly connected.

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

α-orientations for increasing quadrangulations

Recall increasing quadrangulations are planar maps with faces of degree 4 such that: – faces have labels in {1, . . . , 2n − 2} – around labeled vertices, face labels increase in ccw order – around white vertices, face labels increase in cw order Orient each edge so that the minimum incident label is on the left Each black vertex has indegree αh(black) = m − 1, outdegree 1 Each white vertex has indegree αh(white) = 1. This orientation is accessible, in fact strongly connected.

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

α-orientations for increasing quadrangulations

Recall increasing quadrangulations are planar maps with faces of degree 4 such that: – faces have labels in {1, . . . , 2n − 2} – around labeled vertices, face labels increase in ccw order – around white vertices, face labels increase in cw order Orient each edge so that the minimum incident label is on the left Each black vertex has indegree αh(black) = m − 1, outdegree 1 Each white vertex has indegree αh(white) = 1. This orientation is accessible, in fact strongly connected.

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

This is our choice of canonical α to decompose increasing quadrangulations.

• pening of an increasing quadrangulation

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

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) find spanning tree 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) find spanning tree 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) find spanning tree 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pen

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

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) find spanning tree 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pen

1 12 7 5 6 9 10 8 3 2 4 11 11 but forget half-edges

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) find spanning tree 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pen

1 12 7 5 6 9 10 8 3 2 4 11 11 but forget half-edges give labels to edges eliminate root black vertex

• pening of an increasing quadrangulation

endow with min αc-orient (return cycles) find spanning tree 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11

• pen

1 12 7 5 6 9 10 8 3 2 4 11 11 but forget half-edges give labels to edges eliminate root black vertex

• Proposition. The resulting simple Hurwitz trees has n unlabelled vertices, n − 1

labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

From simple Hurwitz trees to increasing quadrangulations

i k k j i k j Cas 1: i k j i i k j Cas 2:

• u

k m ℓ k i k j A local rule to create increasing half edges Two half-edges with same label ⇒ edge and face of degree 4 Iterate the local rules as long as possible...

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

From simple Hurwitz trees to factorizations

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

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 1 2 3 5 6 4 vertex label are useless for the bijection

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 vertex label are useless for the bijection

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 vertex label are useless for the bijection adding buds

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 vertex label are useless for the bijection adding buds Parings and adding buds again

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 vertex label are useless for the bijection adding buds Parings and adding buds again again

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 vertex label are useless for the bijection adding buds Parings and adding buds again again again

From simple Hurwitz trees to factorizations

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

• Lemma. When it stops, there are only white half-edges left.

vertex label are useless for the bijection adding buds Parings and adding buds again again again

From simple Hurwitz trees to factorizations

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

• Lemma. When it stops, there are only white half-edges left.

We connect them to a new black vertex and reload labels. vertex label are useless for the bijection adding buds Parings and adding buds again again again

From simple Hurwitz trees to factorizations

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

• Lemma. When it stops, there are only white half-edges left.

We connect them to a new black vertex and reload labels. 1 1 2 3 5 6 4 1 1 4 5 2 6 3 vertex label are useless for the bijection 7 adding buds Parings and adding buds again again again

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 1 2 3 5 6 4 1 1 4 5 2 6 3 vertex label are useless for the bijection 7 adding buds Parings and adding buds again again again

Theorem[Duchi-Poulalhon-S. 2012] Closure is the reverse bijection between – simple Hurwitz trees of size n, and – increasing quadrangulations, and – simple ramified covers of S by itself with m = 2n − 2 critical values.

From simple Hurwitz trees to factorizations

1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 12 7 5 6 9 10 8 3 2 4 11 1 1 2 3 5 6 4 1 1 4 5 2 6 3 vertex label are useless for the bijection 7 adding buds Parings and adding buds again again again

Theorem[Duchi-Poulalhon-S. 2012] Closure is the reverse bijection between – simple Hurwitz trees of size n, and – increasing quadrangulations, and – simple ramified covers of S by itself with m = 2n − 2 critical values.

R´ esum´ e des 2 premiers ´ episodes

Compter des classes d’´ equivalence de revˆ etements ramifi´ es compter certaines plongements de graphes ⇔ ⇔ compter certains arbres

Plan de l’expos´ e

Plan de l’expos´ e

Revˆ etements ramifi´ es et cartes Cartes et arbres ´ Enum´ eration d’arbres et formule d’Hurwitz Revˆ etements et cartes al´ eatoires

Hurwitz formula for increasing quadrangulations

Theorem[Duchi-Poulalhon-S. 2012] Increasing quadrangulations (size n) are in bijection with simple Hurwitz trees having n unlabelled vertices, n − 1 labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

4 1 1 2 3 5 6 8 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Hurwitz formula for increasing quadrangulations

Theorem[Duchi-Poulalhon-S. 2012] Increasing quadrangulations (size n) are in bijection with simple Hurwitz trees having n unlabelled vertices, n − 1 labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

4 1 1 2 3 5 6 8 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 1 1 2 3 5 6 8 7 (2n − 2)!

Hurwitz formula for increasing quadrangulations

Theorem[Duchi-Poulalhon-S. 2012] Increasing quadrangulations (size n) are in bijection with simple Hurwitz trees having n unlabelled vertices, n − 1 labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

4 1 1 2 3 5 6 8 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 1 1 2 3 5 6 8 7 (2n − 2)! 4 1 1 2 3 5 6 8 7 n

Hurwitz formula for increasing quadrangulations

Theorem[Duchi-Poulalhon-S. 2012] Increasing quadrangulations (size n) are in bijection with simple Hurwitz trees having n unlabelled vertices, n − 1 labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

4 1 1 2 3 5 6 8 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 1 1 2 3 5 6 8 7 (2n − 2)! 4 1 1 2 3 5 6 8 7 n nn−2

Hurwitz formula for increasing quadrangulations

Theorem[Duchi-Poulalhon-S. 2012] Increasing quadrangulations (size n) are in bijection with simple Hurwitz trees having n unlabelled vertices, n − 1 labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

4 1 1 2 3 5 6 8 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 1 1 2 3 5 6 8 7 (2n − 2)! 4 1 1 2 3 5 6 8 7 n nn−2 nn−3

Hurwitz formula for increasing quadrangulations

Theorem[Duchi-Poulalhon-S. 2012] Increasing quadrangulations (size n) are in bijection with simple Hurwitz trees having n unlabelled vertices, n − 1 labeled vertices of degree 2, 2n − 2 edges that increase ccw around labeled vertices.

4 1 1 2 3 5 6 8 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 1 1 2 3 5 6 8 7 (2n − 2)! 4 1 1 2 3 5 6 8 7 n nn−2 nn−3 nn−3 · (2n − 2)!

The number of simple ramified cover of S by itself with m = 2n − 2 critical points is nn−3(2n − 2)!.

4 1 1 2 3 5 6 8 7 nn−2

Hurwitz formula for factorizations in transpositions

• Theorem. Let λ = 1ℓ1, . . . , nℓn be a partition n, and ℓ = P

i ℓi.

The number of m-uples of transpositions (τ1, . . . , τm) such that

• (product cycle type) τ1 · · · τm = σ has cycle type λ
• (transitivity) the associated graph is connected
• (minimality) the number of factors is m = n + ℓ − 2

is nℓ−3 · m! · n! · Y

i≥1

1 ℓi! „ ii i! «ℓi λ = n, factorizations of n-cycles: nn−2 · (n − 1)! λ = 1n, factorizations of the identity: nn−3 · (2n − 2)!

(Hurwitz 1891, Strehl 1996) (Goulden–Jackson 1997) (Lando–Zvonkine 1999) (Bousquet-M´ elou–Schaeffer 2000) (recurrences, Abel identities) (gfs and differential eqns) (geometry of LL mapping) (bijection + inclusion/exclusion)

Proofs:

A formula for general factorizations [BMS00]

• Theorem. Let λ = 1ℓ1, . . . , nℓn be a partition of n, and ℓ = P

i ℓi.

The number of m-uple of permutations (σ1, . . . , σm) such that

• (factorization) σ1 · · · σm = σ with cycle type λ
• (transitivity) σ1, . . . , σm acts transitively on {1, . . . , n}
• (minimality) the total rank of factors is P

i r(σi) = n + ℓ − 2

is m ((m − 1)n − 1)! (mn − (n + ℓ − 2))! · n! · Y

i

1 ℓi! “mi − 1 i ”ℓi Proofs:

(Bousquet-M´ elou–Schaeffer 2000) (Goulden–Serrano 2009) (bijection + inclusion/exclusion)(gfs and differential eqns)

λ = n, factorizations of n-cycles:

1 (mn+1)

`mn+1

n

´ · (n − 1)! λ = 1n, identity factorizations:

m (m−2)n+2 (m−1)n−1 (m−2)n+1

`(m−1)n

n

´ · (n − 1)!

R´ esum´ e des 3 premiers ´ episodes

Compter des classes d’´ equivalence de revˆ etements ramifi´ es compter certaines plongements de graphes ⇔ ⇔ les formules simples appellent des preuves constructives compter certains arbres

Plan de l’expos´ e

Plan de l’expos´ e

Revˆ etements ramifi´ es et cartes Cartes et arbres ´ Enum´ eration d’arbres et formule d’Hurwitz Revˆ etements et cartes al´ eatoires

Quadrangulations croissantes al´ eatoires uniformes

¯ Qn = {quadrangulations croissantes ` a n faces}. Quadrangulation croissante uniforme = variable al´ eatoire Qn ` a valeur dans ¯ Qn avec Pr(Qn = q) = 1 | ¯ Qn| = 1 nn−3(2n − 2)! pour tout q ∈ ¯ Qn

Quadrangulations croissantes al´ eatoires uniformes

¯ Qn = {quadrangulations croissantes ` a n faces}. Quadrangulation croissante uniforme = variable al´ eatoire Qn ` a valeur dans ¯ Qn avec Pr(Qn = q) = 1 | ¯ Qn| = 1 nn−3(2n − 2)! pour tout q ∈ ¯ Qn Comment ´ etudier Qn ?

• le choix de la distribution uniforme combinatoire est le plus imm´

ediat Parall` ele naturel avec la distribution uniforme sur les quadrangulations enracin´ ees: Pr( Qn = q) = 1 | Qn| = 1

2·3n(2n)! (n+2)!n!

pour tout q ∈ Qn

Propri´ et´ es des cartes al´ eatoires uniformes ?

Propri´ et´ es des cartes al´ eatoires uniformes ?

• n est loin d’une discr´

etisation al´ eatoire d’une g´ eom´ etrie euclidienne

Delaunay de points al´ eatoires dans un disque Triangulation uniforme al´ eatoire d’un disque

en physique on lie cela ` a la mod´ elisation discr` ete de la gravit´ e quantique

Quadrangulations uniformes comme surfaces al´ eatoires

Chapuy Schaeffer Marckert L’allure d’une sph` ere al´ eatoire d´ epend un peu de qui dessine... Objectif: Choisir une m´ etrique intrins` eque et d´ ecrire les surfaces ainsi obtenues

´ Etudier les quadrangulations al´ eatoires uniformes

Distribution uniforme sur les quadrangulations ` a n faces, pour n grand 1` ere approche: ´ Etudier le comportement asymptotique de param` etres:

• degr´

e d’un sommet al´ eatoire

• loi 0-1 pour les propri´

et´ es locales

• distance entre 2 sommets al´

eatoires

• longueur d’un plus petit cycle diviseur

⇒ esp´ erance, moments, lois limites discr` etes ou continues, qd n → ∞

Bender, Canfield et al (90’s →) en combinatoire Ambjørn, Watabiki et al (90’s →) en physique

´ Etudier les quadrangulations al´ eatoires uniformes

Distribution uniforme sur les quadrangulations ` a n faces, pour n grand 1` ere approche: ´ Etudier le comportement asymptotique de param` etres:

• degr´

e d’un sommet al´ eatoire

• loi 0-1 pour les propri´

et´ es locales

• distance entre 2 sommets al´

eatoires

• longueur d’un plus petit cycle diviseur

⇒ esp´ erance, moments, lois limites discr` etes ou continues, qd n → ∞

Bender, Canfield et al (90’s →) en combinatoire Ambjørn, Watabiki et al (90’s →) en physique

Exemple: ∆n = distance entre 2 sommets al´ eatoires uniformes de Qn Th´ eor` eme (Chassaing-S. 2004) E(∆n) ∼ c · n1/4 (n−1/4∆n)

d

− → max (serpent Brownien)

´ Etudier les quadrangulations al´ eatoires uniformes

Distribution uniforme sur les quadrangulations ` a n faces, pour n grand 2` eme approche: D´ efinir des surfaces al´ eatoires limites

´ Etudier les quadrangulations al´ eatoires uniformes

Distribution uniforme sur les quadrangulations ` a n faces, pour n grand 2` eme approche: D´ efinir des surfaces al´ eatoires limites – convergence vers une limite d’´ echelle

(Pb pos´ e au s´ eminaire Hypathie en 2002 ` a Lyon)

⇒ la carte Brownienne

Marckert, Mokkadem, Le Gall, Miermont, . . .

´ Etudier les quadrangulations al´ eatoires uniformes

Distribution uniforme sur les quadrangulations ` a n faces, pour n grand 2` eme approche: D´ efinir des surfaces al´ eatoires limites – convergence vers une limite d’´ echelle – convergence vers une limite infinie discr` ete

(Pb pos´ e au s´ eminaire Hypathie en 2002 ` a Lyon) Angel, Schramm, . . .

⇒ la carte Brownienne

Marckert, Mokkadem, Le Gall, Miermont, . . .

⇒ la quadrangulation infinie uniforme (UIPQ)

puis Weill, Curien, Benjamini,... puis Durhus, Chassaing, Krikun, Bettinelli,...

Conclusions

– L’excursion Brownienne d´ ecrit la limite d’´ echelle de toute sorte d’excursions al´ eatoires discr` etes plus ou moins complexes. – L’arbre continu al´ eatoire est limite d’´ echelle de toute sorte d’arbres al´ eatoires discrets plus ou moins complexes. ⇒ On pense qu’il en est de mˆ eme de la carte Brownienne.

Conclusions

– L’excursion Brownienne d´ ecrit la limite d’´ echelle de toute sorte d’excursions al´ eatoires discr` etes plus ou moins complexes. – L’arbre continu al´ eatoire est limite d’´ echelle de toute sorte d’arbres al´ eatoires discrets plus ou moins complexes. ⇒ On pense qu’il en est de mˆ eme de la carte Brownienne. Les r´ esultats de Le Gall et Miermont valent pour des cartes avec des contraintes de degr´ e de faces plus g´ en´ erales (q-angulations,. . . )

Conclusions

– L’excursion Brownienne d´ ecrit la limite d’´ echelle de toute sorte d’excursions al´ eatoires discr` etes plus ou moins complexes. – L’arbre continu al´ eatoire est limite d’´ echelle de toute sorte d’arbres al´ eatoires discrets plus ou moins complexes. ⇒ On pense qu’il en est de mˆ eme de la carte Brownienne. Les r´ esultats de Le Gall et Miermont valent pour des cartes avec des contraintes de degr´ e de faces plus g´ en´ erales (q-angulations,. . . ) Un challenge est de montrer que des objets a priori plus ´ eloign´ es tels que les graphes planaires (non plong´ es) ou les revˆ etements ramifi´ es, sont en fait dans la mˆ eme classe d’universalit´ e.

Conclusions

– L’excursion Brownienne d´ ecrit la limite d’´ echelle de toute sorte d’excursions al´ eatoires discr` etes plus ou moins complexes. – L’arbre continu al´ eatoire est limite d’´ echelle de toute sorte d’arbres al´ eatoires discrets plus ou moins complexes. ⇒ On pense qu’il en est de mˆ eme de la carte Brownienne. Les r´ esultats de Le Gall et Miermont valent pour des cartes avec des contraintes de degr´ e de faces plus g´ en´ erales (q-angulations,. . . ) Un challenge est de montrer que des objets a priori plus ´ eloign´ es tels que les graphes planaires (non plong´ es) ou les revˆ etements ramifi´ es, sont en fait dans la mˆ eme classe d’universalit´ e. On dispose d’un cadre bijectif tr` es g´ en´ eral pour la construction de cartes par recollements d’arbres On obtient ainsi en particulier un codage d’arbres pour les revˆ etements... Il reste ` a utiliser ces constructions pour passer ` a la limite...

(Bernardi-Chapuy-Fusy 2011, Albenque-Poulalhon 2012)