Geometry of a uniform minimal factorization Paul Thevenin CMAP, - - PowerPoint PPT Presentation

Geometry of a uniform minimal factorization

Paul Thevenin

CMAP, Ecole Polytechnique

June 14, 2019

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 1 / 32

1

Minimal factorizations Definitions Geometrical coding of a factorization

2

Connection with the Brownian excursion

3

Conclusion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 2 / 32

Minimal factorizations

1

Minimal factorizations Definitions Geometrical coding of a factorization

2

Connection with the Brownian excursion

3

Conclusion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 3 / 32

Minimal factorizations Definitions

1

Minimal factorizations Definitions Geometrical coding of a factorization

2

Connection with the Brownian excursion

3

Conclusion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 4 / 32

Minimal factorizations Definitions

Definitions

Fix n ≥ 1 Sn the set of permutations of 1, n Tn the set of transpositions of 1, n.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions

Definitions

Fix n ≥ 1 Sn the set of permutations of 1, n Tn the set of transpositions of 1, n. The n-cycle is the permutation cn := (1 2 · · · n).

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions

Definitions

Fix n ≥ 1 Sn the set of permutations of 1, n Tn the set of transpositions of 1, n. The n-cycle is the permutation cn := (1 2 · · · n). Minimal factorization of the n-cycle : an ordered (n − 1)-tuple of transpositions whose product is cn.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions

Definitions

Fix n ≥ 1 Sn the set of permutations of 1, n Tn the set of transpositions of 1, n. The n-cycle is the permutation cn := (1 2 · · · n). Minimal factorization of the n-cycle : an ordered (n − 1)-tuple of transpositions whose product is cn. Mn :=

• (τ1, ..., τn−1) ∈ Tn−1

n

, τ1 · · · τn−1 = (1 2 · · · n)

• ⊂ Sn

is the set of minimal factorizations of cn.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions

Definitions

Fix n ≥ 1 Sn the set of permutations of 1, n Tn the set of transpositions of 1, n. The n-cycle is the permutation cn := (1 2 · · · n). Minimal factorization of the n-cycle : an ordered (n − 1)-tuple of transpositions whose product is cn. Mn :=

• (τ1, ..., τn−1) ∈ Tn−1

n

, τ1 · · · τn−1 = (1 2 · · · n)

• ⊂ Sn

is the set of minimal factorizations of cn. Convention We read transpositions in a factorization from left to right.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions

Definitions

Fix n ≥ 1 Sn the set of permutations of 1, n Tn the set of transpositions of 1, n. The n-cycle is the permutation cn := (1 2 · · · n). Minimal factorization of the n-cycle : an ordered (n − 1)-tuple of transpositions whose product is cn. Mn :=

• (τ1, ..., τn−1) ∈ Tn−1

n

, τ1 · · · τn−1 = (1 2 · · · n)

• ⊂ Sn

is the set of minimal factorizations of cn. Convention We read transpositions in a factorization from left to right. Example : (34)(89)(35)(13)(16)(18)(23)(78) ∈ M9.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions

Brief history and first properties

Called minimal : one needs at least n − 1 transpositions to generate the n-cycle

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions

Brief history and first properties

Called minimal : one needs at least n − 1 transpositions to generate the n-cycle Study of minimal factorizations started in the 50’s.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions

Brief history and first properties

Called minimal : one needs at least n − 1 transpositions to generate the n-cycle Study of minimal factorizations started in the 50’s. Combinatorial approaches : D´ enes (’59) proves that #Mn = nn−2

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions

Brief history and first properties

Called minimal : one needs at least n − 1 transpositions to generate the n-cycle Study of minimal factorizations started in the 50’s. Combinatorial approaches : D´ enes (’59) proves that #Mn = nn−2 Moskowski (’89), Goulden & Pepper (’93) : bijective proofs of this result, using bijections between factorizations and trees.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions

A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Definitions

A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Question : for n large, what does a typical minimal factorization look like ?

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Definitions

A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Question : for n large, what does a typical minimal factorization look like ? typical : here, take a minimal factorization uniformly in Mn.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Definitions

A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Question : for n large, what does a typical minimal factorization look like ? typical : here, take a minimal factorization uniformly in Mn. look like : find a way of representing a minimal factorization geometrically

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Geometrical coding of a factorization

1

Minimal factorizations Definitions Geometrical coding of a factorization

2

Connection with the Brownian excursion

3

Conclusion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 8 / 32

Minimal factorizations Geometrical coding of a factorization

Idea 1 Code a minimal factorization by a set of chords of the unit disk.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 9 / 32

Minimal factorizations Geometrical coding of a factorization

Idea 1 Code a minimal factorization by a set of chords of the unit disk. Each transposition is coded by a chord.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 9 / 32

Minimal factorizations Geometrical coding of a factorization

Idea 1 Code a minimal factorization by a set of chords of the unit disk. Each transposition is coded by a chord. Transposition (a b) ↔ Chord

• e−2iπa/n, e−2iπb/n

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 9 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

(34)(89)(35)(13)(16)(18)(23)(78) 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization

For f ∈ Mn, let L(f) be the set of chords that we obtain. 1 2 3 4 5 6 7 8 9

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 11 / 32

Minimal factorizations Geometrical coding of a factorization

Remark : no intersection between two chords (except maybe at their endpoints). Lamination : set of chords with this property.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 12 / 32

Minimal factorizations Geometrical coding of a factorization

Remark : no intersection between two chords (except maybe at their endpoints). Lamination : set of chords with this property. In fact the chords form a tree (Goulden & Yong, ’02).

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 12 / 32

Minimal factorizations Geometrical coding of a factorization

Remark : no intersection between two chords (except maybe at their endpoints). Lamination : set of chords with this property. In fact the chords form a tree (Goulden & Yong, ’02). We denote by fn a uniform minimal factorization of cn.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 12 / 32

Minimal factorizations Geometrical coding of a factorization

Remark : no intersection between two chords (except maybe at their endpoints). Lamination : set of chords with this property. In fact the chords form a tree (Goulden & Yong, ’02). We denote by fn a uniform minimal factorization of cn. Study of L(fn) ⇒ Study of a random tree.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 12 / 32

Minimal factorizations Geometrical coding of a factorization

Theorem (F´ eray & Kortchemski, ’17) There exists a (random) lamination L∞ such that, in distribution, lim

n→∞L(fn) = L∞

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 13 / 32

Minimal factorizations Geometrical coding of a factorization

Theorem (F´ eray & Kortchemski, ’17) There exists a (random) lamination L∞ such that, in distribution, lim

n→∞L(fn) = L∞

L∞

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 13 / 32

Minimal factorizations Geometrical coding of a factorization

Idea 2 Preserve the structure of the factorization, by keeping track of the order of the transpositions.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 14 / 32

Minimal factorizations Geometrical coding of a factorization

Idea 2 Preserve the structure of the factorization, by keeping track of the order of the transpositions. Take f = (τ1, ..., τn−1) such that τ1 · · · τn−1 = cn.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 14 / 32

Minimal factorizations Geometrical coding of a factorization

Idea 2 Preserve the structure of the factorization, by keeping track of the order of the transpositions. Take f = (τ1, ..., τn−1) such that τ1 · · · τn−1 = cn. Lk(f) : only draw the chords corresponding to τ1, . . . , τk (or all chords, if k ≥ n).

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 14 / 32

Minimal factorizations Geometrical coding of a factorization

Example f ∈ M1000 uniform

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 15 / 32

Minimal factorizations Geometrical coding of a factorization

Example f ∈ M1000 uniform L10(f)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 15 / 32

Minimal factorizations Geometrical coding of a factorization

Example f ∈ M1000 uniform L10(f) L20(f)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 15 / 32

Minimal factorizations Geometrical coding of a factorization

Example f ∈ M1000 uniform L10(f) L20(f) L50(f)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 15 / 32

Minimal factorizations Geometrical coding of a factorization

Example f ∈ M1000 uniform L10(f) L20(f) L50(f) L94(f)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 15 / 32

Minimal factorizations Geometrical coding of a factorization

Example f ∈ M1000 uniform L10(f) L20(f) L50(f) L94(f) L151(f)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 15 / 32

Minimal factorizations Geometrical coding of a factorization

Theorem (T. ’19+) There exists a lamination-valued process (Lc)0≤c≤∞ such that, in distribution, lim

n→∞

• L⌊c√n⌋(fn)
• 0≤c≤∞ = (Lc)0≤c≤∞

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 16 / 32

Minimal factorizations Geometrical coding of a factorization

Theorem (T. ’19+) There exists a lamination-valued process (Lc)0≤c≤∞ such that, in distribution, lim

n→∞

• L⌊c√n⌋(fn)
• 0≤c≤∞ = (Lc)0≤c≤∞

Generalization of a theorem of F´ eray & Kortchemski ’17, who prove this convergence at c fixed.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 16 / 32

Minimal factorizations Geometrical coding of a factorization

Theorem (T. ’19+) There exists a lamination-valued process (Lc)0≤c≤∞ such that, in distribution, lim

n→∞

• L⌊c√n⌋(fn)
• 0≤c≤∞ = (Lc)0≤c≤∞

Generalization of a theorem of F´ eray & Kortchemski ’17, who prove this convergence at c fixed. New lamination-valued process, which interpolates between the unit circle and L∞.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 16 / 32

Minimal factorizations Geometrical coding of a factorization

Realization of this process

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 17 / 32

Minimal factorizations Geometrical coding of a factorization

How to understand this convergence ?

Information on this limit process ⇒ Information of a large typical factorization.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 18 / 32

Minimal factorizations Geometrical coding of a factorization

How to understand this convergence ?

Information on this limit process ⇒ Information of a large typical factorization. ”Large” transpositions start appearing after √n transpositions At order √n, there is only a finite number of large transpositions

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 18 / 32

Connection with the Brownian excursion

1

Minimal factorizations Definitions Geometrical coding of a factorization

2

Connection with the Brownian excursion

3

Conclusion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 19 / 32

Connection with the Brownian excursion

Standard Brownian excursion ❡ : Brownian motion conditioned to reach 0 at time 1 and to stay nonnegative between 0 and 1. ❡

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 20 / 32

Connection with the Brownian excursion

Standard Brownian excursion ❡ : Brownian motion conditioned to reach 0 at time 1 and to stay nonnegative between 0 and 1. A realization of the standard Brownian excursion ❡.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 20 / 32

Connection with the Brownian excursion

The lamination L∞ can be obtained from the standard Brownian excursion :

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 21 / 32

Connection with the Brownian excursion

The lamination L∞ can be obtained from the standard Brownian excursion : Fix a realization of the excursion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 21 / 32

Connection with the Brownian excursion

The lamination L∞ can be obtained from the standard Brownian excursion : Fix a realization of the excursion Given u := (s, t) under the excursion, associate to it a chord in the unit disk.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 21 / 32

Connection with the Brownian excursion

u := (s, t) under the excursion.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 22 / 32

Connection with the Brownian excursion

Define g(u) := sup{s′ < s, ❡(s′) < t}, d(u) := inf{s′ > s, ❡(s′) < t}.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 23 / 32

Connection with the Brownian excursion

Draw the chord [e−2iπg(u), e−2iπd(u)]

e−2iπg(u) e−2iπd(u)

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 24 / 32

Connection with the Brownian excursion

Theorem The lamination obtained by drawing the chords corresponding to each point under the Brownian excursion has the law of L∞.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 25 / 32

Connection with the Brownian excursion

Theorem The lamination obtained by drawing the chords corresponding to each point under the Brownian excursion has the law of L∞.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 25 / 32

Connection with the Brownian excursion

As well, one can obtain the lamination-valued process (Lc)c≥0 from the Brownian excursion.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 26 / 32

Connection with the Brownian excursion

As well, one can obtain the lamination-valued process (Lc)c≥0 from the Brownian excursion. For c ≥ 0, let Nc be a Poisson point process of intensity

2cdsdt d(u)−g(u)

under the excursion.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 26 / 32

Connection with the Brownian excursion

As well, one can obtain the lamination-valued process (Lc)c≥0 from the Brownian excursion. For c ≥ 0, let Nc be a Poisson point process of intensity

2cdsdt d(u)−g(u)

under the excursion. Couple this processes so that (Nc)c≥0 is nondecreasing for the inclusion.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 26 / 32

Connection with the Brownian excursion

As well, one can obtain the lamination-valued process (Lc)c≥0 from the Brownian excursion. For c ≥ 0, let Nc be a Poisson point process of intensity

2cdsdt d(u)−g(u)

under the excursion. Couple this processes so that (Nc)c≥0 is nondecreasing for the inclusion. Denote by Lc the lamination associated to Nc

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 26 / 32

Connection with the Brownian excursion

Theorem (T., ’19+) The process (Lc)c≥0 is the limit of the lamination-valued processes

• btained from minimal factorizations.

lim

n→∞

• L⌊c√n⌋(fn)
• c≥0 = (Lc)c≥0

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 27 / 32

Connection with the Brownian excursion

Key idea to relate these two models : draw the dual tree of this lamination (Goulden & Yong, ’02)

1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

1 3 7 6 9 4 2 5 8

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 28 / 32

Conclusion

1

Minimal factorizations Definitions Geometrical coding of a factorization

2

Connection with the Brownian excursion

3

Conclusion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 29 / 32

Conclusion

Conclusion

Geometric representation of a minimal factorization

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 30 / 32

Conclusion

Conclusion

Geometric representation of a minimal factorization Convergence, in some sense, of a uniform factorization

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 30 / 32

Conclusion

Conclusion

Geometric representation of a minimal factorization Convergence, in some sense, of a uniform factorization Relation with the Brownian excursion

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 30 / 32

Conclusion

Perspectives

Connections with the standard coalescent process on Aldous’CRT.

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 31 / 32

Conclusion

Perspectives

Connections with the standard coalescent process on Aldous’CRT. Connections with L´ evy processes (Gaussian inverse process).

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 31 / 32

Conclusion

Perspectives

Connections with the standard coalescent process on Aldous’CRT. Connections with L´ evy processes (Gaussian inverse process). Possible generalizations to other models of factorizations (not necessarily into transpositions).

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 31 / 32

Conclusion

Thanks !

Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 32 / 32