Random braids: Stabilizing the Garside normal form Vincent Jug & - - PowerPoint PPT Presentation

Random braids: Stabilizing the Garside normal form

Vincent Jugé & Jean Mairesse

Université Paris-Est Marne-la-Vallée (LIGM) – Sorbonne Université & CNRS (LIP6)

23/04/2019

Vincent Jugé & Jean Mairesse Stabilization of random braids

Contents

1

Introduction

2

Random walk in dimer monoids & groups

3

Random walk in braid monoids

4

Conclusion

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid and Write braids in your favorite normal form.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid and Write braids in your favorite normal form.

“Do the words you write converge?” (A. Vershik, „ 2000)

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid and Write braids in your favorite normal form.

“Do the words you write converge?” (A. Vershik, „ 2000)

It should depend on which normal form you consider!

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid and Write braids in your favorite normal form.

“Do the words you write converge?” (A. Vershik, „ 2000)

It should depend on which normal form you consider! “It works with the Markov-Ivanovsky normal form” (A. Vershik & A. Malyutin, 2007)

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid and Write braids in your favorite normal form.

“Do the words you write converge?” (A. Vershik, „ 2000)

It should depend on which normal form you consider! “It works with the Markov-Ivanovsky normal form [but] the stability problem remains open for the Birman–Ko–Lee normal form and the Garside normal form, among others” (A. Vershik & A. Malyutin, 2007)

Vincent Jugé & Jean Mairesse Stabilization of random braids

Questions & motivations

Consider a uniform random walk in a braid monoid and Write braids in your favorite normal form.

“Do the words you write converge?” (A. Vershik, „ 2000)

It should depend on which normal form you consider! “It works with the Markov-Ivanovsky normal form [but] the stability problem remains open for the Birman–Ko–Lee normal form and the Garside normal form, among others” (A. Vershik & A. Malyutin, 2007) “It works with them too!” (V. J. & J. M., this talk)

Vincent Jugé & Jean Mairesse Stabilization of random braids

Contents

1

Introduction

2

Random walk in dimer monoids & groups Dimer monoids & dimer groups Random walk in dimer monoids Random walk in dimer groups

3

Random walk in braid monoids

4

Conclusion

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1 S4 Viennot heap diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1 S4 Viennot heap diagrams

Dimer monoid

M`

n “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy`

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1 S4 Viennot heap diagrams

Dimer monoid

M`

n “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy`

Cartier-Foata normal form: S1S3 ¨ S1S4 ¨ S3 ¨ S2S4 ¨ S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S1 S3 S4 S3 S1 S2 S1 S4 Viennot heap diagrams

Dimer monoid

M`

n “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy`

Left Garside normal form: S1S3 ¨ S1S4 ¨ S3 ¨ S2S4 ¨ S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris?

S3 S4 S3 S1 S2 S1 Viennot heap diagrams MAGNET S1 S4

Dimer monoid

M`

n “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy`

Left Garside normal form: S1S3 ¨ S1S4 ¨ S3 ¨ S2S4 ¨ S1 Right Garside normal form: S3 ¨ S1S4 ¨ S1S3 ¨ S2 ¨ S1S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S1 S3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S1 S3 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S1 S3 S4 S3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S1 S3 S4 S3 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1 S4 Viennot heap diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1 S4 Viennot heap diagrams

Dimer group

Mn “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1 S4 Viennot heap diagrams

Dimer group

Mn “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy Left Garside normal form: S´1

3

¨ S4 ¨ S3 ¨ S´1

2 S´1 4

¨ S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like playing boring Tetris with antimatter?

S3 S4 S3 S2 S1 Viennot heap diagrams MAGNET S4

Dimer group

Mn “ xS1, . . . , Sn | |i ´ j| ě 2 ñ SiSj “ SjSiy Left Garside normal form: S´1

3

¨ S4 ¨ S3 ¨ S´1

2 S´1 4

¨ S1 Right Garside normal form: S´1

3

¨ S4 ¨ S3 ¨ S´1

2

¨ S1S´1

4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. Does the random walk pXkqkě0 converge?

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. Does the random walk pXkqkě0 converge? Do the words GarLpXkqkě0 and GarRpXkqkě0 converge?

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. Does the random walk pXkqkě0 converge? Do the words GarLpXkqkě0 and GarRpXkqkě0 converge? Do their prefixes and suffixes converge?

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. Does the random walk pXkqkě0 converge? Do the words GarLpXkqkě0 and GarRpXkqkě0 converge? Do their prefixes and suffixes converge?

Theorem [Folklore]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in irreducible trace monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tS1, . . . , Snu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. Does the random walk pXkqkě0 converge? Do the words GarLpXkqkě0 and GarRpXkqkě0 converge? Do their prefixes and suffixes converge?

Theorem [Folklore]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq S1 S1 S3 S4 S3 S4

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq S3 S4 S3 S4 MAGNET S1 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq “ tSi : x ě Siu. S3 S4 S3 S4 MAGNET S1 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq “ tSi : x ě Siu. Consequence: if Yk “ Si˘1 for some Si P suffixpGarLpXkqq then S3 S4 S3 S4 S1 S1 S2 S1 S1 S4 S3 S4 S3 S1 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq “ tSi : x ě Siu. Consequence: if Yk “ Si˘1 for some Si P suffixpGarLpXkqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`1qq; S3 S4 S3 S4 S1 S1 S2 S1 S4 S3 S4 S3 S1 S2 S1 distinct suffixes

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq “ tSi : x ě Siu. Consequence: if Yk “ Si˘1 for some Si P suffixpGarLpXkqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`1qq; suffixpGarRpXkqq ‰ suffixpGarRpXk`1qq. S3 S4 S3 S4 S1 S2 MAGNET S1 S3 S4 S3 S1 S2 S1 MAGNET S1 S4 distinct suffixes

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq “ tSi : x ě Siu. Consequence: if Yk “ Si˘1 for some Si P suffixpGarLpXkqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`1qq; suffixpGarRpXkqq ‰ suffixpGarRpXk`1qq.

Prefix-convergence of GarLpXkqkě0

Lemma: for all x P M`

n ,

prefixpGarLpxqq “ tSi : Si ď xu.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Suffix-divergence

Lemma: for all x P M`

n ,

suffixpGarLpxqq Ď suffixpGarRpxqq “ tSi : x ě Siu. Consequence: if Yk “ Si˘1 for some Si P suffixpGarLpXkqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`1qq; suffixpGarRpXkqq ‰ suffixpGarRpXk`1qq.

Prefix-convergence of GarLpXkqkě0

Lemma: for all x P M`

n ,

prefixpGarLpxqq “ tSi : Si ď xu. Consequence: prefixpGarLpXkqq Ď prefixpGarLpXk`1qq.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking traces T “ S1S2 . . . SnSn´1 . . . S1 S1 S2 S3 S4 S3 S2 S1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking traces T “ S1S2 . . . SnSn´1 . . . S1 T

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking traces T “ S1S2 . . . SnSn´1 . . . S1 T

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking traces T “ S1S2 . . . SnSn´1 . . . S1 Lemma: for all x, y P M`

n , prefixpGarRpxTyqq “ prefixpGarRpxTqq.

MAGNET x y T MAGNET x T same prefixes

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking traces T “ S1S2 . . . SnSn´1 . . . S1 Lemma: for all x, y P M`

n , prefixpGarRpxTyqq “ prefixpGarRpxTqq.

Consequence: if Yk . . . Yk`2n´2 “ T then prefixpGarRpXk`2n´1qq “ prefixpGarRpXℓqq for all ℓ ě k ` 2n ´ 1. MAGNET Xk X ´1

k`2n´1Xℓ

T MAGNET Xk T same prefixes

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Random walk

1 Select i.i.d. generators pYkqkě0 in tS˘1

1 , . . . , S˘1 n u.

2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Random walk

1 Select i.i.d. generators pYkqkě0 in tS˘1

1 , . . . , S˘1 n u.

2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk.

Theorem [Folklore]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Random walk

1 Select i.i.d. generators pYkqkě0 in tS˘1

1 , . . . , S˘1 n u.

2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk.

Theorem [Folklore]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗ ñ The proof of suffix-divergence still works!

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in irreducible trace groups

Random walk

1 Select i.i.d. generators pYkqkě0 in tS˘1

1 , . . . , S˘1 n u.

2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk.

Theorem [Folklore]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗ ñ The proof of suffix-divergence still works!

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 S1 S1 S4 S1 S4 S3 S3 S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 S3 S3 S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 (escape rate in free groups) S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 (escape rate in free groups) Lemma 2: for all x P Mn, if Cipxq ě 2, then prefixpGarLpxqq “ prefixpGarLpxS˘1

i

qq. S3 shield S3 S1 S1 S4 S1 S4 S3 S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 (escape rate in free groups) Lemma 2: for all x P Mn, if Cipxq ě 2, then prefixpGarLpxqq “ prefixpGarLpxS˘1

i

qq. S1 S1 S4 S1 S4 S3 S3 S2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarLpXkqkě0

Key ingredient: Cipxq “ #toccurrences of S˘1

i

• r S˘1

i`1 in GarLpxqu.

Lemma: mini CipXkq Ñ `8 (escape rate in free groups) Lemma 2: for all x P Mn, if Cipxq ě 2, then prefixpGarLpxqq “ prefixpGarLpxS˘1

i

qq. S1 S1 S4 S1 S4 S3 S3 S2 prefix protected

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarLpXkqq “ tS1us ą 0

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarLpXkqq “ tS1us ą 0 stable prefix Xℓ After ℓ ě K steps, with proba ą 0

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarLpXkqq “ tS1us ą 0 stable prefix Xℓ After ℓ ě K steps, with proba ą 0 TK`1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarLpXkqq “ tS1us ą 0 S1 S˘1

1

stable prefix Xℓ After ℓ ě K steps, with proba ą 0 TK`1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarLpXkqq “ tS1us ą 0 S1 S˘1

1

forever stable prefix Xℓ After ℓ ě K steps, with proba ą 0 TK`1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 S1 S˘1

1

forever MAGNET stable prefix Xℓ After ℓ ě K steps, with proba ą 0 TK`1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Xk

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 S1 S˘1

1

Xk T MAGNET with proba ą 0 suffix

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Consequence: prefixpGarRpXkqqkě0 converges τ0, τ1, . . . , τk: stopping times

• ld prefix

Xτ2k MAGNET

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Consequence: prefixpGarRpXkqqkě0 converges τ0, τ1, . . . , τk: stopping times new prefix Xτ2k`1 MAGNET

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Consequence: prefixpGarRpXkqqkě0 converges τ0, τ1, . . . , τk: stopping times Xτ2k`2 S1 MAGNET

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Consequence: prefixpGarRpXkqqkě0 converges τ0, τ1, . . . , τk: stopping times Xτ2k`2 S1 X ´1

τ2k`2Xℓ

S1 MAGNET forever with proba ą 0

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Consequence: prefixpGarRpXkqqkě0 converges τ0, τ1, . . . , τk: stopping times Xτ2k`2 S1 X ´1

τ2k`2Xℓ

S1 MAGNET forever with proba ą 0 stable prefix

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups

Prefix-convergence of GarRpXkqkě0

Lemma: Pr@k ě 1, prefixpGarRpXkqq “ tS1us ą 0 Lemma 2: #tk ě 1 : suffixpGarRpXkqq “ tS1uu “ `8 Consequence: prefixpGarRpXkqqkě0 converges τ0, τ1, . . . , τk: stopping times Xτ2k`2 S1 X ´1

τ2k`2Xℓ

S1 MAGNET forever with proba ą 0 stable prefix τ2k`3 “ `8

Vincent Jugé & Jean Mairesse Stabilization of random braids

Contents

1

Introduction

2

Random walk in dimer monoids & groups

3

Random walk in braid monoids Braid monoids & braid groups Random walk in braid monoids

4

Conclusion

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1 σ1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1 σ1 σ1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1 σ1 σ1 σ2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1 σ1 σ1 σ2 σ3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1 σ1 σ1 σ2 σ3 Artin braid diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ1 σ2 σ1 σ3 σ2 σ3 σ1 σ1 σ1 σ2 σ3 Artin braid diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ2 σ1 σ2 σ3 σ2 σ3 σ1 σ1 σ1 σ2 σ3 Artin braid diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ2 σ1 σ2 σ3 σ2 σ3 σ1 σ1 σ1 σ2 σ3 Artin braid diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ2 σ1 σ2 σ3 σ2 σ1 σ3 σ1 σ1 σ2 σ3 Artin braid diagrams

Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise?

σ2 σ1 σ2 σ3 σ2 σ1 σ3 σ1 σ1 σ2 σ3 Artin braid diagrams

Braid monoid

B`

n “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy`

Vincent Jugé & Jean Mairesse Stabilization of random braids

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σ2 σ´1

1

σ´1

2

σ3 σ´1

2

σ1 σ´1

3

σ1 σ´1

1

σ2 σ3 Artin braid diagrams

Braid monoid

B`

n “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy`

Braid group

Bn “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy

Vincent Jugé & Jean Mairesse Stabilization of random braids

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σ2 σ´1

1

σ´1

2

σ3 σ´1

2

σ1 σ´1

3

1 1 σ2 σ3 Artin braid diagrams

Braid monoid

B`

n “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy`

Braid group

Bn “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy

Vincent Jugé & Jean Mairesse Stabilization of random braids

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σ2 σ´1

1

σ´1

2

σ3 σ´1

2

σ1 σ´1

3

σ2 σ3 Artin braid diagrams

Braid monoid

B`

n “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy`

Braid group

Bn “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, |i ´ j| ě 2 ñ σiσj “ σjσiy

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Theorem [Garside 1969]

B`

n is a lattice for both orders ď and ě.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Theorem [Garside 1969]

B`

n is a lattice for both orders ď and ě.

ñ The proof of is based on simple braids

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Theorem [Garside 1969]

B`

n is a lattice for both orders ď and ě.

ñ The proof of is based on simple braids

Simple braids?

Positive braids where strands never cross twice:

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Theorem [Garside 1969]

B`

n is a lattice for both orders ď and ě.

ñ The proof of is based on simple braids

Simple braids?

Positive braids where strands never cross twice: 1 σ1 σ2 σ1σ2 σ2σ1 ∆

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Theorem [Garside 1969]

B`

n is a lattice for both orders ď and ě.

ñ The proof of is based on simple braids

Simple braids?

Positive braids where strands never cross twice: in bijection with Sn; 1 σ1 σ2 σ1σ2 σ2σ1 ∆

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Theorem [Garside 1969]

B`

n is a lattice for both orders ď and ě.

ñ The proof of is based on simple braids

Simple braids?

Positive braids where strands never cross twice: in bijection with Sn; closed under ď- and ě-divisibility. 1 σ1 σ2 σ1σ2 σ2σ1 ∆

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Left Garside normal form

Minimal factorisation of the form β “ β1β2 ¨ ¨ ¨ βk such that @i ď k, βi “ GCDďp∆n, βiβi`1 ¨ ¨ ¨ βkq ∆ σ1σ3 σ1σ2σ3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Left Garside normal form

Minimal factorisation of the form β “ β1β2 ¨ ¨ ¨ βk such that @i ď k, βi “ GCDďp∆n, βiβi`1 ¨ ¨ ¨ βkq ô @i ă k, Lpβi`1q Ď Rpβiq ∆ σ1σ3 σ1σ2σ3

t1, 2, 3u t1, 3u t1u t1, 2, 3u t1, 3u t3u L : R :

Lpβq “ ti : σi ď βu Rpβq “ ti : β ě σiu

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Left Garside normal form

Minimal factorisation of the form β “ β1β2 ¨ ¨ ¨ βk such that @i ď k, βi “ GCDďp∆n, βiβi`1 ¨ ¨ ¨ βkq ô @i ă k, Lpβi`1q Ď Rpβiq

Right Garside normal form

Minimal factorisation of the form β “ β1β2 ¨ ¨ ¨ βℓ such that @i ď ℓ , βi “ GCDěp∆n, β1β2 ¨ ¨ ¨ βiq ô @i ă ℓ , Lpβi`1q Ě Rpβiq σ3 σ1σ3σ2σ1 ∆

t3u t1, 3u t1, 2, 3u t3u t1, 2u t1, 2, 3u L : R :

Lpβq “ ti : σi ď βu Rpβq “ ti : β ě σiu

Vincent Jugé & Jean Mairesse Stabilization of random braids

Simple braids & Garside normal forms

Left Garside normal form

Minimal factorisation of the form β “ β1β2 ¨ ¨ ¨ βk such that @i ď k, βi “ GCDďp∆n, βiβi`1 ¨ ¨ ¨ βkq ô @i ă k, Lpβi`1q Ď Rpβiq

Right Garside normal form

Minimal factorisation of the form β “ β1β2 ¨ ¨ ¨ βk such that @i ď k, βi “ GCDěp∆n, β1β2 ¨ ¨ ¨ βiq ô @i ă k, Lpβi`1q Ě Rpβiq σ3 σ1σ3σ2σ1 ∆

t3u t1, 3u t1, 2, 3u t3u t1, 2u t1, 2, 3u L : R :

Lpβq “ ti : σi ď βu Rpβq “ ti : β ě σiu

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tσ1, . . . , σnu. Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tσ1, . . . , σnu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tσ1, . . . , σnu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. ñ Main difference with dimer monoids: σi∆ “ ∆σn´i

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tσ1, . . . , σnu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. ñ Main difference with dimer monoids: σi∆ “ ∆σn´i

Theorem [Folklore]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗

• Vincent Jugé & Jean Mairesse

Stabilization of random braids

Random walk in braid monoids

Random walk

1 Select i.i.d. generators pYkqkě0 in tσ1, . . . , σnu. 2 Random process pXkqkě0 defined by:

X0 “ 1 and Xk`1 “ XkYk. ñ Main difference with dimer monoids: σi∆ “ ∆σn´i

Theorem [Folklore + J. & M. 2016]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗

• Vincent Jugé & Jean Mairesse

Stabilization of random braids

Random walk in braid monoids

Suffix-divergence of GarLpXkqkě0

Lemma: if YkYk`1 “ σiσi˘1 for some i P RpsuffixpGarLpXkqqq then ∆ σ1σ3

1 P R

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Suffix-divergence of GarLpXkqkě0

Lemma: if YkYk`1 “ σiσi˘1 for some i P RpsuffixpGarLpXkqqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`2qq. ∆ σ1σ3 σ1σ2

1 R R

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Suffix-divergence of GarLpXkqkě0

Lemma: if YkYk`1 “ σiσi˘1 for some i P RpsuffixpGarLpXkqqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`2qq.

Prefix-convergence of GarLpXkqkě0

Lemma: prefixpGarLpXkqq ď prefixpGarLpXk`1qq.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Suffix-divergence of GarLpXkqkě0

Lemma: if YkYk`1 “ σiσi˘1 for some i P RpsuffixpGarLpXkqqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`2qq.

Prefix-convergence of GarLpXkqkě0

Lemma: prefixpGarLpXkqq ď prefixpGarLpXk`1qq.

Suffix-convergence of GarRpXkqkě0

Lemma: for all α, β P B`

n ,

suffixpGarRpα∆βqq “ ∆.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Suffix-divergence of GarLpXkqkě0

Lemma: if YkYk`1 “ σiσi˘1 for some i P RpsuffixpGarLpXkqqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`2qq.

Prefix-convergence of GarLpXkqkě0

Lemma: prefixpGarLpXkqq ď prefixpGarLpXk`1qq.

Suffix-convergence of GarRpXkqkě0

Lemma: for all α, β P B`

n ,

suffixpGarRpα∆βqq “ ∆. Consequence: if Yk . . . Yk`npn`1q{2´1 “ ∆ then suffixpGarRpXℓqq “ ∆ for all ℓ ě k ` npn ` 1q{2.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Suffix-divergence of GarLpXkqkě0

Lemma: if YkYk`1 “ σiσi˘1 for some i P RpsuffixpGarLpXkqqq then suffixpGarLpXkqq ‰ suffixpGarLpXk`2qq.

Prefix-convergence of GarLpXkqkě0

Lemma: prefixpGarLpXkqq ď prefixpGarLpXk`1qq.

Suffix-convergence of GarRpXkqkě0

Lemma: for all α, β P B`

n ,

suffixpGarRpα∆βqq “ ∆. Consequence: if Yk . . . Yk`npn`1q{2´1 “ ∆ then suffixpGarRpXℓqq “ ∆ for all ℓ ě k ` npn ` 1q{2. ñ Also proves that GarLpXkqkě0 prefix-converges

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6;

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u. ñ There exist blocking braids if n is even: β1

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u. ñ There exist blocking braids if n is even: β1 β2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u. ñ There exist blocking braids if n is even: β1 β2

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u. ñ There exist blocking braids if n is even: β1 β2 β3

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u. ñ There exist blocking braids if n is even: β1 β2 β3 β4 β5 β6

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0

Key ingredient: blocking braid B “ β1β2β3β4β5β6 such that GarLpBq “ GarRpBq “ β1 ¨ β2 ¨ β3 ¨ β4 ¨ β5 ¨ β6; #Lpβ4q “ n ´ 2 and β1 “ β6 P tσ1, . . . , σn´1u. ñ There exist blocking braids if n is odd: β3 β2 β1 β4 β5 β6

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0 – continued

Lemma #1: for all α, β P B`

n , if GarRpαBq “ GarRpαq ¨ GarRpBq and

(αBβ is ∆-free or β P tσ1, . . . , σnu), then GarRpαBβq “ GarRpαq ¨ GarRpBβq.

β B α B β same suffixes

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0 – continued

Lemma #1: for all α, β P B`

n , if GarRpαBq “ GarRpαq ¨ GarRpBq and

(αBβ is ∆-free or β P tσ1, . . . , σnu), then GarRpαBβq “ GarRpαq ¨ GarRpBβq. Key ingredient #2: Cpxq “ #toccurrences of a blocking braid in GarRpxqu.

β B α B β same suffixes

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0 – continued

Lemma #1: for all α, β P B`

n , if GarRpαBq “ GarRpαq ¨ GarRpBq and

(αBβ is ∆-free or β P tσ1, . . . , σnu), then GarRpαBβq “ GarRpαq ¨ GarRpBβq. Key ingredient #2: Cpxq “ #toccurrences of a blocking braid in GarRpxqu. Lemma #2: for all α, β P B`

n ,

Cpαβq ď Cpαq ` Cpβq ` K (K = constant)

Uses: Lemma #1 Playing with magnets Tricky inductions

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0 – continued

Lemma #1: for all α, β P B`

n , if GarRpαBq “ GarRpαq ¨ GarRpBq and

(αBβ is ∆-free or β P tσ1, . . . , σnu), then GarRpαBβq “ GarRpαq ¨ GarRpBβq. Key ingredient #2: Cpxq “ #toccurrences of a blocking braid in GarRpxqu. Lemma #2: for all α, β P B`

n ,

Cpαβq ď Cpαq ` Cpβq ` K (K = constant) Lemma #3: ErCpXkqs “ Θpkq

Uses: Lemma #1 Playing with magnets Transience of pXkqkě0 in B`

n {x∆2y

Ugly calculations

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0 – continued

Lemma #1: for all α, β P B`

n , if GarRpαBq “ GarRpαq ¨ GarRpBq and

(αBβ is ∆-free or β P tσ1, . . . , σnu), then GarRpαBβq “ GarRpαq ¨ GarRpBβq. Key ingredient #2: Cpxq “ #toccurrences of a blocking braid in GarRpxqu. Lemma #2: for all α, β P B`

n ,

Cpαβq ď Cpαq ` Cpβq ` K (K = constant) Lemma #3: ErCpXkqs “ Θpkq Kingman subadditive lemma: CpXkq „ ErCpXkqs almost surely

Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in braid monoids

Prefix-convergence of GarRpXkqkě0 – continued

Lemma #1: for all α, β P B`

n , if GarRpαBq “ GarRpαq ¨ GarRpBq and

(αBβ is ∆-free or β P tσ1, . . . , σnu), then GarRpαBβq “ GarRpαq ¨ GarRpBβq. Key ingredient #2: Cpxq “ #toccurrences of a blocking braid in GarRpxqu. Lemma #2: for all α, β P B`

n ,

Cpαβq ď Cpαq ` Cpβq ` K (K = constant) Lemma #3: ErCpXkqs “ Θpkq Kingman subadditive lemma: CpXkq „ ErCpXkqs almost surely

ñ Mission accomplished!

Vincent Jugé & Jean Mairesse Stabilization of random braids

Contents

1

Introduction

2

Random walk in dimer monoids & groups

3

Random walk in braid monoids

4

Conclusion

Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

1 Limit of an infinite-state Markov chain with L1 factors;

M1 M2 M3 M4

. . .

L1 L1 L1 L1

Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

1 Limit of an infinite-state Markov chain with L1 factors; 2 Ergodic process; Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

1 Limit of an infinite-state Markov chain with L1 factors; 2 Ergodic process; 3 Finite penetration distance; Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

1 Limit of an infinite-state Markov chain with L1 factors; 2 Ergodic process; 3 Finite penetration distance; 4 Maximal linear convergence speed.

Stable prefix ˚ ∆ „ αn “ opnq „ βn

Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

1 Limit of an infinite-state Markov chain with L1 factors; 2 Ergodic process; 3 Finite penetration distance; 4 Maximal linear convergence speed.

Theorem [Folklore]

Computing GarRpXk`1q when knowing GarRpXkq and Yk in expected time Opkq.

Vincent Jugé & Jean Mairesse Stabilization of random braids

What is our limit object?

1 Limit of an infinite-state Markov chain with L1 factors; 2 Ergodic process; 3 Finite penetration distance; 4 Maximal linear convergence speed.

Theorem [Folklore + J. & M. 2016]

Computing GarRpXk`1q when knowing GarRpXkq and Yk in expected time opkq.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Seeking non-trivial limits

Delete ∆s

Unsatisfactory result: prefixpGarLpXkqqkě0 and suffixpGarRpXkqqkě0 must converge

(a.s. towards ∆)

Vincent Jugé & Jean Mairesse Stabilization of random braids

Seeking non-trivial limits

Delete ∆s

Unsatisfactory result: prefixpGarLpXkqqkě0 and suffixpGarRpXkqqkě0 must converge

(a.s. towards ∆)

Theorem [Folklore + J. & M. 2016]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗

• Vincent Jugé & Jean Mairesse

Stabilization of random braids

Seeking non-trivial limits

Delete ∆s

Unsatisfactory result: prefixpGarLpXkqqkě0 and suffixpGarRpXkqqkě0 must converge

(a.s. towards ∆)

Idea: move to B`

n {x∆y

(i.e. move ∆s to the right and delete them)

Theorem [Folklore + J. & M. 2016]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Vincent Jugé & Jean Mairesse Stabilization of random braids

Seeking non-trivial limits

Delete ∆s

Unsatisfactory result: prefixpGarLpXkqqkě0 and suffixpGarRpXkqqkě0 must converge

(a.s. towards ∆)

Idea: move to B`

n {x∆y

(i.e. move ∆s to the right and delete them)

Theorem [Folklore + J. & M. 2016]

Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Random walk in irreducible trace monoids

Theorem [Folklore] Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Vincent Jugé & Jean Mairesse Stabilization of random braids

This looks like déjà vu. . .

Random walk in irreducible trace groups

Theorem [Folklore] Convergence of the words GarLpXkqkě0 GarRpXkqkě0 prefix-

• suffix-

✗ ✗

Vincent Jugé & Jean Mairesse Stabilization of random braids

Vincent Jugé & Jean Mairesse Stabilization of random braids

From braid monoids to other monoids

Key ingredients

1 Length-preserving, left-right symmetric relations 2 Two-way Garside family F: closed under ď, ě, LCMď and LCMě 3 Positive escape rate Vincent Jugé & Jean Mairesse Stabilization of random braids

From braid monoids to other monoids

Key ingredients

1 Length-preserving, left-right symmetric relations 2 Two-way Garside family F: closed under ď, ě, LCMď and LCMě 3 Positive escape rate 4 Connectivity of the Charney graph G “ pS, Eq, where:

S “ tproper subsets of tσ1, . . . , σn´1uu and E “ tpX, Y q | Dβ P F s.t. X “ Lpβq and Y “ Rpβqu.

Vincent Jugé & Jean Mairesse Stabilization of random braids

From braid monoids to other monoids

Key ingredients

1 Length-preserving, left-right symmetric relations 2 Two-way Garside family F: closed under ď, ě, LCMď and LCMě 3 Positive escape rate 4 Connectivity of the Charney graph G “ pS, Eq, where:

S “ tproper subsets of tσ1, . . . , σn´1uu and E “ tpX, Y q | Dβ P F s.t. X “ Lpβq and Y “ Rpβqu. Extensions to other monoids with Garside families Finite family in dual braid monoids (of every spherical type)

Vincent Jugé & Jean Mairesse Stabilization of random braids

From braid monoids to other monoids

Key ingredients

1 Length-preserving, left-right symmetric relations 2 Two-way Garside family F: closed under ď, ě, LCMď and LCMě 3 Positive escape rate 4 Connectivity of the Charney graph G “ pS, Eq, where:

S “ tproper subsets of tσ1, . . . , σn´1uu and E “ tpX, Y q | Dβ P F s.t. X “ Lpβq and Y “ Rpβqu. Extensions to other monoids with Garside families Finite family in dual braid monoids (of every spherical type) Finite family in Artin–Tits monoids of FC type Infinite family in all Artin-Tits monoids

Vincent Jugé & Jean Mairesse Stabilization of random braids

From braid monoids to other monoids

Key ingredients

1 Length-preserving, left-right symmetric relations 2 Two-way Garside family F: closed under ě and LCMď 3 Positive escape rate 4 Connectivity of the Charney graph G “ pS, Eq, where:

S “ tproper subsets of tσ1, . . . , σn´1uu and E “ tpX, Y q | Dβ P F s.t. X “ Lpβq and Y “ Rpβqu. Extensions to other monoids with Garside families Finite family in dual braid monoids (of every spherical type) Finite family in Artin–Tits monoids of FC type Infinite family in all Artin-Tits monoids (without elements σ2

i )

Finite, one-way Garside family in all Artin–Tits monoids

Vincent Jugé & Jean Mairesse Stabilization of random braids

From (dual) braid monoids to groups

Which Garside normal form do we choose?

Vincent Jugé & Jean Mairesse Stabilization of random braids

From (dual) braid monoids to groups

Which Garside normal form do we choose?

Left Garside NF: α “ ∆uβ with β P B`

n in left NF;

Vincent Jugé & Jean Mairesse Stabilization of random braids

From (dual) braid monoids to groups

Which Garside normal form do we choose?

Left Garside NF: α “ ∆uβ with β P B`

n in left NF;

Left Garside NF∆: α “ β∆u with β P B`

n in left NF;

Vincent Jugé & Jean Mairesse Stabilization of random braids

From (dual) braid monoids to groups

Which Garside normal form do we choose?

Left Garside NF: α “ ∆uβ with β P B`

n in left NF;

Left Garside NF∆: α “ β∆u with β P B`

n in left NF;

Right Garside NF: α “ β∆u with β P B`

n in right NF;

Vincent Jugé & Jean Mairesse Stabilization of random braids

From (dual) braid monoids to groups

Which Garside normal form do we choose?

Left Garside NF: α “ ∆uβ with β P B`

n in left NF;

Left Garside NF∆: α “ β∆u with β P B`

n in left NF;

Right Garside NF: α “ β∆u with β P B`

n in right NF;

Symmetric Garside NF: α “ β´1γ with β, γ P B`

n in left NF.

Vincent Jugé & Jean Mairesse Stabilization of random braids

From (dual) braid monoids to groups

Which Garside normal form do we choose?

Left Garside NF: α “ ∆uβ with β P B`

n in left NF;

Left Garside NF∆: α “ β∆u with β P B`

n in left NF;

Right Garside NF: α “ β∆u with β P B`

n in right NF;

Symmetric Garside NF: α “ β´1γ with β, γ P B`

n in left NF.

ñ Similar or identical results in non-degenerate cases.

Vincent Jugé & Jean Mairesse Stabilization of random braids

Bibliography

• E. Artin, Theorie der Zöpfe, Math. Sem. Univ. Hamburg (1926)
• F. Garside, The braid group and other groups, Quart. J. Math. Oxford (1969)
• V. Kaimanovich, A. Vershik, Random Walks on discrete groups: boundary and entropy,
• Ann. Probab. (1983)
• W. Thurston, Finite state algorithms for the braid group, Circulated notes (1988)
• S. Gaubert, J. Mairesse, Task resource models and (max,+) automata, Cambridge

University Press (1998)

• A. Vershik, S. Nechaev, R. Bikbov, Statistical properties of braid groups in locally free

approximation, Comm. Math. Phys. (2000)

• A. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Russian
• Math. Surveys (2000)
• A. Malyutin, The Poisson-Furstenberg boundary of the locally free group, Journal of

Mathematical Sciences (2005)

• J. Mairesse, F. Mathéus, Randomly growing braid on three strands and the manta ray,
• Ann. Appl. Probab. (2007)
• A. Vershik, A. Malyutin, Boundaries of braid groups and the Markov-Ivanovsky normal

form, Izvestiya: Mathematics (2008)

• P. Dehornoy, F. Digne, E. Godelle, D. Krammer, J. Michel, Foundations of Garside theory

(2013)

• P. Dehornoy, M. Dyer, C. Hohlweg, Garside families in Artin–Tits monoids and low

elements in Coxeter groups, C. R. Math. (2015)

• V. Jugé, Combinatorics of braids, PhD Thesis (2016)

Vincent Jugé & Jean Mairesse Stabilization of random braids

Thank you very much for your attention! Barka Rahmat Asante Imela Danke German Igbo Mossi Swahili Tajik

Vincent Jugé & Jean Mairesse Stabilization of random braids

Thank you very much for your attention! Barka Rahmat Asante Imela Danke German Igbo Mossi Swahili Tajik Questions?

Vincent Jugé & Jean Mairesse Stabilization of random braids