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

Random walk in dimer monoids Random walk 1 Select i.i.d. generators p Y k q k ě 0 in t S 1 , . . . , S n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Does the random walk p X k q k ě 0 converge? Do the words Gar L p X k q k ě 0 and Gar R p X k q k ě 0 converge? Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Random walk 1 Select i.i.d. generators p Y k q k ě 0 in t S 1 , . . . , S n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Does the random walk p X k q k ě 0 converge? Do the words Gar L p X k q k ě 0 and Gar R p X k q k ě 0 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 p Y k q k ě 0 in t S 1 , . . . , S n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Does the random walk p X k q k ě 0 converge? Do the words Gar L p X k q k ě 0 and Gar R p X k q k ě 0 converge? Do their prefixes and suffixes converge? Theorem [Folklore] Convergence of the words Gar L p X k q k ě 0 Gar R p X k q k ě 0 prefix- � � suffix- ✗ ✗ Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in irreducible trace monoids Random walk 1 Select i.i.d. generators p Y k q k ě 0 in t S 1 , . . . , S n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Does the random walk p X k q k ě 0 converge? Do the words Gar L p X k q k ě 0 and Gar R p X k q k ě 0 converge? Do their prefixes and suffixes converge? Theorem [Folklore] Convergence of the words Gar L p X k q k ě 0 Gar R p X k q k ě 0 prefix- � � suffix- ✗ ✗ Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq S 4 S 3 S 1 S 4 S 1 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq MAGNET S 1 S 4 S 1 S 3 S 4 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq “ t S i : x ě S i u . MAGNET S 1 S 4 S 1 S 3 S 4 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq “ t S i : x ě S i u . Consequence : if Y k “ S i ˘ 1 for some S i P suffix p Gar L p X k qq then S 1 S 2 S 4 S 2 S 4 S 3 S 3 S 1 S 4 S 1 S 4 S 1 S 3 S 1 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq “ t S i : x ě S i u . Consequence : if Y k “ S i ˘ 1 for some S i P suffix p Gar L p X k qq then suffix p Gar L p X k qq ‰ suffix p Gar L p X k ` 1 qq ; S 1 distinct S 2 S 4 S 2 S 4 suffixes S 3 S 3 S 1 S 4 S 1 S 4 S 1 S 3 S 1 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq “ t S i : x ě S i u . Consequence : if Y k “ S i ˘ 1 for some S i P suffix p Gar L p X k qq then suffix p Gar L p X k qq ‰ suffix p Gar L p X k ` 1 qq ; suffix p Gar R p X k qq ‰ suffix p Gar R p X k ` 1 qq . MAGNET S 1 S 4 MAGNET distinct S 2 S 4 S 2 suffixes S 1 S 3 S 1 S 3 S 1 S 4 S 1 S 4 S 3 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq “ t S i : x ě S i u . Consequence : if Y k “ S i ˘ 1 for some S i P suffix p Gar L p X k qq then suffix p Gar L p X k qq ‰ suffix p Gar L p X k ` 1 qq ; suffix p Gar R p X k qq ‰ suffix p Gar R p X k ` 1 qq . Prefix-convergence of Gar L p X k q k ě 0 Lemma : for all x P M ` n , prefix p Gar L p x qq “ t S i : S i ď x u . Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Suffix-divergence Lemma : for all x P M ` n , suffix p Gar L p x qq Ď suffix p Gar R p x qq “ t S i : x ě S i u . Consequence : if Y k “ S i ˘ 1 for some S i P suffix p Gar L p X k qq then suffix p Gar L p X k qq ‰ suffix p Gar L p X k ` 1 qq ; suffix p Gar R p X k qq ‰ suffix p Gar R p X k ` 1 qq . Prefix-convergence of Gar L p X k q k ě 0 Lemma : for all x P M ` n , prefix p Gar L p x qq “ t S i : S i ď x u . Consequence : prefix p Gar L p X k qq Ď prefix p Gar L p X k ` 1 qq . Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Prefix-convergence of Gar R p X k q k ě 0 Key ingredient : blocking traces T “ S 1 S 2 . . . S n S n ´ 1 . . . S 1 S 1 S 2 S 3 S 4 S 3 S 2 S 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Prefix-convergence of Gar R p X k q k ě 0 Key ingredient : blocking traces T “ S 1 S 2 . . . S n S n ´ 1 . . . S 1 T Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Prefix-convergence of Gar R p X k q k ě 0 Key ingredient : blocking traces T “ S 1 S 2 . . . S n S n ´ 1 . . . S 1 T Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Prefix-convergence of Gar R p X k q k ě 0 Key ingredient : blocking traces T “ S 1 S 2 . . . S n S n ´ 1 . . . S 1 Lemma : for all x , y P M ` n , prefix p Gar R p x T y qq “ prefix p Gar R p x T qq . MAGNET y MAGNET T T same x x prefixes Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer monoids Prefix-convergence of Gar R p X k q k ě 0 Key ingredient : blocking traces T “ S 1 S 2 . . . S n S n ´ 1 . . . S 1 Lemma : for all x , y P M ` n , prefix p Gar R p x T y qq “ prefix p Gar R p x T qq . Consequence : if Y k . . . Y k ` 2 n ´ 2 “ T then prefix p Gar R p X k ` 2 n ´ 1 qq “ prefix p Gar R p X ℓ qq for all ℓ ě k ` 2 n ´ 1. MAGNET X ´ 1 k ` 2 n ´ 1 X ℓ MAGNET T T same X k X k prefixes Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Random walk 1 Select i.i.d. generators p Y k q k ě 0 in t S ˘ 1 1 , . . . , S ˘ 1 n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Random walk 1 Select i.i.d. generators p Y k q k ě 0 in t S ˘ 1 1 , . . . , S ˘ 1 n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Theorem [Folklore] Convergence of the words Gar L p X k q k ě 0 Gar R p X k q k ě 0 prefix- � � suffix- ✗ ✗ Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Random walk 1 Select i.i.d. generators p Y k q k ě 0 in t S ˘ 1 1 , . . . , S ˘ 1 n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Theorem [Folklore] Convergence of the words Gar L p X k q k ě 0 Gar R p X k q k ě 0 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 p Y k q k ě 0 in t S ˘ 1 1 , . . . , S ˘ 1 n u . 2 Random process p X k q k ě 0 defined by: X 0 “ 1 and X k ` 1 “ X k Y k . Theorem [Folklore] Convergence of the words Gar L p X k q k ě 0 Gar R p X k q k ě 0 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 Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 S 1 S 3 S 2 S 4 S 3 S 1 S 4 S 1 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 S 3 S 2 S 3 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 S 3 S 2 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 (escape rate in free groups) S 3 S 2 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 (escape rate in free groups) Lemma 2 : for all x P M n , if C i p x q ě 2, then prefix p Gar L p x qq “ prefix p Gar L p x S ˘ 1 qq . i S 3 S 1 S 3 shield S 2 S 4 S 3 S 1 S 4 S 1 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 (escape rate in free groups) Lemma 2 : for all x P M n , if C i p x q ě 2, then prefix p Gar L p x qq “ prefix p Gar L p x S ˘ 1 qq . i S 1 S 2 S 4 S 3 S 1 S 4 S 1 S 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar L p X k q k ě 0 Key ingredient : C i p x q “ # t occurrences of S ˘ 1 or S ˘ 1 i ` 1 in Gar L p x qu . i Lemma : min i C i p X k q Ñ `8 (escape rate in free groups) Lemma 2 : for all x P M n , if C i p x q ě 2, then prefix p Gar L p x qq “ prefix p Gar L p x S ˘ 1 qq . i S 1 S 2 S 4 S 3 S 1 S 4 prefix S 1 S 3 protected Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar L p X k qq “ t S 1 us ą 0 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar L p X k qq “ t S 1 us ą 0 After ℓ ě K steps, X ℓ with proba ą 0 stable prefix Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar L p X k qq “ t S 1 us ą 0 After ℓ ě K steps, X ℓ with proba ą 0 stable prefix T K ` 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar L p X k qq “ t S 1 us ą 0 After ℓ ě K steps, X ℓ with proba ą 0 stable prefix S ˘ 1 1 T K ` 1 S 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar L p X k qq “ t S 1 us ą 0 After ℓ ě K steps, X ℓ with proba ą 0 stable prefix S ˘ 1 1 T K ` 1 S 1 forever Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 After ℓ ě K steps, X ℓ with proba ą 0 stable prefix S ˘ 1 1 T K ` 1 MAGNET S 1 forever Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 X k Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 MAGNET S 1 suffix with proba ą 0 T S ˘ 1 1 X k Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 Consequence : prefix p Gar R p X k qq k ě 0 converges MAGNET τ 0 , τ 1 , . . . , τ k : X τ 2 k stopping times old prefix Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 Consequence : prefix p Gar R p X k qq k ě 0 converges MAGNET τ 0 , τ 1 , . . . , τ k : X τ 2 k ` 1 stopping times new prefix Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 Consequence : prefix p Gar R p X k qq k ě 0 converges MAGNET S 1 τ 0 , τ 1 , . . . , τ k : X τ 2 k ` 2 stopping times Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 Consequence : prefix p Gar R p X k qq k ě 0 converges MAGNET X ´ 1 τ 2 k ` 2 X ℓ with proba ą 0 S 1 forever S 1 τ 0 , τ 1 , . . . , τ k : X τ 2 k ` 2 stopping times Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 Consequence : prefix p Gar R p X k qq k ě 0 converges MAGNET X ´ 1 τ 2 k ` 2 X ℓ with proba ą 0 S 1 forever S 1 τ 0 , τ 1 , . . . , τ k : X τ 2 k ` 2 stopping times stable prefix Vincent Jugé & Jean Mairesse Stabilization of random braids

Random walk in dimer groups Prefix-convergence of Gar R p X k q k ě 0 Lemma : P r@ k ě 1 , prefix p Gar R p X k qq “ t S 1 us ą 0 Lemma 2 : # t k ě 1 : suffix p Gar R p X k qq “ t S 1 uu “ `8 Consequence : prefix p Gar R p X k qq k ě 0 converges MAGNET τ 2 k ` 3 “ `8 X ´ 1 τ 2 k ` 2 X ℓ with proba ą 0 S 1 forever S 1 τ 0 , τ 1 , . . . , τ k : X τ 2 k ` 2 stopping times stable prefix Vincent Jugé & Jean Mairesse Stabilization of random braids

Contents Introduction 1 Random walk in dimer monoids & groups 2 Random walk in braid monoids 3 Braid monoids & braid groups Random walk in braid monoids Conclusion 4 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 σ 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 σ 1 σ 1 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 Artin braid diagrams Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 1 σ 2 σ 1 σ 3 σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 Artin braid diagrams Vincent Jugé & Jean Mairesse Stabilization of random braids

Do you like braiding your hair clockwise? σ 2 σ 1 σ 2 σ 3 σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 Artin braid diagrams Vincent Jugé & Jean Mairesse Stabilization of random braids