Drawing heaps uniformly at random Samy Abbes 1 , Sbastien Gouzel 2,3 - PowerPoint PPT Presentation

Heaps of pieces and trace monoids Dimer monoid Heap of pieces Alphabet: Pieces: a b c d Σ “t a , b , c , d u Free monoid: Vertical heaps: Σ ˚ “t 1 , a , b , c , d , a 2 , ab , ac , ad , ba ,... u d Independence relation: a d a c I “tp a , c q , p c , a q , p a , d q , p d , a q , p b , d q , p d , b qu a c b Dimer monoid: Horizontal layout: M p Σ , I q“x a , b , c , d | ac “ ca , ad “ da , bd “ db y ` a b a c b d c c Dependency graph S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces viewed from their places Heap of pieces Petri net Vertical heaps of pieces: a c c a b d a d a c a c b b c d S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces viewed from their places Heap of pieces Petri net Vertical heaps of pieces: a c c a b d 1 1 a d a c a c b b 1 2 3 1 2 3 2 2 Place views: c a 3 3 b c c c c a b d a c d d a c c b b c 1 2 3 1 2 3 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces viewed from their places Heap of pieces Petri net Vertical heaps of pieces: a c c a b d 1 1 a d a c a c b b 1 2 3 1 2 3 2 2 Place views: c a 3 3 b c c c c a b d a c d d a c c b b c 1 2 3 1 2 3 Heap of pieces ô Consistent place views S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces viewed from their places Heap of pieces Petri net Vertical heaps of pieces: a 1 1 b 1 2 3 1 2 3 2 2 Place views: c 3 3 d a ? c 1 2 3 1 2 3 Heap of pieces ô Consistent place views S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces viewed from their places Heap of pieces Petri net Vertical heaps of disconnected pieces: a ˝ a ‚ c d d d 1 4 a ˝ a ‚ a ˝ a ‚ a ˝ a ‚ c b a c 1 2 3 4 1 2 3 4 Place views: b 2 3 a a d a d a a c c d a c c a b b d a 1 2 3 4 1 2 3 4 Heap of pieces ô Consistent place views S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces viewed from their places Heap of pieces Petri net Vertical heaps of disconnected pieces: d 1 4 a c 1 2 3 4 1 2 3 4 Place views: b 2 3 a b c d b c d a 1 2 3 4 1 2 3 4 Heap of pieces ô Consistent place views S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and Cartier-Foata normal forms Heap of pieces Vertical heaps of pieces: a c c b d a d a c a c b S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and Cartier-Foata normal forms Heap of pieces Vertical heaps of pieces: a c c b d a d a c a c b Cartier-Foata factorisations: ac c b d ad ac ac b S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and Cartier-Foata normal forms Cliques ( C ) Heap of pieces Horizontal heaps: Vertical heaps of pieces: a b a c c b d c d a d a c a c a d a c b b d Cartier-Foata factorisations: ac c b d ad ac ac b S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and Cartier-Foata normal forms Cliques ( C ) Heap of pieces Horizontal heaps: Vertical heaps of pieces: a b a c c b d c d a d a c a c a d a c b b d Cartier-Foata factorisations: ac c b d Local conditions on ad ac consecutive cliques in heaps ac b S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and left divisibility Heap of pieces a c b b ď a a d a c a c S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and left divisibility Heap of pieces Place views a a c b b c c b b ď ď a b a b d a a d a c c a c c a c a c S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and left divisibility Heap of pieces Place views a a c b b c c b b ď ď a b a b d a a d a c c a c c a c a c Cartier-Foata ď 1 ac ď b b ď a ad ď ac ac + upper commutativity ( bd P C ) S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and left divisibility Heap of pieces Place views a a c b b c c b b ď ď a b a b d a a d a c c a c c a c a c Cartier-Foata Combinatorial properties a ^ b (and a _ b ) exist ď 1 ac h p a q ď k ô a P C k ď b b ď a ad maximality criterion: ď ac ac + upper commutativity a ( bd P C ) S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and left divisibility Heap of pieces Place views a a c b b c c b b ď ď a b a b d a a d a c c a c c a c a c Cartier-Foata Combinatorial properties a ^ b (and a _ b ) exist ď 1 ac h p a q ď k ô a P C k ď b b ď a ad maximality criterion: ď ac ac ℓ . . . + upper commutativity Ž t x ď a : h p x q ď k u ( bd P C ) k S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Heaps of pieces and left divisibility Heap of pieces Place views a a c b b c c b b ď ď a b a b d a a d a c c a c c a c a c Cartier-Foata Combinatorial properties a ^ b (and a _ b ) exist ď 1 ac h p a q ď k ô a P C k ď b b ď a ad maximality criterion: ď ac ac ℓ . . . + upper commutativity C k p a q ( bd P C ) k S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Contents Introduction 1 Trace monoids and heaps 2 First convergence results 3 Bernoulli distributions 4 Going beyond. . . 5 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Probabilistic and topological setting Probabilistic setting Two notions of length: 1 # pieces: | a | 2 # floors: h p a q M k “ t heaps of size k u M k « regular language Ď Σ ˚ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Probabilistic and topological setting Topological setting Probabilistic setting µ k Ý Ñ µ 8 ô P µ k r a ď x s Ñ P µ 8 r a ď x s Two notions of length: 1 # pieces: | a | 2 # floors: h p a q M k “ t heaps of size k u M k « regular language Ď Σ ˚ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Probabilistic and topological setting Topological setting Probabilistic setting µ k Ý Ñ µ 8 ô P µ k r a ď x s Ñ P µ 8 r a ď x s Two notions of length: w Ý Ñ µ 8 ô µ k pò a q Ñ µ 8 pò a q 1 # pieces: | a | µ k w µ k Ý Ñ µ 8 ô with ò a “ t x : a ď x u 2 # floors: h p a q M k “ t heaps of size k u M k « regular language Ď Σ ˚ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Probabilistic and topological setting Topological setting Probabilistic setting µ k Ý Ñ µ 8 ô P µ k r a ď x s Ñ P µ 8 r a ď x s Two notions of length: w Ý Ñ µ 8 ô µ k pò a q Ñ µ 8 pò a q 1 # pieces: | a | µ k w µ k Ý Ñ µ 8 ô with ò a “ t x : a ď x u 2 # floors: h p a q Embed M ` with the topology tò a u M k “ t heaps of size k u Make M ` complete M k « regular language Ď Σ ˚ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Probabilistic and topological setting Topological setting Probabilistic setting w Ý Ñ µ 8 ô P µ k r a ď x s Ñ P µ 8 r a ď x s Two notions of length: µ k w Ý Ñ µ 8 ô µ k pò a q Ñ µ 8 pò a q 1 # pieces: | a | µ k w µ k Ý Ñ µ 8 ô with ò a “ t x : a ď x u 2 # floors: h p a q Embed M ` with the topology tò a u M k “ t heaps of size k u Make M ` complete M k « regular language Ď Σ ˚ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Probabilistic and topological setting Topological setting Probabilistic setting w Ý Ñ µ 8 ô P µ k r a ď x s Ñ P µ 8 r a ď x s Two notions of length: µ k w Ý Ñ µ 8 ô µ k pò a q Ñ µ 8 pò a q 1 # pieces: | a | µ k w µ k Ý Ñ µ 8 ô with ò a “ t x : a ď x u 2 # floors: h p a q Embed M ` with the topology tò a u M k “ t heaps of size k u Make M ` complete M k « regular language Ď Σ ˚ Theorem (S. Abbes & J. Mairesse 2015) The uniform distribution on M k converges weakly in M ` when k Ñ `8 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof α P M ` z | α | ¨ ř γ P C p´ z q | γ | G p z q H p z q “ ř S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof γ P C p´ 1 q | γ | z | γα | G p z q H p z q “ ř ř α P M ` S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof γ P C p´ 1 q | γ | z | γα | G p z q H p z q “ ř ř α P M ` γ P C 1 γ ď θ p´ 1 q | γ | z | θ | “ ř ř θ P M ` where θ “ γα S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof γ P C p´ 1 q | γ | z | γα | G p z q H p z q “ ř ř α P M ` γ P C 1 γ ď θ p´ 1 q | γ | z | θ | “ ř ř θ P M ` θ P M ` z | θ | ř S Ď L p θ q p´ 1 q | S | “ ř where θ “ γα , L p θ q “ t x P Σ : x ď θ u and γ “ Ž S S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof γ P C p´ 1 q | γ | z | γα | G p z q H p z q “ ř ř α P M ` γ P C 1 γ ď θ p´ 1 q | γ | z | θ | “ ř ř θ P M ` θ P M ` z | θ | 1 L p θ q“H “ ř where θ “ γα , L p θ q “ t x P Σ : x ď θ u and γ “ Ž S S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof γ P C p´ 1 q | γ | z | γα | G p z q H p z q “ ř ř α P M ` γ P C 1 γ ď θ p´ 1 q | γ | z | θ | “ ř ř θ P M ` θ P M ` z | θ | 1 θ “ 1 “ ř where θ “ γα , L p θ q “ t x P Σ : x ď θ u and γ “ Ž S S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Proof γ P C p´ 1 q | γ | z | γα | G p z q H p z q “ ř ř α P M ` γ P C 1 γ ď θ p´ 1 q | γ | z | θ | “ ř ř θ P M ` θ P M ` z | θ | 1 θ “ 1 “ 1 “ ř where θ “ γα , L p θ q “ t x P Σ : x ď θ u and γ “ Ž S S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Generating series and Möbius polynomial α P M ` z | α | “ ř k ě 0 λ k z k and H p z q “ ř γ P C p´ z q | γ | G p z q “ ř Proposition (P. Cartier & D. Foata 1969) G p z q H p z q “ 1 Corollary (D. Krob, J. Mairesse & I. Michos 2001) H p z q has a smallest positive root p such that: p H p z q “ 0 ^ | z | ď p q ô z “ p 0 ă p ď 1 and there exists constants Λ ą 0 and ℓ P N such that λ k „ Λ p ´ k k ℓ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k 2 x ÞÑ ax maps M k to pò a q X M k `| a | bijectively S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k 2 x ÞÑ ax maps M k to pò a q X M k `| a | bijectively λ k ´| a | Ñ p ´| a | 3 µ k pò a q “ λ k S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k 2 x ÞÑ ax maps M k to pò a q X M k `| a | bijectively λ k ´| a | Ñ p ´| a | 3 µ k pò a q “ λ k 4 M ` is compact and tHu Y tò a u is closed under X S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | and length p a q “ h p a q Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k 2 x ÞÑ ax maps M k to pò a q X M k `| a | bijectively λ k ´| a | Ñ p ´| a | 3 µ k pò a q “ λ k 4 M ` is compact and tHu Y tò a u is closed under X Proof of the theorem – length p a q “ h p a q 5 Split ò a into sets M ` p b q “ t x : b “ C h p a q p x qu S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | and length p a q “ h p a q Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k 2 x ÞÑ ax maps M k to pò a q X M k `| a | bijectively λ k ´| a | Ñ p ´| a | 3 µ k pò a q “ λ k 4 M ` is compact and tHu Y tò a u is closed under X Proof of the theorem – length p a q “ h p a q 5 Split ò a into sets M ` p b q “ t x : b “ C h p a q p x qu b k ℓ b for some Λ b , q b and ℓ b 6 Prove that # p M ` p b q X M k q „ Λ b q k S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Weak convergence: length p a q “ | a | and length p a q “ h p a q Proof of the theorem – length p a q “ | a | 1 µ k : S ÞÑ # p S X M k q λ k 2 x ÞÑ ax maps M k to pò a q X M k `| a | bijectively λ k ´| a | Ñ p ´| a | 3 µ k pò a q “ λ k 4 M ` is compact and tHu Y tò a u is closed under X Proof of the theorem – length p a q “ h p a q 5 Split ò a into sets M ` p b q “ t x : b “ C h p a q p x qu b k ℓ b for some Λ b , q b and ℓ b 6 Prove that # p M ` p b q X M k q „ Λ b q k 7 Complete the proof as above Caution: lim µ k pò a q does not depend only on h p a q ! S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Contents Introduction 1 Trace monoids and heaps 2 First convergence results 3 Bernoulli distributions 4 Going beyond. . . 5 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions A distribution µ on M ` is . . . Bernoulli if µ pò ab q “ µ pò a q µ pò b q S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions A distribution µ on M ` is . . . Bernoulli if µ pò ab q “ µ pò a q µ pò b q µ pò a 1 a 2 . . . a k q “ ν a 1 ν a 2 . . . ν a k S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions A distribution µ on M ` is . . . Bernoulli if µ pò ab q “ µ pò a q µ pò b q µ pò a 1 a 2 . . . a k q “ ν a 1 ν a 2 . . . ν a k Uniform Bernoulli if µ pò a q “ ν | a | ν 1 “ ν 2 “ . . . “ ν S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions A distribution µ on M ` is . . . Bernoulli if µ pò ab q “ µ pò a q µ pò b q µ pò a 1 a 2 . . . a k q “ ν a 1 ν a 2 . . . ν a k Uniform Bernoulli with parameter ν if µ pò a q “ ν | a | ν 1 “ ν 2 “ . . . “ ν S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions A distribution µ on M ` is . . . Bernoulli if µ pò ab q “ µ pò a q µ pò b q µ pò a 1 a 2 . . . a k q “ ν a 1 ν a 2 . . . ν a k Uniform Bernoulli with parameter ν if µ pò a q “ ν | a | ν 1 “ ν 2 “ . . . “ ν Finite uniform Bernoulli if ν ă p H p z q ą 0 for all z P p 0 , p q S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions A distribution µ on M ` is . . . Bernoulli if µ pò ab q “ µ pò a q µ pò b q µ pò a 1 a 2 . . . a k q “ ν a 1 ν a 2 . . . ν a k Uniform Bernoulli with parameter ν if µ pò a q “ ν | a | ν 1 “ ν 2 “ . . . “ ν Finite uniform Bernoulli if ν ă p H p z q ą 0 for all z P p 0 , p q µ pB M ` q “ 0 µ pt a uq “ H p ν q ν | a | S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µ pt x uq “ S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µ pt x uq “ ν pò x q x S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µ pt x uq “ ν pò x q´ ν pò x a q ´ ν pò x b q ´ ν pò x c q ´ ν pò x d q x c x a x d x b x S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µ pt x uq “ ν pò x q´ ν pò x a q ´ ν pò x b q ´ ν pò x c q ´ ν pò x d q` ν pò x ac q ` ν pò x ad q ` ν pò x bd q x ac x ad x bd x c x a x d x b x S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µ pt x uq “ ν | x | p 1 ´ ν | a | ´ ν | b | ´ ν | c | ´ ν | d | ` ν | ac | ` ν | ad | ` ν | bd | q x ac x ad x bd x c x a x d x b x S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Bernoulli distributions Proving that finite uniform ô µ pt x uq “ H p ν q ν | x | γ ν | x γ | “ ν | x | H p ν q ř γ ν | γ | “ ν | x | H p ν q G p ν q ð µ pò x q “ H p ν q ř ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µ pt x uq “ ν | x | p 1 ´ ν | a | ´ ν | b | ´ ν | c | ´ ν | d | ` ν | ac | ` ν | ad | ` ν | bd | q “ ν | x | H p ν q x ac x ad x bd x c x a x d x b x S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Simulating finite, uniform Bernoulli distributions S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Simulating finite, uniform Bernoulli distributions Approach #1: Pick the length first Pick a target length k with probability λ k ν k H p ν q Pick a trace uniformly at random in t a P M ` | | a | “ k u S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Simulating finite, uniform Bernoulli distributions Approach #1: Pick the length first Pick a target length k with probability λ k ν k H p ν q Pick a trace uniformly at random in t a P M ` | | a | “ k u Approach #2: Pick the ground floor first Order the generators from g 1 to g n and choose whether g i ď a Pick the upper floors recursively S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Simulating finite, uniform Bernoulli distributions Approach #1: Pick the length first Pick a target length k with probability λ k ν k H p ν q Pick a trace uniformly at random in t a P M ` | | a | “ k u Approach #2: Pick the ground floor first Order the generators from g 1 to g n and choose whether g i ď a (based on t g j | 1 ď j ă i , g j ď a u and on the previous floor) Pick the upper floors recursively S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Simulating Bernoulli distributions Approach #1: Pick the length first Pick a target length k with probability λ k ν k H p ν q Pick a trace uniformly at random in t a P M ` | | a | “ k u Approach #2: Pick the ground floor first (Markov chain) Order the generators from g 1 to g n and choose whether g i ď a (based on t g j | 1 ď j ă i , g j ď a u and on the previous floor) Pick the upper floors recursively S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 Ť a ď b , h p a q“ h p b q Ò b S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 Ť a ď b , h p a q“ h p b q Ò b ν | a | “ ř µ pÒ b q 1 a ď b , h p a q“ h p b q S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | a γ | µ pÒ a q “ ř Ť a ď b , h p a q“ h p b q Ò b ν | a | “ ř µ pÒ b q 1 a ď b , h p a q“ h p b q S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | a γ | µ pÒ a q “ ř Ť a ď b , h p a q“ h p b q Ò b ν | a | “ ř µ pÒ b q 1 a ď b , h p a q“ h p b q “ ν | a | ř γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | γ | S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | a γ | µ pÒ a q “ ř Ť a ď b , h p a q“ h p b q Ò b ν | a | “ ř µ pÒ b q 1 a ď b , h p a q“ h p b q “ ν | a | ř γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | γ | “ ν | a | H a p ν q S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | a γ | µ pÒ a q “ ř Ť a ď b , h p a q“ h p b q Ò b ν | a | “ ř µ pÒ b q 1 a ď b , h p a q“ h p b q “ ν | a | ř γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | γ | “ ν | a | H a h p a q p ν q a h p a q . . . a = a 2 a 1 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains Monoid cylinder Cartier-Foata cylinder ò a “ t b P M ` | a ď b u Ò a “ t b P M ` | a “ C h p a q p b qu Möbius inversion formula and Markov simulation Ò a “ 9 γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | a γ | µ pÒ a q “ ř Ť a ď b , h p a q“ h p b q Ò b ν | a | “ ř µ pÒ b q 1 a ď b , h p a q“ h p b q “ ν | a | ř γ P C p´ 1 q | γ | 1 h p a q“ h p a γ q ν | γ | “ ν | a | H a h p a q p ν q “ P r Θ ν 1 “ a 1 , . . . , Θ ν h p a q “ a h p a q s 1 “ a s “ ν | a | H a p ν q P r Θ ν a h p a q i “ a s “ ν | b | H b p ν q P r Θ ν i ` 1 “ b | Θ ν H a p ν q 1 a Ñ b . . . a = a 2 b ô a “ C 1 p ab q a Ñ b ô a 1 a S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p Convergence of p Θ ν i q when ν Ñ p , with limit P r Θ p 1 “ a s “ p | a | H a p p q P r Θ p i ` 1 “ b | Θ p i “ a s “ p | b | H b p p q H a p p q 1 a Ñ b 1 H a p p q‰ 0 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p Convergence of p Θ ν i q when ν Ñ p , with limit P r Θ p 1 “ a s “ p | a | H a p p q P r Θ p i ` 1 “ b | Θ p i “ a s “ p | b | H b p p q H a p p q 1 a Ñ b 1 H a p p q‰ 0 Trivial supercritical parameter: ν “ 1 Possible only if M ` “ N n (i.e. p “ 1). . . S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p Convergence of p Θ ν i q when ν Ñ p , with limit P r Θ p 1 “ a s “ p | a | H a p p q P r Θ p i ` 1 “ b | Θ p i “ a s “ p | b | H b p p q H a p p q 1 a Ñ b 1 H a p p q‰ 0 Trivial supercritical parameter: ν “ 1 Possible only if M ` “ N n (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p Convergence of p Θ ν i q when ν Ñ p , with limit P r Θ p 1 “ a s “ p | a | H a p p q P r Θ p i ` 1 “ b | Θ p i “ a s “ p | b | H b p p q H a p p q 1 a Ñ b 1 H a p p q‰ 0 Trivial supercritical parameter: ν “ 1 Possible only if M ` “ N n (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! Consider the Garside matrix M ν with M ν a , b “ 1 a Ñ b ν | b | S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p Convergence of p Θ ν i q when ν Ñ p , with limit P r Θ p 1 “ a s “ p | a | H a p p q P r Θ p i ` 1 “ b | Θ p i “ a s “ p | b | H b p p q H a p p q 1 a Ñ b 1 H a p p q‰ 0 Trivial supercritical parameter: ν “ 1 Possible only if M ` “ N n (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! Consider the Garside matrix M ν with M ν a , b “ 1 a Ñ b ν | b | 1 ě µ p M ` q “ H p ν q G p ν q ě 0, hence H p ν q “ 0 and µ pB M ` q “ 1 S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random

Infinite uniform Bernoulli distributions as Markov chains Critical parameter: ν “ p Convergence of p Θ ν i q when ν Ñ p , with limit P r Θ p 1 “ a s “ p | a | H a p p q P r Θ p i ` 1 “ b | Θ p i “ a s “ p | b | H b p p q H a p p q 1 a Ñ b 1 H a p p q‰ 0 Trivial supercritical parameter: ν “ 1 Possible only if M ` “ N n (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! Consider the Garside matrix M ν with M ν a , b “ 1 a Ñ b ν | b | 1 ě µ p M ` q “ H p ν q G p ν q ě 0, hence H p ν q “ 0 and µ pB M ` q “ 1 M ν and M p are stochastic Perron matrices if M ` is irreducible S. Abbes , S. Gouëzel, V. Jugé & J. Mairesse Drawing heaps uniformly at random