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

Drawing heaps uniformly at random

Samy Abbes1, Sébastien Gouëzel2,3, Vincent Jugé2,4 & Jean Mairesse2,5

1: Paris 7 (IRIF) — 2: CNRS — 3: Nantes (LMJL) — 4: ENS Cachan (LSV) — 5: Paris 6 (LIP6)

25/05/2016

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Contents

1

Introduction

2

Trace monoids and heaps

3

First convergence results

4

Bernoulli distributions

5

Going beyond. . .

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with S “ pwx ` yqω ` pwx ` yq˚wzω, and w “ ac ` ca, x “ bd ` db, y “ cdab, z “ badc

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with S “ pwx ` yqω ` pwx ` yq˚wzω, and w “ ac ` ca, x “ bd ` db, y “ cdab, z “ badc Set of infinite concurrent executions: S Ď Σω{ ”, with

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with S “ pwx ` yqω ` pwx ` yq˚wzω, and w “ ac ` ca, x “ bd ` db, y “ cdab, z “ badc Set of infinite concurrent executions: S Ď Σω{ ”, with ac ” ca, ad ” da, bd ” db

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with S “ pwx ` yqω ` pwx ` yq˚wzω, and w “ ac ` ca, x “ bd ` db, y “ cdab, z “ badc Set of infinite concurrent executions: S Ď Σω{ ”, with uv ” vu ô p ‚u Y u‚ q X p ‚v Y v‚ q “ H

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a b c d Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with S “ pwx ` yqω ` pwx ` yq˚wzω, and w “ ac ` ca, x “ bd ` db, y “ cdab, z “ badc Set of infinite concurrent executions: S Ď Σω{ ”, with uv ” vu ô p ‚u Y u‚ q X p ‚v Y v‚ q “ H Independence relation: I “ tpu, vq | p ‚u Y u‚ q X p ‚v Y v‚ q “ Hu I “ tpa, cq, pc, aq, pa, dq, pd, aq, pb, dq, pd, bqu

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite one-bounded Petri net with . . . a d b c Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S Ď Σω, with S “ pwx ` yqω ` pwx ` yq˚wzω, and w “ ac ` ca, x “ bd ` db, y “ cdab, z “ badc Set of infinite concurrent executions: S Ď Σω{ ”, with uv ” vu ô p ‚u Y u‚ q X p ‚v Y v‚ q “ H Independence relation: I “ tpu, vq | p ‚u Y u‚ q X p ‚v Y v‚ q “ Hu I “ tpa, cq, pc, aq, pa, dq, pd, aq, pb, dq, pd, bqu

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net with . . . a d b c Set of transitions: Σ “ ta, b, c, du

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net with . . . a d b c Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S “ Σω

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net with . . . a d b c Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S “ Σω Set of infinite concurrent executions: S “ Σω{ ”, with uv ” vu ô u‚ X v‚ “ H

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net with . . . a d b c Set of transitions: Σ “ ta, b, c, du Set of infinite sequential executions: S “ Σω Set of infinite concurrent executions: S “ Σω{ ”, with uv ” vu ô u‚ X v‚ “ H Independence relation: I “ tpu, vq | p ‚u Y u‚ q X p ‚v Y v‚ q “ Hu I “ tpa, cq, pc, aq, pa, dq, pd, aq, pb, dq, pd, bqu

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net . . . with a b c d Can you pick one of its infinite concurrent exectutions uniformly at random?

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net . . . with a b c d Can you pick one of its infinite concurrent exectutions uniformly at random?

1 Define your preferred notion of trace length

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net . . . with a b c d Can you pick one of its infinite concurrent exectutions uniformly at random?

1 Define your preferred notion of trace length 2 Study uniform distributions on traces of length k

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Petri nets and dependency graphs

Consider your favorite full, symmetric one-bounded Petri net . . . with a b c d Can you pick one of its infinite concurrent exectutions uniformly at random?

1 Define your preferred notion of trace length 2 Study uniform distributions on traces of length k 3 Look for suitable convergence properties when

k Ñ `8

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Contents

1

Introduction

2

Trace monoids and heaps

3

First convergence results

4

Bernoulli distributions

5

Going beyond. . .

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and trace monoids

Heap of pieces Pieces: a b c d Trace monoid Alphabet:

Σ“ta,b,c,du

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and trace monoids

Heap of pieces Pieces: a b c d Purely vertical heaps: a c a d b c d a c b d c d a a b Trace monoid Alphabet:

Σ“ta,b,c,du

Free monoid:

Σ˚“t1,a,b,c,d,a2,ab,ac,ad,ba,...u

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and trace monoids

Heap of pieces Pieces: a b c d Purely vertical heaps: a c a d b c d a c b d c d a a b Horizontal layout: a b c c Trace monoid Alphabet:

Σ“ta,b,c,du

Free monoid:

Σ˚“t1,a,b,c,d,a2,ab,ac,ad,ba,...u

Independence relation:

I“tpa,cq,pc,aq,pa,dq,pd,aq,pb,dq,pd,bqu

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and trace monoids

Heap of pieces Pieces: a b c d Vertical heaps: a c a d b c d a Horizontal layout: a b c c Trace monoid Alphabet:

Σ“ta,b,c,du

Free monoid:

Σ˚“t1,a,b,c,d,a2,ab,ac,ad,ba,...u

Independence relation:

I“tpa,cq,pc,aq,pa,dq,pd,aq,pb,dq,pd,bqu

Trace monoid:

MpΣ,Iq“xa,b,c,d|ac“ca,ad“da,bd“dby`

a d b c Dependency graph

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and trace monoids

Heap of pieces Pieces: a b c d Vertical heaps: a c a d b c d a Horizontal layout: a b c c Trace monoid Alphabet:

Σ“ta,b,c,du

Free monoid:

Σ˚“t1,a,b,c,d,a2,ab,ac,ad,ba,...u

Independence relation:

I“tpa,cq,pc,aq,pa,dq,pd,aq,pb,dq,pd,bqu

Trace monoid:

MpΣ,Iq“xa,b,c,d|ac“ca,ad“da,bd“dby`

a d b c Dependency graph

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and trace monoids

Heap of pieces Pieces: a b c d Vertical heaps: a c a d b c d a Horizontal layout: a b c c Dimer monoid Alphabet:

Σ“ta,b,c,du

Free monoid:

Σ˚“t1,a,b,c,d,a2,ab,ac,ad,ba,...u

Independence relation:

I“tpa,cq,pc,aq,pa,dq,pd,aq,pb,dq,pd,bqu

Dimer monoid:

MpΣ,Iq“xa,b,c,d|ac“ca,ad“da,bd“dby`

a d b c Dependency graph

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces viewed from their places

Petri net a b c d Heap of pieces Vertical heaps of pieces: a c a d b a c b a c d c

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces viewed from their places

Petri net

1 2 3 1 2 3

a b c d Heap of pieces Vertical heaps of pieces:

1 2 3

a c a d b a c

1 2 3

b a c d c Place views:

1 2 3 1 2 3

a a b a c b c c d c b a b c c c d c

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces viewed from their places

Petri net

1 2 3 1 2 3

a b c d Heap of pieces Vertical heaps of pieces:

1 2 3

a c a d b a c

1 2 3

b a c d c Place views:

1 2 3 1 2 3

a a b a c b c c d c b a b c c c d c 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

Petri net

1 2 3 1 2 3

a b c d Heap of pieces Vertical heaps of pieces:

1 2 3 1 2 3

Place views:

1 2 3 1 2 3

a ? c 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

Petri net

1 2 4 3

a b c d Heap of pieces Vertical heaps of disconnected pieces:

1 2 3 4

a˝ c a‚ a˝ a‚ d a˝ a‚

1 2 3 4

b a˝ a‚ d c Place views:

1 2 3 4 1 2 3 4

a a a c c d a a d a b a b c d c a d 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

Petri net

1 2 4 3

a b c d Heap of pieces Vertical heaps of disconnected pieces:

1 2 3 4 1 2 3 4

Place views:

1 2 3 4 1 2 3 4

a b c d b c d a 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 a d b a c b a c d c

• 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 a d b a c b a c d c Cartier-Foata factorisations: ac ad b ac b ac d c

• 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 a d b a c b a c d c Cartier-Foata factorisations: ac ad b ac b ac d c Cliques (C) Horizontal heaps: a b c d a c a d b d

• 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 a d b a c b a c d c Cartier-Foata factorisations: ac ad b ac b ac d c Cliques (C) Horizontal heaps: a b c d a c a d b d Local conditions on consecutive cliques in heaps

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and left divisibility

Heap of pieces a c a b ď a c a b d a c

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and left divisibility

Heap of pieces a c a b ď a c a b d a c Place views a a b c b c ď a a b a c b c c d c

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and left divisibility

Heap of pieces a c a b ď a c a b d a c Place views a a b c b c ď a a b a c b c c d c Cartier-Foata ac a b 1 ď ď ď ď ac ad b 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 a c a b ď a c a b d a c Place views a a b c b c ď a a b a c b c c d c Cartier-Foata ac a b 1 ď ď ď ď ac ad b ac + upper commutativity (bd P C) Combinatorial properties a ^ b (and a _ b) exist hpaq ď k ô a P Ck maximality criterion: a

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and left divisibility

Heap of pieces a c a b ď a c a b d a c Place views a a b c b c ď a a b a c b c c d c Cartier-Foata ac a b 1 ď ď ď ď ac ad b ac + upper commutativity (bd P C) Combinatorial properties a ^ b (and a _ b) exist hpaq ď k ô a P Ck maximality criterion: k ℓ . . . Žtx ď a : hpxq ď ku

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Heaps of pieces and left divisibility

Heap of pieces a c a b ď a c a b d a c Place views a a b c b c ď a a b a c b c c d c Cartier-Foata ac a b 1 ď ď ď ď ac ad b ac + upper commutativity (bd P C) Combinatorial properties a ^ b (and a _ b) exist hpaq ď k ô a P Ck maximality criterion: k ℓ . . . Ckpaq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Contents

1

Introduction

2

Trace monoids and heaps

3

First convergence results

4

Bernoulli distributions

5

Going beyond. . .

• 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: hpaq

Mk “ theaps of size ku Mk « regular language Ď Σ˚

• 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: hpaq

Mk “ theaps of size ku Mk « regular language Ď Σ˚ Topological setting µk Ý Ñ µ8 ô Pµkra ď xs Ñ Pµ8ra ď xs

• 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: hpaq

Mk “ theaps of size ku Mk « regular language Ď Σ˚ Topological setting µk Ý Ñ µ8 ô Pµkra ď xs Ñ Pµ8ra ď xs µk

w

Ý Ñ µ8 ô µkpò aq Ñ µ8pò aq µk

w

Ý Ñ µ8 ô with ò a “ tx : a ď xu

• 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: hpaq

Mk “ theaps of size ku Mk « regular language Ď Σ˚ Topological setting µk Ý Ñ µ8 ô Pµkra ď xs Ñ Pµ8ra ď xs µk

w

Ý Ñ µ8 ô µkpò aq Ñ µ8pò aq µk

w

Ý Ñ µ8 ô with ò a “ tx : a ď xu Embed M` with the topology tò au Make M` complete

• 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: hpaq

Mk “ theaps of size ku Mk « regular language Ď Σ˚ Topological setting µk

w

Ý Ñ µ8 ô Pµkra ď xs Ñ Pµ8ra ď xs µk

w

Ý Ñ µ8 ô µkpò aq Ñ µ8pò aq µk

w

Ý Ñ µ8 ô with ò a “ tx : a ď xu Embed M` with the topology tò au Make M` complete

• 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: hpaq

Mk “ theaps of size ku Mk « regular language Ď Σ˚ Topological setting µk

w

Ý Ñ µ8 ô Pµkra ď xs Ñ Pµ8ra ď xs µk

w

Ý Ñ µ8 ô µkpò aq Ñ µ8pò aq µk

w

Ý Ñ µ8 ô with ò a “ tx : a ď xu Embed M` with the topology tò au Make M` complete

Theorem (S. Abbes & J. Mairesse 2015)

The uniform distribution on Mk converges weakly in M` when k Ñ `8

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM` z|α| ¨ ř γPCp´zq|γ|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM`

ř

γPCp´1q|γ|z|γα|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM`

ř

γPCp´1q|γ|z|γα|

“ ř

θPM`

ř

γPC 1γďθp´1q|γ|z|θ|

where θ “ γα

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM`

ř

γPCp´1q|γ|z|γα|

“ ř

θPM`

ř

γPC 1γďθp´1q|γ|z|θ|

“ ř

θPM` z|θ| ř SĎLpθqp´1q|S|

where θ “ γα, Lpθq “ tx P Σ : x ď θu and γ “ Ž S

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM`

ř

γPCp´1q|γ|z|γα|

“ ř

θPM`

ř

γPC 1γďθp´1q|γ|z|θ|

“ ř

θPM` z|θ|1Lpθq“H

where θ “ γα, Lpθq “ tx P Σ : x ď θu and γ “ Ž S

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM`

ř

γPCp´1q|γ|z|γα|

“ ř

θPM`

ř

γPC 1γďθp´1q|γ|z|θ|

“ ř

θPM` z|θ|1θ“1

where θ “ γα, Lpθq “ tx P Σ : x ď θu and γ “ Ž S

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1 Proof GpzqHpzq “ ř

αPM`

ř

γPCp´1q|γ|z|γα|

“ ř

θPM`

ř

γPC 1γďθp´1q|γ|z|θ|

“ ř

θPM` z|θ|1θ“1 “ 1

where θ “ γα, Lpθq “ tx P Σ : x ď θu and γ “ Ž S

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Generating series and Möbius polynomial Gpzq “ ř

αPM` z|α| “ ř kě0 λkzk and Hpzq “ ř γPCp´zq|γ|

Proposition (P. Cartier & D. Foata 1969)

GpzqHpzq “ 1

Corollary (D. Krob, J. Mairesse & I. Michos 2001)

Hpzq has a smallest positive root p such that: pHpzq “ 0 ^ |z| ď pq ô z “ p 0 ă p ď 1 and there exists constants Λ ą 0 and ℓ P N such that λk „ Λp´kkℓ

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

2 x ÞÑ ax maps Mk to pò aq X Mk`|a| bijectively

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

2 x ÞÑ ax maps Mk to pò aq X Mk`|a| bijectively 3 µkpò aq “

λk´|a| λk

Ñ p´|a|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a|

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

2 x ÞÑ ax maps Mk to pò aq X Mk`|a| bijectively 3 µkpò aq “

λk´|a| λk

Ñ p´|a|

4 M` is compact and tHu Y tò au is closed under X

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a| and lengthpaq “ hpaq

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

2 x ÞÑ ax maps Mk to pò aq X Mk`|a| bijectively 3 µkpò aq “

λk´|a| λk

Ñ p´|a|

4 M` is compact and tHu Y tò au is closed under X

Proof of the theorem – lengthpaq “ hpaq

5 Split ò a into sets M`pbq “ tx : b “ Chpaqpxqu

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a| and lengthpaq “ hpaq

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

2 x ÞÑ ax maps Mk to pò aq X Mk`|a| bijectively 3 µkpò aq “

λk´|a| λk

Ñ p´|a|

4 M` is compact and tHu Y tò au is closed under X

Proof of the theorem – lengthpaq “ hpaq

5 Split ò a into sets M`pbq “ tx : b “ Chpaqpxqu 6 Prove that #pM`pbq X Mkq „ Λbqk

bkℓb for some Λb, qb and ℓb

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Weak convergence: lengthpaq “ |a| and lengthpaq “ hpaq

Proof of the theorem – lengthpaq “ |a|

1 µk : S ÞÑ #pSXMkq

λk

2 x ÞÑ ax maps Mk to pò aq X Mk`|a| bijectively 3 µkpò aq “

λk´|a| λk

Ñ p´|a|

4 M` is compact and tHu Y tò au is closed under X

Proof of the theorem – lengthpaq “ hpaq

5 Split ò a into sets M`pbq “ tx : b “ Chpaqpxqu 6 Prove that #pM`pbq X Mkq „ Λbqk

bkℓb for some Λb, qb and ℓb

7 Complete the proof as above

Caution: lim µkpò aq does not depend only on hpaq!

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Contents

1

Introduction

2

Trace monoids and heaps

3

First convergence results

4

Bernoulli distributions

5

Going beyond. . .

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

A distribution µ on M` is . . . Bernoulli if µpò abq “ µpò aqµpò bq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

A distribution µ on M` is . . . Bernoulli if µpò abq “ µpò aqµpò bq µpò a1a2 . . . akq “ νa1νa2 . . . νak

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

A distribution µ on M` is . . . Bernoulli if µpò abq “ µpò aqµpò bq µpò a1a2 . . . akq “ νa1νa2 . . . νak Uniform Bernoulli if µpò aq “ ν|a| ν1 “ ν2 “ . . . “ ν

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

A distribution µ on M` is . . . Bernoulli if µpò abq “ µpò aqµpò bq µpò a1a2 . . . akq “ νa1νa2 . . . νak Uniform Bernoulli with parameter ν if µpò aq “ ν|a| ν1 “ ν2 “ . . . “ ν

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

A distribution µ on M` is . . . Bernoulli if µpò abq “ µpò aqµpò bq µpò a1a2 . . . akq “ νa1νa2 . . . νak Uniform Bernoulli with parameter ν if µpò aq “ ν|a| ν1 “ ν2 “ . . . “ ν Finite uniform Bernoulli if ν ă p Hpzq ą 0 for all z P p0, pq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

A distribution µ on M` is . . . Bernoulli if µpò abq “ µpò aqµpò bq µpò a1a2 . . . akq “ νa1νa2 . . . νak Uniform Bernoulli with parameter ν if µpò aq “ ν|a| ν1 “ ν2 “ . . . “ ν Finite uniform Bernoulli if ν ă p Hpzq ą 0 for all z P p0, pq µpBM`q “ 0 µptauq “ Hpνqν|a|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

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 µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µptxuq “

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µptxuq “ νpò xq x

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µptxuq “ νpò xq´νpò xaq ´ νpò xbq ´ νpò xcq ´ νpò xdq x xc xa xd xb

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µptxuq “ νpò xq´νpò xaq ´ νpò xbq ´ νpò xcq ´ νpò xdq` νpò xacq ` νpò xadq ` νpò xbdq x xc xa xd xb xac xad xbd

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µptxuq “ ν|x|p1 ´ ν|a| ´ ν|b| ´ ν|c| ´ ν|d| ` ν|ac| ` ν|ad| ` ν|bd|q x xc xa xd xb xac xad xbd

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Bernoulli distributions

Proving that finite uniform ô µptxuq “ Hpνqν|x| ð µpò xq “ Hpνq ř

γ ν|xγ| “ ν|x|Hpνq ř γ ν|γ| “ ν|x|HpνqGpνq

ñ Proof #1: At most one measure works! ñ Proof #2: Using inclusion-exclusion: µptxuq “ ν|x|p1 ´ ν|a| ´ ν|b| ´ ν|c| ´ ν|d| ` ν|ac| ` ν|ad| ` ν|bd|q “ ν|x|Hpνq x xc xa xd xb xac xad xbd

• 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 λkνkHpνq Pick a trace uniformly at random in ta P M` | |a| “ ku

• 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 λkνkHpνq Pick a trace uniformly at random in ta P M` | |a| “ ku Approach #2: Pick the ground floor first Order the generators from g1 to gn and choose whether gi 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 λkνkHpνq Pick a trace uniformly at random in ta P M` | |a| “ ku Approach #2: Pick the ground floor first Order the generators from g1 to gn and choose whether gi ď a (based on tgj | 1 ď j ă i, gj ď au 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 λkνkHpνq Pick a trace uniformly at random in ta P M` | |a| “ ku Approach #2: Pick the ground floor first (Markov chain) Order the generators from g1 to gn and choose whether gi ď a (based on tgj | 1 ď j ă i, gj ď au 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 ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

ν|a| “ ř µpÒ bq1aďb,hpaq“hpbq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

ν|a| “ ř µpÒ bq1aďb,hpaq“hpbq µpÒ aq “ ř

γPCp´1q|γ|1hpaq“hpaγqν|aγ|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

ν|a| “ ř µpÒ bq1aďb,hpaq“hpbq µpÒ aq “ ř

γPCp´1q|γ|1hpaq“hpaγqν|aγ|

“ ν|a| ř

γPCp´1q|γ|1hpaq“hpaγqν|γ|

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

ν|a| “ ř µpÒ bq1aďb,hpaq“hpbq µpÒ aq “ ř

γPCp´1q|γ|1hpaq“hpaγqν|aγ|

“ ν|a| ř

γPCp´1q|γ|1hpaq“hpaγqν|γ|

“ ν|a|Hapνq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

ν|a| “ ř µpÒ bq1aďb,hpaq“hpbq a = a1 a2 . . . ahpaq µpÒ aq “ ř

γPCp´1q|γ|1hpaq“hpaγqν|aγ|

“ ν|a| ř

γPCp´1q|γ|1hpaq“hpaγqν|γ|

“ ν|a|Hahpaqpνq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Finite uniform Bernoulli distributions as Markov chains

Monoid cylinder ò a “ tb P M` | a ď bu Cartier-Foata cylinder Ò a “ tb P M` | a “ Chpaqpbqu Möbius inversion formula and Markov simulation Ò a “ 9 Ť

aďb,hpaq“hpbq Ò b

ν|a| “ ř µpÒ bq1aďb,hpaq“hpbq a = a1 a2 . . . ahpaq µpÒ aq “ ř

γPCp´1q|γ|1hpaq“hpaγqν|aγ|

“ ν|a| ř

γPCp´1q|γ|1hpaq“hpaγqν|γ|

“ ν|a|Hahpaqpνq “ PrΘν

1 “ a1, . . . , Θν hpaq “ ahpaqs

PrΘν

1 “ as “ ν|a|Hapνq

PrΘν

i`1 “ b | Θν i “ as “ ν|b| Hbpνq Hapνq1aÑb

a b a Ñ b ô ô a “ C1pabq

• 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 pΘν

i q when ν Ñ p, with limit

PrΘp

1 “ as “ p|a|Happq

PrΘp

i`1 “ b | Θp i “ as “ p|b| Hbppq Happq1aÑb1Happq‰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 pΘν

i q when ν Ñ p, with limit

PrΘp

1 “ as “ p|a|Happq

PrΘp

i`1 “ b | Θp i “ as “ p|b| Hbppq Happq1aÑb1Happq‰0

Trivial supercritical parameter: ν “ 1 Possible only if M` “ Nn (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 pΘν

i q when ν Ñ p, with limit

PrΘp

1 “ as “ p|a|Happq

PrΘp

i`1 “ b | Θp i “ as “ p|b| Hbppq Happq1aÑb1Happq‰0

Trivial supercritical parameter: ν “ 1 Possible only if M` “ Nn (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 pΘν

i q when ν Ñ p, with limit

PrΘp

1 “ as “ p|a|Happq

PrΘp

i`1 “ b | Θp i “ as “ p|b| Hbppq Happq1aÑb1Happq‰0

Trivial supercritical parameter: ν “ 1 Possible only if M` “ Nn (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! Consider the Garside matrix Mν with Mν

a,b “ 1aÑbν|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 pΘν

i q when ν Ñ p, with limit

PrΘp

1 “ as “ p|a|Happq

PrΘp

i`1 “ b | Θp i “ as “ p|b| Hbppq Happq1aÑb1Happq‰0

Trivial supercritical parameter: ν “ 1 Possible only if M` “ Nn (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! Consider the Garside matrix Mν with Mν

a,b “ 1aÑbν|b|

1 ě µpM`q “ HpνqGpνq ě 0, hence Hpνq “ 0 and µpBM`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 pΘν

i q when ν Ñ p, with limit

PrΘp

1 “ as “ p|a|Happq

PrΘp

i`1 “ b | Θp i “ as “ p|b| Hbppq Happq1aÑb1Happq‰0

Trivial supercritical parameter: ν “ 1 Possible only if M` “ Nn (i.e. p “ 1). . . Non-trivial supercritical parameter: p ă ν ă 1 No such distribution exists! Consider the Garside matrix Mν with Mν

a,b “ 1aÑbν|b|

1 ě µpM`q “ HpνqGpνq ě 0, hence Hpνq “ 0 and µpBM`q “ 1 Mν and Mp are stochastic Perron matrices if M` is irreducible

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

Contents

1

Introduction

2

Trace monoids and heaps

3

First convergence results

4

Bernoulli distributions

5

Going beyond. . .

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From uniform to non-uniform Bernoulli measures

Generalisations from the uniform case Collection of parameters: pνaq P p0, 1sn

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From uniform to non-uniform Bernoulli measures

Generalisations from the uniform case Collection of parameters: pνaq P p0, 1sn Multiplicative function: ν : a1 . . . ak ÞÑ νa1 . . . νak Möbius polynomial: Hapνq “ ř

γ 1aγPCp´1q|γ|νpγq

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From uniform to non-uniform Bernoulli measures

Generalisations from the uniform case Collection of parameters: pνaq P p0, 1sn Multiplicative function: ν : a1 . . . ak ÞÑ νa1 . . . νak Möbius polynomial: Hapνq “ ř

γ 1aγPCp´1q|γ|νpγq

Subcritical domain: D “ tν | H1pxνq ą 0 when 0 ď x ď 1u Critical domain: BD X p0, 1sn

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From uniform to non-uniform Bernoulli measures

Generalisations from the uniform case Collection of parameters: pνaq P p0, 1sn Multiplicative function: ν : a1 . . . ak ÞÑ νa1 . . . νak Möbius polynomial: Hapνq “ ř

γ 1aγPCp´1q|γ|νpγq

Subcritical domain: D “ tν | H1pxνq ą 0 when 0 ď x ď 1u Critical domain: BD X p0, 1sn Markov chain: PrΘν

1 “ as “ νpaqHapνq

Markov chain: PrΘν

i`1 “ b | Θν i “ as “ νpbqHbpνq Hapνq1aÑb1Hapνq‰0

1 1 νa νb N ˚ N “ xa, by` 1 1 νa νb N2 “ xa, b | ab “ bay` νa νb νc 1 1 xa, b, c | ac “ cay`

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From uniform to non-uniform Bernoulli measures

Generalisations from the uniform case Collection of parameters: pνaq P p0, 1sn Multiplicative function: ν : a1 . . . ak ÞÑ νa1 . . . νak Möbius polynomial: Hapνq “ ř

γ 1aγPCp´1q|γ|νpγq

Subcritical domain: D “ tν | H1pxνq ą 0 when 0 ď x ď 1u Critical domain: BD X p0, 1sn Markov chain: PrΘν

1 “ as “ νpaqHapνq

Markov chain: PrΘν

i`1 “ b | Θν i “ as “ νpbqHbpνq Hapνq1aÑb1Hapνq‰0

No supercritical Bernoulli measures!

1 1 νa νb N ˚ N “ xa, by` 1 1 νa νb N2 “ xa, b | ab “ bay` νa νb νc 1 1 xa, b, c | ac “ cay`

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From weak convergence to central limit theorems

Some key ingredients: ν: tuple pν1, . . . , νnq P p0, `8qn

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From weak convergence to central limit theorems

Some key ingredients: ν: tuple pν1, . . . , νnq P p0, `8qn µk: ν-uniform distribution on tx P M` | |x| “ ku

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From weak convergence to central limit theorems

Some key ingredients: ν: tuple pν1, . . . , νnq P p0, `8qn µk: ν-uniform distribution on tx P M` | |x| “ ku }x}a: # occurrences of a in the Cartier-Foata word of x

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From weak convergence to central limit theorems

Some key ingredients: ν: tuple pν1, . . . , νnq P p0, `8qn µk: ν-uniform distribution on tx P M` | |x| “ ku }x}a: # occurrences of a in the Cartier-Foata word of x Ak: law of }x}a when x is distributed according to µk

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From weak convergence to central limit theorems

Some key ingredients: ν: tuple pν1, . . . , νnq P p0, `8qn µk: ν-uniform distribution on tx P M` | |x| “ ku }x}a: # occurrences of a in the Cartier-Foata word of x Ak: law of }x}a when x is distributed according to µk

Central limit Theorem (S. A., S. G., V. J. & J. M. 2016+)

There exists constants ρ and σ2 ą 0 such that

1

Ak k L

Ý Ñ ρ

2 ?

k ´

Ak k ´ ρ

¯

L

Ý Ñ Np0, σ2q if M` is irreducible

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

D

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy`

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

B D

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y`

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

Artin–Tits D B

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Artin–Tits monoid: ℓpi, jq “ 2 ñ σiσj “ σjσi

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

Artin–Tits D B

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Artin–Tits monoid: ℓpi, jq “ 3 ñ σiσjσi “ σjσiσj

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

Artin–Tits D B

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Artin–Tits monoid: ℓpi, jq “ 4 ñ σiσjσiσj “ σjσiσjσi

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

Artin–Tits D B

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Artin–Tits monoid: ℓpi, jq “ ` 8 ñ no relation!

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

T D B Artin–Tits

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Trace monoid: Artin–Tits with ℓpi, jq P t2, `8u

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

Left-Garside D B Artin–Tits T

Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Trace monoid: Artin–Tits with ℓpi, jq P t2, `8u Left-Garside monoid: cancellative, ď-lower semi-lattice, finite generating

family Σ closed under suffix and under _

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

From trace monoids to Artin–Tits and left-Garside monoids

D B Artin–Tits T Left-Garside

Theorem (— 2016+)

1 Weak conv. in all A–T monoids 2 CLT 1 in all A–T monoids 3 CLT 2 in irreducible A–T monoids

+some extensions to left-Garside monoids Dimer monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσiy` Braid monoid: xσi | i ‰ j ˘ 1 ñ σiσj “ σjσi, σiσi`1σi “ σi`1σiσi`1y` Artin–Tits monoid: xσi | rσiσjsℓpi,jq “ rσjσisℓpi,jqy` Trace monoid: Artin–Tits with ℓpi, jq P t2, `8u Left-Garside monoid: cancellative, ď-lower semi-lattice, finite generating

family Σ closed under suffix and under _

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random

And then?

Some directions of research Generalisation to all left-Garside monoids Generalisation to trace groups Sampling elements in regular languages L X Mk instead of Mk Identifying nice Markov chains when lengthpaq “ hpbaq Your favorite one (tell me now!)

• S. Abbes, S. Gouëzel, V. Jugé & J. Mairesse

Drawing heaps uniformly at random