Studying some Markov chains using representation theory of monoids - - PowerPoint PPT Presentation

1 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Studying some Markov chains using representation theory of monoids Nicolas M. Thi´ ery

Laboratoire de Recherche en Informatique Universit´ e Paris Sud 11

AL´ EA 2014, March 18th of 2014 Joint work with: Arvind Ayyer, Benjamin Steinberg, Anne Schilling arXiv: 1305.1697, 1401.4250

2 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Highlight of the talk

Some Markov chains The Tsetlin library Directed sandpile models Why are they nicely behaved? Approach 1: Triangularization Approach 2: monoids, representation theory, characters Intermezzo: a monoid on trees Conclusion

3 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

A first example: the Tsetlin library

Configuration: n books on a shelf Operation Ti: move the i-th book to the right

3 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

A first example: the Tsetlin library

Configuration: n books on a shelf Operation Ti: move the i-th book to the right

231 321 132 312 123 213 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3

3 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

A first example: the Tsetlin library

Configuration: n books on a shelf Operation Ti: move the i-th book to the right

A typical self-optimizing model for:

• Cache handling
• Prioritizing

3 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

A first example: the Tsetlin library

Configuration: n books on a shelf Operation Ti: move the i-th book to the right

A typical self-optimizing model for:

• Cache handling
• Prioritizing

Problem

• Average behavior?
• How fast does it stabilize?

4 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Controlling the behavior of the Tsetlin library?

Markov chain description

• Configuration space Ω: all permutations of the books
• Transition operator Ti: taking book i with probability xi

4 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Controlling the behavior of the Tsetlin library?

Markov chain description

• Configuration space Ω: all permutations of the books
• Transition operator Ti: taking book i with probability xi
• Transition operator T = x1T1 + · · · + xnTn

4 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Controlling the behavior of the Tsetlin library?

Markov chain description

• Configuration space Ω: all permutations of the books
• Transition operator Ti: taking book i with probability xi
• Transition operator T = x1T1 + · · · + xnTn
• Stationary distribution?

1-Eigenvector of T

4 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Controlling the behavior of the Tsetlin library?

Markov chain description

• Configuration space Ω: all permutations of the books
• Transition operator Ti: taking book i with probability xi
• Transition operator T = x1T1 + · · · + xnTn
• Stationary distribution?

1-Eigenvector of T

• Spectrum?

Eigenvalues of T

4 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Controlling the behavior of the Tsetlin library?

Markov chain description

• Configuration space Ω: all permutations of the books
• Transition operator Ti: taking book i with probability xi
• Transition operator T = x1T1 + · · · + xnTn
• Stationary distribution?

1-Eigenvector of T

• Spectrum?

Eigenvalues of T

Theorem (Brown, Bidigare ’99)

Each S ⊆ {1, . . . , n} contributes the eigenvalue

i∈S xi with

multiplicity the number of derangements of S.

5 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Abelian sandpile models / chip-firing games

• A graph G
• Configuration: distribution of grains of sand at each site
• Grains fall in at random
• Grains topple to the neighbor sites
• Grains fall off at sinks
• Prototypical model for the phenomenon of self-organized

criticality, like a heap of sand

6 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Directed sandpile Models

• A tree, with edges pointing toward its root
• Configuration: distribution of grains of sand at each site
• Grains fall in at random (leaves only or everywhere)
• Grains topple down at random (one by one or all at once)
• Grains fall off at the sink (=root)

6 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Directed sandpile Models

• A tree, with edges pointing toward its root
• Configuration: distribution of grains of sand at each site
• Grains fall in at random (leaves only or everywhere)
• Grains topple down at random (one by one or all at once)
• Grains fall off at the sink (=root)
• System with reservoirs in nonequilibrium statistical physics

7 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Directed sandpile model on a line with thresholds 1

010 110 100 000 111 011 001 101 σ1 τ3 τ1 τ2 τ3 τ2 τ1 σ1 σ1 τ2 τ3 σ1 τ2 σ1 τ3 τ1 σ1 τ1 τ1 σ1 τ1 τ2 τ1 τ2 τ3 τ1 τ3 τ2 τ3 τ3 τ2 σ1

8 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Directed sandpile models are very nicely behaved

Proposition (Ayyer, Schilling, Steinberg, T. ’13)

The transition graph is strongly connected Equivalently the Markov chain is ergodic

8 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Directed sandpile models are very nicely behaved

Proposition (Ayyer, Schilling, Steinberg, T. ’13)

The transition graph is strongly connected Equivalently the Markov chain is ergodic

Theorem (ASST’13)

Characteristic polynomial of the transition matrix: det(Mτ − λ1) =

• S⊆V

(λ − (yS + xS))TSc where Sc = V \ S and TS =

v∈S Tv

8 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Directed sandpile models are very nicely behaved

Proposition (Ayyer, Schilling, Steinberg, T. ’13)

The transition graph is strongly connected Equivalently the Markov chain is ergodic

Theorem (ASST’13)

Characteristic polynomial of the transition matrix: det(Mτ − λ1) =

• S⊆V

(λ − (yS + xS))TSc where Sc = V \ S and TS =

v∈S Tv

Theorem (ASST’13)

Mixing time: at most 2(nT +c−1)

p

9 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchline

Those models have exceptionally nice eigenvalues

9 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchline

Those models have exceptionally nice eigenvalues In fact quite a few Markov chains have similar behaviors:

• Promotion Markov chains [Ayyer, Klee, Schilling ’12]
• Nonabelian directed sandpile models
• Generalizations of the Tsetlin library (multibook, ...)
• Walks on longest words of finite Coxeter groups
• Half-regular bands

9 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchline

Those models have exceptionally nice eigenvalues In fact quite a few Markov chains have similar behaviors:

• Promotion Markov chains [Ayyer, Klee, Schilling ’12]
• Nonabelian directed sandpile models
• Generalizations of the Tsetlin library (multibook, ...)
• Walks on longest words of finite Coxeter groups
• Half-regular bands

Is there some uniform explanation?

9 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchline

Those models have exceptionally nice eigenvalues In fact quite a few Markov chains have similar behaviors:

• Promotion Markov chains [Ayyer, Klee, Schilling ’12]
• Nonabelian directed sandpile models
• Generalizations of the Tsetlin library (multibook, ...)
• Walks on longest words of finite Coxeter groups
• Half-regular bands

Is there some uniform explanation? Yes

9 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchline

Those models have exceptionally nice eigenvalues In fact quite a few Markov chains have similar behaviors:

• Promotion Markov chains [Ayyer, Klee, Schilling ’12]
• Nonabelian directed sandpile models
• Generalizations of the Tsetlin library (multibook, ...)
• Walks on longest words of finite Coxeter groups
• Half-regular bands

Is there some uniform explanation? Yes: representation theory of R-trivial monoids!

10 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Decomposition of the configuration space (lumping)

231 321 132 312 123 213 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3

11 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Let’s train on a simpler example

123 213 132 312 231 321 π2 π1 π1 π2 π2 π1 π2 π1 π2 π1 π2 π1

12 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Decomposition of the configuration space (lumping)

231 321 132 312 123 213 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3

12 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Decomposition of the configuration space (lumping)

231 321 132 312 123 213 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3 231 312 123 1 1 2 2 3 3 1 1 2 2 3 3

12 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Decomposition of the configuration space (lumping)

231 321 132 312 123 213 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3 231 312 123 1 1 2 2 3 3 1 1 2 2 3 3 231 1 2 3

13 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Explanation: triangularization

Our Markov chains are nicely behaved because

• The transition operators Ti can be

simultaneously triangularized!

13 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Explanation: triangularization

Our Markov chains are nicely behaved because

• The transition operators Ti can be

simultaneously triangularized!

• Divide, and conquer subspaces of dimension 1!

13 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Explanation: triangularization

Our Markov chains are nicely behaved because

• The transition operators Ti can be

simultaneously triangularized!

• Divide, and conquer subspaces of dimension 1!

Problems

• Proving that it is triangularizable?
• Constructing a triangularization?

13 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Explanation: triangularization

Our Markov chains are nicely behaved because

• The transition operators Ti can be

simultaneously triangularized!

• Divide, and conquer subspaces of dimension 1!

Problems

• Proving that it is triangularizable?
• Constructing a triangularization?

Or just be lazy

Learn a bit of representation theory and use characters.

14 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Approach 2: monoids, representation theory, characters

Definition (Transition monoid of a Markov chain / automaton)

Ti : Ω → Ω transition operators of the Markov chain

14 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Approach 2: monoids, representation theory, characters

Definition (Transition monoid of a Markov chain / automaton)

Ti : Ω → Ω transition operators of the Markov chain Monoid: (M, ◦) = Ti A finite monoid of functions Similar to a permutation group, except for invertibility

15 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The left Cayley graph for the 1D sandpile model

010 110 100 000 111 011 001 101 σ1 τ3 τ1 τ2 τ3 τ2 τ1 σ1 σ1 τ2 τ3 σ1 τ2 σ1 τ3 τ1 σ1 τ1 τ1 σ1 τ1 τ2 τ1 τ2 τ3 τ1 τ3 τ2 τ3 τ3 τ2 σ1

15 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The left Cayley graph for the 1D sandpile model

010 110 100 000 111 011 001 101 σ1 τ3 τ1 τ2 τ3 τ2 τ1 σ1 σ1 τ2 τ3 σ1 τ2 σ1 τ3 τ1 σ1 τ1 τ1 σ1 τ1 τ2 τ1 τ2 τ3 τ1 τ3 τ2 τ3 τ3 τ2 σ1

id τ3 τ3τ2 τ3τ2τ1 000 τ3τ2σ1 001 τ3τ1 τ3τ1τ2 τ3τ1σ1 010 τ3σ1 τ3σ1τ2 τ3σ1σ1 011 τ2 τ2τ3 τ2τ3τ1 τ2τ3τ1τ2 τ2τ3σ1 τ2τ3σ1τ2 τ2τ1 τ2τ1σ1 100 τ2σ1 τ2σ1σ1 101 τ1 τ1τ3 τ1τ3σ1 τ1τ3σ1τ2 τ1τ2 τ1τ2τ3 τ1σ1 τ1σ1σ1 110 σ1 σ1τ3 σ1τ3σ1 σ1τ3σ1τ2 σ1τ2 σ1τ2τ3 σ1σ1 σ1σ1σ1 111 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 τ3 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 σ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ1 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2 τ2

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

• This graph is acyclic: R-triviality

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

• This graph is acyclic: R-triviality

= ⇒ Triangularizability = ⇒ Nice eigenvalues

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

• This graph is acyclic: R-triviality

= ⇒ Triangularizability = ⇒ Nice eigenvalues

• What this captures: loss of information

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

• This graph is acyclic: R-triviality

= ⇒ Triangularizability = ⇒ Nice eigenvalues

• What this captures: loss of information
• Not too deep

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

• This graph is acyclic: R-triviality

= ⇒ Triangularizability = ⇒ Nice eigenvalues

• What this captures: loss of information
• Not too deep

= ⇒ Bound on the rates of convergence / mixing time

16 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

The right Cayley graph for the 1D sandpile model

id τ2τ1σ1 100 τ3σ1 τ3τ2 τ3τ1 τ2τ3τ1τ2 τ3σ1τ2 τ3σ1σ1 011 τ2σ1σ1 101 τ2σ1 τ2τ3 τ2τ1 τ3τ2σ1 001 σ1τ2 σ1τ3 τ3τ1σ1 010 τ3τ1τ2 σ1σ1 τ1σ1 τ1τ3 τ1τ2 τ3 τ1 τ2 τ3τ2τ1 000 τ1σ1σ1 110 σ1 σ1τ3σ1τ2 τ1τ3σ1 τ1τ3σ1τ2 σ1τ3σ1 τ1τ2τ3 τ2τ3σ1 τ2τ3τ1 σ1τ2τ3 σ1σ1σ1 111 τ2τ3σ1τ2 τ3 τ3 σ1 τ1 τ2 τ3 τ1 σ1 τ2 σ1 τ2 σ1 σ1 τ1 τ2 τ3 σ1 τ1 σ1 τ1 τ2 σ1 τ3 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ1 τ2 τ3 τ3 τ1 τ3 τ2 τ1 τ3 τ1 τ3 σ1 σ1 σ1 τ3 τ1 τ1 τ3 τ3 τ2 τ3 τ3 σ1 τ2 τ3 τ2 τ1 τ2 τ1 τ2 τ1 σ1 σ1 τ1 τ2 τ3 τ1 τ3 σ1 τ2 σ1 σ1 τ1 σ1 τ3 τ3 τ2 τ2 σ1 τ3 τ1 τ2 τ2 τ3 τ1 σ1 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ3 σ1 τ2 τ1 τ2 τ1 τ3 τ2 τ2 σ1 τ3 τ2 σ1 σ1 τ1 τ2 τ3 τ1 τ2 τ2 τ2 τ1 τ2 σ1 τ1 τ2 τ3 σ1 τ1 τ2 τ2 σ1 τ1 τ1 τ3 τ3 σ1 τ3 τ1 τ1 σ1 τ1 τ2 τ3 τ1 σ1 σ1 σ1 τ2 τ3

• This graph is acyclic: R-triviality

= ⇒ Triangularizability = ⇒ Nice eigenvalues

• What this captures: loss of information
• Not too deep

= ⇒ Bound on the rates of convergence / mixing time

• A form of coupling from the past

17 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Strategy

Combinatorial point of view

• Show that M is R-trivial

⇒ the representation matrix are simultaneously triangularizable

17 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Strategy

Combinatorial point of view

• Show that M is R-trivial

⇒ the representation matrix are simultaneously triangularizable

• Eigenvalues indexed by a lattice of subsets of the generators

17 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Strategy

Combinatorial point of view

• Show that M is R-trivial

⇒ the representation matrix are simultaneously triangularizable

• Eigenvalues indexed by a lattice of subsets of the generators
• Count fixed points
• Recover multiplicities from M¨
• bius inversion on the lattice

17 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Strategy

Combinatorial point of view

• Show that M is R-trivial

⇒ the representation matrix are simultaneously triangularizable

• Eigenvalues indexed by a lattice of subsets of the generators
• Count fixed points
• Recover multiplicities from M¨
• bius inversion on the lattice

Representation theory point of view

• Simple modules are of dimension 1

17 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Strategy

Combinatorial point of view

• Show that M is R-trivial

⇒ the representation matrix are simultaneously triangularizable

• Eigenvalues indexed by a lattice of subsets of the generators
• Count fixed points
• Recover multiplicities from M¨
• bius inversion on the lattice

Representation theory point of view

• Simple modules are of dimension 1
• Compute the character of a transformation module

17 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Strategy

Combinatorial point of view

• Show that M is R-trivial

⇒ the representation matrix are simultaneously triangularizable

• Eigenvalues indexed by a lattice of subsets of the generators
• Count fixed points
• Recover multiplicities from M¨
• bius inversion on the lattice

Representation theory point of view

• Simple modules are of dimension 1
• Compute the character of a transformation module
• Recover the composition factors using the character table

18 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

This is effective!

GAP, Semigroupe

• Transformation monoids

Sage

• Character calculation for R-trivial (aperiodic) monoids
• Eigenvalues calculation
• Status: functional but not yet integrated
• Feel free to ask for a demo

18 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

This is effective!

GAP, Semigroupe

• Transformation monoids

Sage

• Character calculation for R-trivial (aperiodic) monoids
• Eigenvalues calculation
• Status: functional but not yet integrated
• Feel free to ask for a demo

Speaking of which

Interested in sharing code for studying Finite State Markov chains? Let’s talk! (today 2pm?)

19 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

How to prove R-triviality?

19 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

How to prove R-triviality?

Theorem (ASST’14)

Let M be a monoid generated by A := {x1 < · · · < xn} such that a2 = a for a ∈ A bab = ba for a < b ∈ A Then, M is R-trivial

19 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

How to prove R-triviality?

Theorem (ASST’14)

Let M be a monoid generated by A := {x1 < · · · < xn} such that a2 = a for a ∈ A bab = ba for a < b ∈ A Then, M is R-trivial

Examples

Basically all our monoids!

19 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

How to prove R-triviality?

Theorem (ASST’14)

Let M be a monoid generated by A := {x1 < · · · < xn} such that a2 = a for a ∈ A bab = ba for a < b ∈ A Then, M is R-trivial

Examples

Basically all our monoids!

Remark (Connection with the plactic monoid)

Take the (half) plactic monoid: bac = bca, for a < b ≤ c Set the generators to be idempotent: a2 = a Byproduct: bab = bba = ba

19 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

How to prove R-triviality?

Theorem (ASST’14)

Let M be a monoid generated by A := {x1 < · · · < xn} such that a2 = a for a ∈ A bab = ba for a < b ∈ A Then, M is R-trivial

Examples

Basically all our monoids!

Remark (Connection with the plactic monoid)

Take the (half) plactic monoid: bac = bca, for a < b ≤ c Set the generators to be idempotent: a2 = a Byproduct: bab = bba = ba

Proof of the theorem.

Consider the worst case!

20 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Definition (Free tree monoid Mn)

The monoid Mn generated by A := {x1 < · · · < xn} satisfying a2 = a for a ∈ A bab = ba for a < b ∈ A

20 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Definition (Free tree monoid Mn)

The monoid Mn generated by A := {x1 < · · · < xn} satisfying a2 = a for a ∈ A bab = ba for a < b ∈ A

Proposition (ASST’14)

Mn is a monoid on a certain class of trees (#A007018)

20 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Definition (Free tree monoid Mn)

The monoid Mn generated by A := {x1 < · · · < xn} satisfying a2 = a for a ∈ A bab = ba for a < b ∈ A

Proposition (ASST’14)

Mn is a monoid on a certain class of trees (#A007018)

Proof: reduced words + bijection with trees.

x3x2x4x1x2x4

20 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Definition (Free tree monoid Mn)

The monoid Mn generated by A := {x1 < · · · < xn} satisfying a2 = a for a ∈ A bab = ba for a < b ∈ A

Proposition (ASST’14)

Mn is a monoid on a certain class of trees (#A007018)

Proof: reduced words + bijection with trees.

x3x2x4x1x2x4 = x3x2x4x1x2

20 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Definition (Free tree monoid Mn)

The monoid Mn generated by A := {x1 < · · · < xn} satisfying a2 = a for a ∈ A bab = ba for a < b ∈ A

Proposition (ASST’14)

Mn is a monoid on a certain class of trees (#A007018)

Proof: reduced words + bijection with trees.

x3x2x4x1x2x4 = x3x2x4x1x2 ← →

1 2 3 4 x3 x2 x4 x1 x2

21 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Left Cayley graph Right Cayley graph

• x
• y

x

• x

y x

• y
• x

y

• x

y x y y x y x x x y y

• x
• x

y

• x

y x

• y
• y

x

• x

y x y x x y x, y y x, y

Proposition (ASST’14)

Mn is R-trivial

Proof.

Nice explicit Knuth-Bendix completion of the relations

21 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Free tree monoids

Left Cayley graph Right Cayley graph

• x
• y

x

• x

y x

• y
• x

y

• x

y x y y x y x x x y y

• x
• x

y

• x

y x

• y
• y

x

• x

y x y x x y x, y y x, y

Proposition (ASST’14)

Mn is R-trivial

Proof.

Nice explicit Knuth-Bendix completion of the relations

Problem

Description of the stationary distribution of the left Cayley graph?

22 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Finite state Markov chains and Representation Theory

The idea of decomposing the configuration space is not new!

Using representation theory of groups

• Diaconis et al.
• Nice combinatorics (symmetric functions, ...)

22 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Finite state Markov chains and Representation Theory

The idea of decomposing the configuration space is not new!

Using representation theory of groups

• Diaconis et al.
• Nice combinatorics (symmetric functions, ...)

Using representation theory of monoids

• Any finite state Markov chain can be seen as a representation
• f a monoid M
• M can be chosen to be a group iff the uniform distribution is

stationary.

23 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Finite State Markov chains and Representation Theory

Using representation theory of right regular bands

• Tsetlin library, Hyperplane arrangements, ...
• Bidigare, Hanlon, Rockmore ’99, Brown ’00, Saliola, ...
• Revived the interest for representation theory of monoids

23 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Finite State Markov chains and Representation Theory

Using representation theory of right regular bands

• Tsetlin library, Hyperplane arrangements, ...
• Bidigare, Hanlon, Rockmore ’99, Brown ’00, Saliola, ...
• Revived the interest for representation theory of monoids

Using representation of R-trivial monoids?

• Steinberg ’06, ...
• Not semi-simple.

23 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Finite State Markov chains and Representation Theory

Using representation theory of right regular bands

• Tsetlin library, Hyperplane arrangements, ...
• Bidigare, Hanlon, Rockmore ’99, Brown ’00, Saliola, ...
• Revived the interest for representation theory of monoids

Using representation of R-trivial monoids?

• Steinberg ’06, ...
• Not semi-simple. But simple modules of dimension 1!
• Nice combinatorics

24 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchlines

• A (useful?) class of Markov chains with very nice behavior

24 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchlines

• A (useful?) class of Markov chains with very nice behavior

Explanation: representation theory of monoids

• Representation theory of monoids =

⇒ Nice combinatorics

• Tool: computer exploration

24 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchlines

• A (useful?) class of Markov chains with very nice behavior

Explanation: representation theory of monoids

• Representation theory of monoids =

⇒ Nice combinatorics

• Tool: computer exploration

Announcements

• Tentative European H2020 Sage proposal

Want to participate?

24 / 24

Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion

Punchlines

• A (useful?) class of Markov chains with very nice behavior

Explanation: representation theory of monoids

• Representation theory of monoids =

⇒ Nice combinatorics

• Tool: computer exploration

Announcements

• Tentative European H2020 Sage proposal

Want to participate?

• Maˆ

ıtre de conf´ erence position GALAC Team: Graphes Algorithmes et Combinatoire Laboratoire de Recherche en Informatique Universit´ e Paris Sud