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 1 / 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 2 / 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 T i : 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 T i : move the i -th book to the right 123 3 3 213 2 3 2 3 1 1 132 2 2 312 3 3 1 2 1 2 231 1 1 321 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 T i : 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 T i : 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? 3 / 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 T i : taking book i with probability x i 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 T i : taking book i with probability x i • Transition operator T = x 1 T 1 + · · · + x n T n 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 T i : taking book i with probability x i • Transition operator T = x 1 T 1 + · · · + x n T n • 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 T i : taking book i with probability x i • Transition operator T = x 1 T 1 + · · · + x n T n • 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 T i : taking book i with probability x i • Transition operator T = x 1 T 1 + · · · + x n T n • Stationary distribution? 1-Eigenvector of T • Spectrum? Eigenvalues of T Theorem (Brown, Bidigare ’99) Each S ⊆ { 1 , . . . , n } contributes the eigenvalue � i ∈ S x i with multiplicity the number of derangements of S. 4 / 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 5 / 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 6 / 24
Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion Directed sandpile model on a line with thresholds 1 σ 1 111 τ 2 σ 1 τ 1 σ 1 τ 3 σ 1 τ 3 τ 1 τ 2 110 011 101 τ 1 τ 1 τ 2 σ 1 σ 1 τ 2 τ 3 τ 3 σ 1 τ 2 τ 3 τ 3 τ 2 τ 2 010 100 001 τ 1 τ 1 τ 1 σ 1 τ 3 τ 3 τ 2 000 7 / 24 τ 1
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: � ( λ − ( y S + x S )) T Sc det( M τ − λ 1) = S ⊆ V where S c = V \ S and T S = � v ∈ S T v 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: � ( λ − ( y S + x S )) T Sc det( M τ − λ 1) = S ⊆ V where S c = V \ S and T S = � v ∈ S T v Theorem (ASST’13) Mixing time: at most 2( n T + c − 1) p 8 / 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! 9 / 24
Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion Decomposition of the configuration space (lumping) 123 3 3 213 2 3 2 3 1 1 132 2 3 2 312 3 1 2 1 2 231 1 1 321 10 / 24
Some Markov chains Why are they nicely behaved? Intermezzo: a monoid on trees Conclusion Let’s train on a simpler example 321 π 1 π 2 π 1 π 2 231 312 π 2 π 1 π 2 π 1 213 132 π 1 π 2 π 1 π 2 123 11 / 24
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