Uniform Sampling of Subshifts of Finite Type Ir` ene Marcovici With the support of the European INTEGER project Institut ´ Elie Cartan de Lorraine, Universit´ e de Lorraine, Nancy, France AofA’15, Strobl Monday 8 June 2015 Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Subshifts of finite type Let us color the vertices of the lattice Z d using a finite number of colors , with the constraint that some pairs of colors are not allowed for adjacent sites . Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Subshifts of finite type Let us color the vertices of the lattice Z d using a finite number of colors , with the constraint that some pairs of colors are not allowed for adjacent sites . Questions What do typical configurations look like? Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Subshifts of finite type Let us color the vertices of the lattice Z d using a finite number of colors , with the constraint that some pairs of colors are not allowed for adjacent sites . Questions What do typical configurations look like? How to define “the uniform distribution” on infinite allowed colorings? Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Subshifts of finite type Let us color the vertices of the lattice Z d using a finite number of colors , with the constraint that some pairs of colors are not allowed for adjacent sites . Questions What do typical configurations look like? How to define “the uniform distribution” on infinite allowed colorings? How to sample configurations uniformly ? Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Subshifts of finite type Let us color the vertices of the lattice Z d using a finite number of colors , with the constraint that some pairs of colors are not allowed for adjacent sites . Questions What do typical configurations look like? How to define “the uniform distribution” on infinite allowed colorings? How to sample configurations uniformly ? Given the set A of colors and the (finite) list of constraints, the set Σ of allowed configurations is called a: subshift of finite type (SFT) . Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Subshifts of finite type Let us color the vertices of the lattice Z d using a finite number of colors , with the constraint that some pairs of colors are not allowed for adjacent sites . Questions What do typical configurations look like? How to define “the uniform distribution” on infinite allowed colorings? How to sample configurations uniformly ? Given the set A of colors and the (finite) list of constraints, the set Σ of allowed configurations is called a: subshift of finite type (SFT) . It is a subset of A Z d , which is shift-invariant. Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Example Fibonacci / golden mean / hard-core (or hard-square) subshift Set of configurations without two consecutive black squares. Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Example Fibonacci / golden mean / hard-core (or hard-square) subshift Set of configurations without two consecutive black squares. A two-dimensional configuration Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Example Fibonacci / golden mean / hard-core (or hard-square) subshift Set of configurations without two consecutive black squares. A one-dimensional configuration A two-dimensional configuration Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Example Fibonacci / golden mean / hard-core (or hard-square) subshift Set of configurations without two consecutive black squares. A one-dimensional configuration 0 1 A two-dimensional Graph of allowed transitions configuration in one-dimension Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
One-dimensional SFT Let A be an alphabet with n letters, and let A ∈ M n ( { 0 , 1 } ). Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
One-dimensional SFT Let A be an alphabet with n letters, and let A ∈ M n ( { 0 , 1 } ). One-dimensional subshift of finite type The subshift of finite type associated to A is the set Σ A of words w ∈ A Z such that if A i , j = 0, w does not contain the pattern ij . � 1 if ij is an allowed pattern , A i , j = 0 if ij is a forbidden pattern . Σ A = { w ∈ A Z ; ∀ k ∈ Z , A w k , w k +1 = 1 } . Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
One-dimensional SFT Let A be an alphabet with n letters, and let A ∈ M n ( { 0 , 1 } ). One-dimensional subshift of finite type The subshift of finite type associated to A is the set Σ A of words w ∈ A Z such that if A i , j = 0, w does not contain the pattern ij . � 1 if ij is an allowed pattern , A i , j = 0 if ij is a forbidden pattern . Σ A = { w ∈ A Z ; ∀ k ∈ Z , A w k , w k +1 = 1 } . In what follows, we assume that the matrix A is irreducible and aperiodic. Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
The Parry measure From Perron-Frobenius theory, the matrix A has a real eigenvalue λ > 0 such that | µ | ≤ λ for any other eigenvalue µ . Furthermore, there is a unique choice of r 1 , . . . , r n ≥ 0 such that � n i =1 r i = 1 and r 1 r 1 . . . . A = λ . . . r n r n Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
The Parry measure From Perron-Frobenius theory, the matrix A has a real eigenvalue λ > 0 such that | µ | ≤ λ for any other eigenvalue µ . Furthermore, there is a unique choice of r 1 , . . . , r n ≥ 0 such that � n i =1 r i = 1 and r 1 r 1 . . . . A = λ . . . r n r n Definition of the Parry measure The Parry measure is the (shift-invariant) Markov measure π on A Z of transition matrix P defined, for any i , j ∈ A , by r j P i , j = A i , j . λ r i Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
The Parry measure From Perron-Frobenius theory, the matrix A has a real eigenvalue λ > 0 such that | µ | ≤ λ for any other eigenvalue µ . Furthermore, there is a unique choice of r 1 , . . . , r n ≥ 0 such that � n i =1 r i = 1 and r 1 r 1 . . . . A = λ . . . r n r n Definition of the Parry measure The Parry measure is the (shift-invariant) Markov measure π on A Z of transition matrix P defined, for any i , j ∈ A , by r j P i , j = A i , j . λ r i For a word a 1 . . . a k ∈ A k , π ( a 1 . . . a k ) = π ( a 1 ) P a 1 , a 2 . . . P a k − 1 , a k . Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
The Parry measure The Parry measure π is “the uniform distribution” on Σ A . Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
The Parry measure The Parry measure π is “the uniform distribution” on Σ A . Proposition Let µ k be the uniform measure on allowed patterns of length 2 k + 1, centered at position 0. The sequence µ k converges (weakly) to π on A Z . Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Markov-uniform property Proposition The Parry measure is Markov-uniform : for given k ≥ 1 and a , b ∈ A , the value π ( awb ) does not depend on the word w ∈ A k such that awb is allowed. Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Markov-uniform property Proposition The Parry measure is Markov-uniform : for given k ≥ 1 and a , b ∈ A , the value π ( awb ) does not depend on the word w ∈ A k such that awb is allowed. r j Proof. By definition, P i , j = A i , j λ r i . If awb is allowed, then: π ( awb ) = π a P a , w 1 P w 1 , w 2 . . . P w k − 1 , w k P w k , b Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Markov-uniform property Proposition The Parry measure is Markov-uniform : for given k ≥ 1 and a , b ∈ A , the value π ( awb ) does not depend on the word w ∈ A k such that awb is allowed. r j Proof. By definition, P i , j = A i , j λ r i . If awb is allowed, then: π ( awb ) = π a P a , w 1 P w 1 , w 2 . . . P w k − 1 , w k P w k , b r w 1 r w 2 r w k r b = π a · · · λ r a λ r w 1 λ r w k − 1 λ r w k Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Markov-uniform property Proposition The Parry measure is Markov-uniform : for given k ≥ 1 and a , b ∈ A , the value π ( awb ) does not depend on the word w ∈ A k such that awb is allowed. r j Proof. By definition, P i , j = A i , j λ r i . If awb is allowed, then: π ( awb ) = π a P a , w 1 P w 1 , w 2 . . . P w k − 1 , w k P w k , b r w 1 r w 2 r w k r b = π a · · · λ r a λ r w 1 λ r w k − 1 λ r w k π a r b = . λ k +1 r a Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
Measure of maximal entropy Theorem Let M Σ A be the set of translation invariant measures on the SFT Σ A , and let π ∈ M Σ A . The following properties are equivalent. (i) π is the Parry measure associated to Σ A , (ii) π is a Markov-uniform measure on Σ A , (iii) π is the measure of maximal entropy of Σ A , (iv) the entropy of π is equal to the topological entropy h (Σ A ). Ir` ene Marcovici Uniform Sampling of Subshifts of Finite Type
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