Probabilistic cellular automata with memory two Ir` ene Marcovici - PowerPoint PPT Presentation

Probabilistic cellular automata with memory two Ir` ene Marcovici Joint work with J´ erˆ ome Casse (NYU Shanghai) Institut ´ Elie Cartan de Lorraine, Univ. de Lorraine, Nancy ALEA in Europe Workshop Vienna, October 13, 2017 Ir` ene Marcovici Probabilistic cellular automata with memory two

Content 1 Introductory example (the 8-vertex model) 2 Invariant product measure and ergodicity 3 Directional reversibility 4 Horizontal Zig-zag Markov Chains 5 A TASEP model Ir` ene Marcovici Probabilistic cellular automata with memory two

Content 1. Introductory example (the 8-vertex model) Ir` ene Marcovici Probabilistic cellular automata with memory two

Probabilistic cellular automata with memory two η t +1 η t η t − 1 Finite symbol set: S Ir` ene Marcovici Probabilistic cellular automata with memory two

Probabilistic cellular automata with memory two η t +1 η t η t − 1 Finite symbol set: S Ir` ene Marcovici Probabilistic cellular automata with memory two

Probabilistic cellular automata with memory two d η t +1 a c η t b η t − 1 n n − 1 n + 1 Finite symbol set: S Ir` ene Marcovici Probabilistic cellular automata with memory two

Probabilistic cellular automata with memory two d η t +1 a c η t b η t − 1 n n − 1 n + 1 Finite symbol set: S For any a , b , c ∈ S , T ( a , b , c ; · ) is a probability distribution on S . The value η t +1 ( n ) is equal to d with probability T ( a , b , c ; d ). Conditionnally to η t and η t − 1 , the values ( η t +1 ( n )) n ∈ Z t +1 are independent. Ir` ene Marcovici Probabilistic cellular automata with memory two

Probabilistic cellular automata with memory two η t +2 d η t +1 a c η t b η t − 1 n n − 1 n + 1 Finite symbol set: S For any a , b , c ∈ S , T ( a , b , c ; · ) is a probability distribution on S . The value η t +1 ( n ) is equal to d with probability T ( a , b , c ; d ). Conditionnally to η t and η t − 1 , the values ( η t +1 ( n )) n ∈ Z t +1 are independent. Ir` ene Marcovici Probabilistic cellular automata with memory two

Probabilistic cellular automata with memory two η t +2 d η t +1 a c η t b η t − 1 n n − 1 n + 1 Finite symbol set: S For any a , b , c ∈ S , T ( a , b , c ; · ) is a probability distribution on S . The value η t +1 ( n ) is equal to d with probability T ( a , b , c ; d ). Conditionnally to η t and η t − 1 , the values ( η t +1 ( n )) n ∈ Z t +1 are independent. Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The 8-vertex PCA of parameters p , r ∈ (0 , 1): T (0 , 0 , 1; · ) = T (1 , 0 , 0; · ) = B ( p ) , T (0 , 1 , 1; · ) = T (1 , 1 , 0; · ) = B (1 − p ) , T (0 , 1 , 0; · ) = T (1 , 1 , 1; · ) = B ( r ) , T (1 , 0 , 1; · ) = T (0 , 0 , 0; · ) = B (1 − r ) . Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The 8-vertex PCA of parameters p , r ∈ (0 , 1): T (0 , 0 , 1; · ) = T (1 , 0 , 0; · ) = B ( p ) , T (0 , 1 , 1; · ) = T (1 , 1 , 0; · ) = B (1 − p ) , T (0 , 1 , 0; · ) = T (1 , 1 , 1; · ) = B ( r ) , T (1 , 0 , 1; · ) = T (0 , 0 , 0; · ) = B (1 − r ) . r = 0 . 2 and p = 0 . 9 Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The 8-vertex PCA of parameters p , r ∈ (0 , 1): T (0 , 0 , 1; · ) = T (1 , 0 , 0; · ) = B ( p ) , T (0 , 1 , 1; · ) = T (1 , 1 , 0; · ) = B (1 − p ) , T (0 , 1 , 0; · ) = T (1 , 1 , 1; · ) = B ( r ) , T (1 , 0 , 1; · ) = T (0 , 0 , 0; · ) = B (1 − r ) . As a special case, for p = r , we have: T ( a , b , c ; · ) = p δ a + b + c mod 2 + (1 − p ) δ a + b + c +1 mod 2 . p = r = 0 . 2 Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA (1) (3) (5) (7) (2) (4) (6) (8) Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA (1) (3) (5) (7) (2) (4) (6) (8) Arrow pointing up iff same colour. 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 1 Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA W ( O ) = � x ∈ V n w type( x ) W ( O ) P ( O ) = O ∈ On W ( O ) . � Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA W ( O ) = � x ∈ V n w type( x ) W ( O ) P ( O ) = O ∈ On W ( O ) . � w 1 = w 2 = a w 3 = w 4 = b w 5 = w 6 = c w 7 = w 8 = d (1) (3) (5) (7) (2) (4) (6) (8) Hypothesis: b + d = a + c . Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA T (0 , 0 , 1; · ) = T (1 , 0 , 0; · ) = B ( p ) , W ( O ) = � x ∈ V n w type( x ) T (0 , 1 , 1; · ) = T (1 , 1 , 0; · ) = B (1 − p ) , T (0 , 1 , 0; · ) = T (1 , 1 , 1; · ) = B ( r ) , W ( O ) P ( O ) = O ∈ On W ( O ) . � T (1 , 0 , 1; · ) = T (0 , 0 , 0; · ) = B (1 − r ) . w 1 = w 2 = a w 3 = w 4 = b w 5 = w 6 = c w 7 = w 8 = d (1) (3) (5) (7) (2) (4) (6) (8) Hypothesis: b + d = a + c . PCA with r = b / ( b + d ), p = a / ( a + c ) Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA T (0 , 0 , 1; · ) = T (1 , 0 , 0; · ) = B ( p ) , W ( O ) = � x ∈ V n w type( x ) T (0 , 1 , 1; · ) = T (1 , 1 , 0; · ) = B (1 − p ) , T (0 , 1 , 0; · ) = T (1 , 1 , 1; · ) = B ( r ) , W ( O ) P ( O ) = O ∈ On W ( O ) . � T (1 , 0 , 1; · ) = T (0 , 0 , 0; · ) = B (1 − r ) . w 1 = w 2 = a w 3 = w 4 = b w 5 = w 6 = c w 7 = w 8 = d (1) (3) (5) (7) (2) (4) (6) (8) Hypothesis: b + d = a + c . PCA with r = b / ( b + d ), p = a / ( a + c ) Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The uniform Horizontal Zig-zag Product Measure (HZPM) is invariant. What are the conditions on the transition kernel T for having an invariant HZPM? Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The uniform Horizontal Zig-zag Product Measure (HZPM) is invariant. What are the conditions on the transition kernel T for having an invariant HZPM? Does the PCA asymptotically forget its initial condition (ergodicity)? Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The uniform Horizontal Zig-zag Product Measure (HZPM) is invariant. What are the conditions on the transition kernel T for having an invariant HZPM? Does the PCA asymptotically forget its initial condition (ergodicity)? What are the directional properties of the stationary space-time diagram (reversibility, i.i.d. lines...)? Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA The uniform Horizontal Zig-zag Product Measure (HZPM) is invariant. What are the conditions on the transition kernel T for having an invariant HZPM? Does the PCA asymptotically forget its initial condition (ergodicity)? What are the directional properties of the stationary space-time diagram (reversibility, i.i.d. lines...)? Ir` ene Marcovici Probabilistic cellular automata with memory two

Example: the 8-vertex PCA p = r = 0 . 2 Multi-directional reversibility Ir` ene Marcovici Probabilistic cellular automata with memory two

Content 2. Invariant product measure and ergodicity Ir` ene Marcovici Probabilistic cellular automata with memory two

Condition for having an invariant HZPM Theorem Let A be a PCA with transition kernel T and let p be a probability vector on S . The HZPM π p is invariant for A if and only if � ∀ a , c , d ∈ S , p ( d ) = p ( b ) T ( a , b , c ; d ) . b ∈ S d η t +1 η t a c η t − 1 b n n − 1 n + 1 Ir` ene Marcovici Probabilistic cellular automata with memory two

Condition for having an invariant HZPM r ℓ Ir` ene Marcovici Probabilistic cellular automata with memory two

Condition for having an invariant HZPM r ℓ Ir` ene Marcovici Probabilistic cellular automata with memory two

Condition for having an invariant HZPM r ℓ Ir` ene Marcovici Probabilistic cellular automata with memory two

Condition for having an invariant HZPM b ′ b ′ b ′ b ′ 0 1 2 3 a ′ a ′ a ′ a ′ a ′ 0 1 2 3 4 r ℓ b 0 b 1 b 2 b 3 a 0 a 1 a 2 a 3 a 4 For given boundary conditions ℓ, r , probability transition: P ( ℓ, r ) (( a 0 , b 0 , a 1 , b 1 , . . . , b k − 1 , a k ) , ( a ′ 0 , b ′ 0 , a ′ 1 , b ′ 1 , . . . , b ′ k − 1 , a ′ k )) For any ℓ, r , the product measure with parameter p is invariant. Ir` ene Marcovici Probabilistic cellular automata with memory two

Ergodicity There exists θ ( ℓ, r ) < 1 such that for any probability distributions ν, ν ′ on S 2 k +1 , we have: || P ( ℓ, r ) ν − P ( ℓ, r ) ν ′ || 1 ≤ θ ( ℓ, r ) || ν − ν ′ || 1 . Ir` ene Marcovici Probabilistic cellular automata with memory two

Ergodicity There exists θ ( ℓ, r ) < 1 such that for any probability distributions ν, ν ′ on S 2 k +1 , we have: || P ( ℓ, r ) ν − P ( ℓ, r ) ν ′ || 1 ≤ θ ( ℓ, r ) || ν − ν ′ || 1 . Define: θ = max { θ ( ℓ, r ) : ( ℓ, r ) ∈ S 2 } Ir` ene Marcovici Probabilistic cellular automata with memory two