Probabilistic Cellular Automata: a statistical mechanics point of ... - IECL

Probabilistic Cellular Automata: a statistical mechanics point of view Pierre-Yves Louis Universit´ e de Poitiers

LIAFA Paris, 10 janvier 2013

Outline

Outline

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics Motivation of the class of reversible PCA dynamics Finitely many interacting sites (finite volume) Infinitely-many interacting sites (infinite volume) 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

2 / 32

Framework

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

3 / 32

Framework

Probabilistic Cellular Automata Terminology:

PCA = random CA = stochastic CA = locally interacting Markov chains

• G = (V , E ) is a graph whose nodes are the elementary entities usually G = Zd or G = Λ b Zd • S a finite space (possible states of an entity) e.g. S = {0, 1}, S = {−1, +1}... • η := (ηk )k∈Λ is called a configuration • local updating stochastic rule: pk (s|η), k ∈ G , s ∈ S, η ∈ S V • pk (· |σ) is a probability on S • ∃ Vk b V

pk (s|η) = pk (s|ηVk )

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

4 / 32

Framework

Mathematical Probability framework • it is a stochastic Markov process (σ(n))n∈N with discrete-time on the state space S Λ • challenge: understand the dynamical evolution in a statistical point of view • A state of the system is a probability measure on the configurations space SV . • P is said to be a (finite volume) PCA (synchronous) dynamics if Y PΛτ (η, σ) = pk (σk |ηΛ τΛc ) k∈Λ

• P is said to be a (infinite volume) PCA dynamics if O P(η, σ) = pk (σk |η) k∈Zd

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

5 / 32

Framework

Some historial references following this point of view • Piatetski-Shapiro • Toom, Vasilyev, Stavskaja, Mitjushin, Kurdumov, Pirogov, 1970–1978; Dawson, 1974 • Kozlov, Vasilyev, 1980 • K¨ unsch, 1984; Malyshev,Ignatyuk, 1987 • Georges,Le Doussal, 1989 • Lebowitz, Maes, Speer, 1990 – Maes, Shlosman 1991

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

6 / 32

Framework

Some remarks

• no general theory, no general results • only few general results from Markov Processes general theory • statistical point of view on the dynamical evolution, long-time behaviour • state of the system is a probability distribution µ on S Λ • candidates for equilibrium/steady states reached in the long-time behaviour are stationary distributions µ which are probability distributions invariant w.r.t. dynamics i.e. P(σ(0) = ·) = µ(·) ⇒ ∀n > 1, P(σ(n) = ·) = µ(·) or, equivalently µP = µ

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

7 / 32

Framework

Detailed balance

• The Markov dynamics is said to be reversible if there exists a probability distribution µ such that P(dσ|η)µ(dη) = P(dη|σ)µ(dσ). It means, that the distribution of the process, starting with µ as initial distribution is invariant under time-reversal. • In particular, such a distribution is a stationary distribution.

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

8 / 32

Framework

Example: Stavskaja Model on Z S = {0, 1}, G = Z, Vk = {k − 1, k}  1 if σk = σk−1 = 1, pk (1|σ) = ε ∈ [0, 1] otherwise. • finitely-many interacting sites Λ b Z: since σ ≡ 1 is absorbing, long-time behaviour is absorption in δ+1 • infinitely-many interacting sites Λ = Z, it exists ε∗ > 0 such that • with ε > ε∗ , the dynamics is ergodic ∀µ starting distribution, lim Pµ (σ(n) = ·) = δ+1 n→∞

• with ε < ε∗ , the dynamics is non-ergodic, in particular

lim Pδ0 (σ(n) = ·) = µε (·) 6= δ+1 .

n→∞

and every stationary distribution translation-invariant is of the form αµε + (1 − α)δ+1 ,

α ∈ [0, 1].

See [Leontovitch & Vaserstein, 1970] and more new results [Mendon¸ca 2011] P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

9 / 32

Framework

Questions

• May be the non-trivial µε is due to the deterministic component of

the updating rule ? • How are these stationary distributions like ? • Why do we care about infinitely-many interacting entities ?

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

10 / 32

Static Probability models for interacting sites: Gibbs measures

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

11 / 32

Static Probability models for interacting sites: Gibbs measures

Static probability models for interacting sites

Aim: try to find candidates and characterise the equilibrium distributions of these d dynamics, on the infinite space S Z E = S Λ , Λ b Zd , S = {−1, +1}, |E | = 2|Λ| P Energy HΛ (η) = − i∼j Ji,j ηi ηj

PτΛ ({ω : σ(ω) = (ηk )k∈Λ }) =

1 −βHΛ (ηΛ τΛc ) e , ZΛτ

ZΛτ =

X

e −βHΛ (ηΛ τΛc )

ηΛ ∈E

with the inverse temperature parameter β > 0. Ising potential ϕi,j (σ) = Ji,j σi σj when i ∼ j.

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

12 / 32

Static Probability models for interacting sites: Gibbs measures

Static probability models for interacting sites: Gibbs distributions • finitely-many interacting sites µτΛ (σΛ ) =

1 −βHΛ (σΛ τΛc ) e ZΛτ

Remark: Compatibility property : Λ2 ⊂ Λ1 τΛ1c σΛ1 \Λ2

c (σΛ2 |σΛ1 \Λ2 ) = µΛ2 µτΛΛ1 1

(σΛ2 )

• infinitely-many interacting sites, ∀Λ b Zd , ∀τ boundary condition, µ( (·)Λ | τΛc ) = µτΛ ( · ) d

Main motivation: construct distributions on S Z , which are “locally” the same. When the number of interacting sites goes to infinity (thermodynamic limit), there may be different limiting distributions: phase transition. P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

13 / 32

Static Probability models for interacting sites: Gibbs measures

Relation IPS/Gibbs measure

d

Let µ be a Gibbs measure on S Z w.r.t. a potential (ϕA )A⊂Zd . d it exists a continuous-time Markov process on S Z with a sequential updating rule (interacting particle system dynamics), such that R = G(ϕ),

Si ⊂ G(ϕ).

Ergodicity of the dynamics ; Phase transition w.r.t. (ϕA )A⊂Zd .

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

14 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

15 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics

Specificities of the synchronous updating

• the existence of some reversible steady state costs more than for a

sequential updating dynamics: the reversed probability kernel is required to be of product form • for some given Gibbs measure µ, on the contrary to Markov dynamics

with sequential updating, there is no canonical way to construct a PCA dynamics, which would admit µ as stationary measure.

Proposition (Dawson, 1975) Let µ be a Gibbs measure with respect to some nearest neighbour 2 potential on {0, 1}Z . There is no translation invariant PCA, which could have µ as reversible measure.

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

16 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics

Associated space potential

Theorem (P. Dai Pra / C. Maes, 1992) d

Let P = (pk )k∈Zd be a PCA dynamics on S Z , translation invariant and purely stochastics pk (s|σ) > 0 d

if ∃ϕ potential on S Z s.t. G(ϕ) ∩ Si 6= ∅ then Si ⊂ G(ϕ).

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

17 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Motivation of the class of reversible PCA dynamics

Outline

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics Motivation of the class of reversible PCA dynamics Finitely many interacting sites (finite volume) Infinitely-many interacting sites (infinite volume) 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

18 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Motivation of the class of reversible PCA dynamics

Motivation for another PCA family of dynamical models

• There are techniques to write down Markov processes associated to a given potential, such that the equilibrium distributions are characterised as Gibbs distributions w.r.t this potential: mainly Hastings-Metropolis algorithm (flip dynamics) and Gibbs-sampler. • considering the case of one site Λ = {k}, the Gibbs-sampler for the Ising potential gives   P exp (β j∼k K(j−k)σk σj +βhσk )) P µ{k} (σk |σVk c ) = j +h))  2 cosh(β( j∼k K(j−k)σ  P 1 = 2 1 + σk tanh(β( j∼k K(j − k)σj + h))

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

19 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Motivation of the class of reversible PCA dynamics

d

A family of reversible PCA dynamics on {−1, +1}Z Define, for k ∈ Zd , s ∈ S, η ∈ S Z d pk ( s |η) =

X

1 1 + s tanh β( K(j − k)ηj + h) 2 j

where • β > 0 (tunning the spatial and temporal dependence) • K : Zd → R is symmetric and with finite range Vk := {j ∈ Zd , K(j − k) 6= 0} b Zd • h∈R

Note, it is a purely stochastic dynamics. d

∀k, ∀s ∈ S, ∀η ∈ S Z ,

pk (s|η) > 0 d

It is the form of a general reversible PCA dynamics on {−1, +1}Z . P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

20 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Finitely many interacting sites (finite volume)

Outline

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics Motivation of the class of reversible PCA dynamics Finitely many interacting sites (finite volume) Infinitely-many interacting sites (infinite volume) 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

21 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Finitely many interacting sites (finite volume)

Identification of the finite-volume steady states For the PCA PΛτ on S Λ with fixed boundary condition τ there exists a unique stationary (reversible) prob. measure νΛτ given by   P X 1 Y βhηk τ K(j − k)(ηΛ τΛc )j + βh e βηk j∈Λc K(k−j)τj e cosh β νΛ (ηΛ ) = τ WΛ d k∈Λ

j∈Z

and with periodic boundary condition: νΛper (ηΛ ) =

X 1 Y βhηk e cosh(β K(j − k)(ηΛper )k 0 ) per WΛ d k∈Λ

j∈Z

1 − V ∩Λ6=∅ ϕV (η per ) k k e with WΛper X K(j − k)ηj + βh) ϕ{k} (η) = −βhηk ϕVk (η) = −log cosh(β rewrite νΛper (ηΛ ) =

P

j∈Vk

νΛper

is the finite volume Gibbs measure with periodic boundary associated with the

potential ϕ. P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

22 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Finitely many interacting sites (finite volume)

Effect of the synchronous updating scheme on the stationary distribution As consequence: when the update probability pk (s|η) is applied

• with sequential updating scheme: it is the Hastings-Metropolis Markov chain (coincide with the Gibbs sampler), the dynamics converges toward the Gibbs measure associated with the n.n. Ising potential

• with parallel updating scheme: PCA case, the dynamics converges (for periodic boundary conditions) towards the Gibbs measure related to the potential ϕ X ϕVk (η) = −log cosh(β ηj ) j∈Vk

Fact: For some given probability updating rule, changing the updating scheme modifies the nature of the stationary measure.

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

23 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Infinitely-many interacting sites (infinite volume)

Outline

1 Framework 2 Static Probability models for interacting sites: Gibbs measures 3 Loss of ergodicity at infinite volume for a potential-related family of

PCA dynamics Motivation of the class of reversible PCA dynamics Finitely many interacting sites (finite volume) Infinitely-many interacting sites (infinite volume) 4 Synchronisation for PCA with spin space [0, 1]

P.-Y. Louis (L.M.A. Poitiers)

PCA

LIAFA Paris, 10 janvier 2013

24 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Infinitely-many interacting sites (infinite volume)

Characterisation of stationary meas. as Gibbs measures (Kozlov-Vasilyev, 1980; K¨ unsch 1984; Dai Pra 1992; Dai Pra, Louis, Rœlly 2002) Relations between R, S and G(ϕ) : R = G(ϕP ) ∩ S,

P.-Y. Louis (L.M.A. Poitiers)

PCA

Ri = Si .

LIAFA Paris, 10 janvier 2013

25 / 32

Loss of ergodicity at infinite volume for a potential-related family of PCA dynamics Infinitely-many interacting sites (infinite volume)

Ergodicity criterion for P.C.A. Dobrushin-Vasershtein Criterion If γ := sup

X

j sup pk (+1|σVk ) − pk (+1|σVk )