Ergodicity of Noisy Cellular Automata:
The Coupling Method and Beyond
Irène Marcovici(B)
Institut Élie Cartan de Lorraine, Université de Lorraine, Nancy, France
[email protected]
Abstract. When perturbating a cellular automaton by a random noise
(positive probability of error, for each cell independently), the system
is generally expected to be ergodic, meaning that during its evolution,
it eventually forgets about its initial condition. For a high noise, this
can be shown by coupling. However, for a small noise, ergodicity is often
very difficult to prove. We present extensions of the coupling method to
small noises when the cellular automaton has some specific properties
(hardcore exclusion, nilpotency, permutivity).
Consider a set of cells indexed by Z, each cell containing a letter from a
finite symbol set S. A cellular automaton (CA) is a dynamical system which acts
locally and synchronously on the configuration space S Z . When the updates are
random, we obtain a probabilistic cellular automaton (PCA): at each time step,
the new content of each cell is randomly chosen, independently of the others,
according to a distribution given by the states in a finite neighbourhood of the
cell. Examples of PCA are given by noisy CA: the updates are governed by a
deterministic rule, which is perturbated by errors with a positive probability.
A PCA is said to be ergodic if it forgets its initial condition, meaning that
it has a unique and attractive invariant measure. A variety of tools have been
developed to study the ergodicity of PCA. But most of them only allow to handle
PCA for which the transition probability to any state given any neighbourhood
states is large enough. In particular, ergodicity is often very difficult to prove for
noisy CA, even in cases where it appears clear from heuristics or simulations.
In Sect. 1, we recall the coupling method and the notion of envelope PCA,
which gives a general framework to prove ergodicity for the high noise regime.
Details and more references can be found in a joint publication with A. Bušić
and J. Mairesse [1]. The next sections are devoted to three examples of families
of CA for which some specific tools allow to prove the ergodicity for small noise.
In Sect. 2, we present new results on the noisy version of the hardcore CA.
These results stem from a joint work with J.B. Martin, motivated by the study
of a percolation game, together with A.E. Holroyd [3]. Then, in Sects. 3 and
4, we consider perturbations of nilpotent and permutive CA. These sections
are based on a work initiated with S. Taati and carried on together with M.
Sablik. It is interesting to note that nilpotent CA and permutive CA present
opposite behaviours within the rich zoology of CA. While nilpotent CA reach in
c Springer International Publishing Switzerland 2016
A. Beckmann et al. (Eds.): CiE 2016, LNCS 9709, pp. 153–163, 2016.
DOI: 10.1007/978-3-319-40189-8 16
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bounded time a configuration where a single symbol remains, bipermutive CA
are expected to converge to the uniform distribution on S Z from a large class of
initial distributions (randomization phenomenon).
1
Definitions
Let S be a finite set of symbols. We equip the set S Z of configurations with the
product topology. For a finite subset K ⊂ Z and an element y ∈ S K , the set
[yK ] = {z ∈ S Z ; ∀k ∈ K, zk = yk } is called a cylinder of base K, and we denote
by C(K) the set of all cylinders of base K. The set of probability distributions
on S Z for the Borelian σ-algebra is denoted by M(S Z ). For a distribution μ ∈
M(S Z ), we denote by μ[yK ] the probability of the cylinder [yK ].
Let m ≥ 1 be an integer, and let n1 , . . . , nm ∈ Z. A local rule of neighbourhood
N = {n1 , . . . , nm } is a function f : S m → S. The cellular automaton (CA) of
local rule f is the function F : S Z → S Z defined by:
∀x ∈ S Z , ∀k ∈ Z, F (x)k = f (xk+n1 , . . . , xk+nm ).
We denote by σ : S Z → S Z the shift map, defined by σ(x)k = xk−1 . By
Curtis-Hedlund-Lyndon theorem, CA are exactly continuous functions that commute with σ.
For probabilistic cellular automata (PCA), the local rule is a function ϕ :
S m → M(S), where M(S) denotes the set of probability distributions on S.
From a configuration x ∈ S Z , cell k is updated by a symbol chosen according to
the distribution ϕ(xk+n1 , . . . , xk+nm ), independently for different cells. A PCA
can be viewed as a Markov chain on S Z . The evolution of a PCA is described by
a family of random variables (X t )t≥0 , where X t represents the configuration at
time t when iterating the dynamics from the (deterministic or random) initial
condition X 0 . Formally, the PCA of local rule ϕ can also be seen as the function
Φ : M(S Z ) → M(S Z )
μ → Φμ
defined on cylinder sets by:
Φμ[yK ] =
[xK+N ]∈C(K+N )
μ[xK+N ]
ϕ(xk+n1 , . . . , xk+nm )(yk ),
k∈K
for any probability distribution μ ∈ M(S Z ). If the initial configuration X 0 has
distribution μ0 , then at time t, configuration X t has distribution Φt μ0 .
Definition 1. Let Φ be a PCA. The distribution π ∈ M(S Z ) is invariant for Φ
if Φπ = Φ. The PCA Φ is ergodic if it has a unique invariant distribution π and
if for any initial distribution μ0 ∈ M(S Z ), (Φt μ0 )t≥0 converges (weakly) to π.
Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond
155
In this article, we will focus on specific families of PCA obtained when adding
random and independent errors in the updates of a deterministic CA.
Let F be a deterministic CA, and let Φ be a PCA with same symbol set and
same neighbourhood as F . We say that Φ is an ε-perturbation of F if its local
function is such that for all x1 , . . . , xm ∈ S, ϕ(x1 , . . . , xm )(f (x1 , . . . , xm )) ≥ 1−ε,
meaning that there is a deviation from F with probability at most ε.
Let us consider a map R : S → M(S), and let F be a CA of local rule f .
The noisy version of the CA F with noise R is the PCA of local rule ϕ = R ◦ f .
Starting from a configuration x ∈ S Z , the CA F is first applied, and then, for
each cell k independently, the symbol at cell k is replaced by a symbol distributed
according to the distribution R(F (x)k ). We denote simply by Ri,j the probability
R(i)(j) for a symbol i to be changed into symbol j. The noise is said to be
positive if for all i, j ∈ S, Ri,j > 0. The matrix R is a stochastic
matrix. The
noise preserves the uniform distribution on S if for all j ∈ S, i∈S Ri,j = 1,
meaning that the matrix R is doubly stochastic.
We will also pay special attention to elementary PCA, of symbol set S =
{0, 1} and neighbourhood N = {0, 1}. They are characterized by four parameters: the probabilities θij = ϕ(i, j)(1) to update a cell by the symbol 1 if its
neighbourhood is in state i, j, for i, j ∈ S 2 . If we further assume that θ01 = θ10 ,
the general tools that have been developed to prove the ergodicity allow to handle more than 90 % of the volume of the cube defined by the parameter space [2].
But for example, the noisy hardcore CA we study in Sect. 3 belongs to an open
domain of the cube where none of those criteria is valid.
2
2.1
The Coupling Method
The Envelope PCA
Intuitively, a PCA is ergodic if it “forgets” its initial condition. In some cases,
it is possible to prove the ergodicity in a constructive way, by making evolve
simultaneously the trajectories from different initial conditions, using a common
source of randomness, and showing that the evolutions of all these trajectories
are asymptotically the same.
The envelope PCA allows to systematize this idea of coupling. Instead of
running the PCA from different initial configurations, we define a new PCA on
an extended alphabet, containing a symbol ? representing sites whose values are
not known (i.e. which may differ between the different copies) and we run it
from a single initial configuration containing only the symbol ?. Each time we
are able to make the different copies match on a cell, the symbol ? is replaced by
the state q ∈ S on which the different copies agree. An evolution of the envelope
PCA thus encodes a coupling of different copies of the original PCA, with a
symbol ? denoting sites where the copies disagree. If the density of symbols ?
converges to 0 when time goes to infinity, it means that the PCA is ergodic.
Let us assume that Φ is a PCA defined on a binary symbol set S = {0, 1},
and let S̃ = {0, 1, ?}. We define a partial order on S̃ by 0 ≺ ? 1. The envelope
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PCA Φ̃ of Φ is the PCA of neighbourhood N and local function ϕ̃ : S̃ m → M(S̃)
defined for q ∈ S by
ϕ̃(y1 , . . . , ym )(q) = min{ϕ(x1 , . . . , xm )(q) ; x1 y1 , . . . , xm ym },
where in the expression above, x1 , . . . , xm are taken in S. The probability of a
transition to the symbol ? is then given by:
ϕ̃(y1 , . . . , ym )(?) = 1 − ϕ̃(y1 , . . . , ym )(0) − ϕ̃(y1 , . . . , ym )(1).
From a configuration y ∈ S̃ Z , cell k is thus updated by the symbol q ∈ S with the
minimum of the probabilities of transition to the symbol q for Φ, taken over all
the values of the neighbourhood of cell k that are compatible with the unknown
cells of y. With the remaining probability, the cell is updated by a ?.
Proposition 1. If the density Φ̃t δ?Z [?] of symbols ? at time t starting from the
initial configuration ?Z converges to 0 as t → ∞, then the PCA Φ is ergodic.
The fact that symbols ? die out is equivalent to the ergodicity of the envelope
PCA Φ̃, but the ergodicity of the original PCA Φ does not imply the ergodicity
of Φ̃. Note also that the definition of the envelope PCA can be extended to sets
of symbols having more than two elements [1].
2.2
Ergodicity Criterion
In this section, we still consider a binary PCA Φ and its envelope PCA Φ̃. In
the evolution of the envelope PCA, at each time step, a cell is updated by the
symbol ? only if it has at least one neighbour in state ?, and in that case, it
becomes a ? with probability at most:
p? = ϕ̃(?, . . . , ?)(?)
= 1 − min ϕ(x1 , . . . , xm )(0) −
x1 ,...,xm ∈S
=
=
max
ϕ(x1 , . . . , xm )(0) −
max
ϕ(x1 , . . . , xm )(1) −
x1 ,...,xm ∈S
x1 ,...,xm ∈S
min
x1 ,...,xm ∈S
ϕ(x1 , . . . , xm )(1)
min
ϕ(x1 , . . . , xm )(0)
min
ϕ(x1 , . . . , xm )(1).
x1 ,...,xm ∈S
x1 ,...,xm ∈S
This quantity measures how much the probability transitions depend on the
value of the neighbourhood.
Let us consider the oriented graph G describing the dependences between
sites in the space-time diagram of the PCA. The set of vertices of G is Z × N,
and there is an edge from (k, t) to (, t+1) if k ∈ +N . For a given parameter p ∈
(0, 1), the directed site percolation on G consists in declaring each site to be open
with probability p, independently for different sites. One can show that there is a
critical value pc (N ) ∈ (0, 1), such that if p < pc (N ), then there is almost surely
no infinite open (oriented) component (note that pc (N ) ≥ 1/Card N ).
By dominating the process of symbols ? in the space-time diagram of the
envelope PCA by a directed site percolation of parameter p? , one proves that if
p? < pc (N ), then the symbols ? die out. Next proposition follows.
Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond
157
Proposition 2. Let pc (N ) be the critical value of the two-dimensional directed
site percolation of neighbourhood N . If p? < pc (N ), then Φ̃t δ?Z [?] −−−−→ 0, so
t→+∞
that the PCA Φ is ergodic.
For a noisy CA with noise R on a binary symbol set, one can check that
unless the CA is constant, p? = |R0,1 − R1,1 |, which shows ergodicity for R0,1
and R1,1 close enough to each other.
For elementary PCA, we have pc (N ) ≈ 0.7. If p? = maxi,j∈S 2 θij −
mini,j∈S 2 θij is smaller than this critical value, then the PCA is ergodic.
2.3
Coupling from the Past
If the PCA Φ̃ is ergodic, that is, if Φ̃t δ?Z [?] −−−−→ 0, then the unique invariant
t→+∞
distribution of π can be sampled exactly by coupling from the past (Propp-Wilson
method). We define the update function h : S̃ m × (0, 1) → S̃ of Φ̃ by:
⎧
⎪
⎨0 if 0 ≤ u ≤ ϕ̃(y1 , . . . , yk )(0),
h(y1 , . . . , ym , u) = 1 if 1 − ϕ̃(y1 , . . . , yk )(1) ≤ u ≤ 1,
⎪
⎩
? otherwise.
This function has the property that if (Uk )k∈Z is a family of independent
random variables, uniformly distributed on (0, 1), then for any y ∈ S̃ Z , the
configuration (Zk )k∈Z defined by Zk = h(yk+n1 , . . . , yk+nm , Uk ) is distributed
according to Φ̃δy .
Let K be a finite subset of Z. We draw a sequence (uk,−t )k∈Z,t≥0 of independent random samples, uniformly distributed on (0, 1). We iterate the envelope
PCA Φ̃ from time −T to time 0, starting with the configuration ?Z at time −T ,
and always using the sample uk,−t to update cell k at time −t, with the help of
the update function h. If for some time T , the resulting configuration obtained
at time 0 with this procedure is such that there are no symbols ? on the cells of
K, then the symbols observed on K are distributed according to the marginal
of the distribution π on K.
After having iterated the envelope PCA from some time −T , the effect of
starting from time −(T + 1) can only be to change some ? in the space time
diagram into symbols 0 and 1 (once a symbol 0 or 1 appears at a cell (k, −t) of
the space time diagram, it is fixed). The fact that Φ̃t δ?Z [?] → 0 ensures that there
exists almost surely a time T such that at time t = 0, there are no more symbols
? on the cells of K. Note that when implementing this sampling procedure, it is
enough to consider only the cells that are in the dependence cone of K.
3
The Noisy Hardcore CA
Let us consider the elementary PCA Φ defined by the parameters θ00 = 1 −
p, θ01 = θ10 = θ11 = q, with 0 < p + q ≤ 1. This PCA is the noisy version of
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I. Marcovici
the deterministic CA of local rule f (i, j) = (1 − i)(1 − j), with a noise defined
by R1,0 = p and R0,1 = q. The rule f can be seen as an exclusion rule: a cell
becomes a 1 if it has no neighbour in state 1, and a 0 otherwise. For Φ, after
applying that deterministic dynamics, each 1 is changed into a 0 with probability
p, and each 0 is changed into a 1 with probability q, independently.
Until recently, the question of ergodicity of Φ was a repertoried open problem.
This PCA is closely related to the enumeration of directed lattice animals, which
are classical objects in combinatorics. It also appears in the study of a percolation
game [3]. With the notations of Sect. 2.2, we have p? = 1 − p − q. Thus, the
criterion of Proposition 2 provides the ergodicity of Φ for p + q > 0.3 or so, but is
of no help for smaller values of p + q. In that case, the comparison with oriented
percolation is too rough to prove that symbols ? die out. Nevertheless, one can
prove the following (see [3] for complete proof in the case p = 0 or q = 0).
Proposition 3. The noisy hardcore CA, that is, the elementary PCA Φ defined
by θ00 = 1 − p, θ01 = θ10 = θ11 = q, is ergodic for any parameters p + q < 1.
Proof (sketch). The local function of Φ̃ is given by the following probability
transitions (time is going up, and symbol ∗ represents any element of S̃).
1
0
0
with probability 1 − p
with probability p
0
1
0
1
∗
with probability q
with probability 1 − q
∗
1
1
?
0
?
0
?
with probability q
with probability 1 − p − q
with probability p
?
?
0
The envelope PCA Φ̃ is itself the noisy version of the deterministic CA F̃
(which is the deterministic CA obtained when taking p = q = 0 in the table
above), with a noise R̃ that changes any symbol into a 0 with probability p, and
into a 1 with probability q.
For a given configuration in S̃ Z , let us weight the occurrences of the symbols
? as follows:
– if a ? is followed by the pattern 01, then it receives weight 3;
– if a ? is followed by a 0 and then by something other than a 1, it receives
weight 2;
– otherwise, a ? receives weight 1.
One can prove that the weight can only decrease under the action of the
deterministic CA F̃ . Precisely, if μ is a shift-invariant and reflection-invariant
distribution on S̃ Z , then F̃ μ[?01] + F̃ μ[?0] + F̃ μ[?] ≤ μ[?01] + μ[?0] + μ[?].
We now add the random noise R̃, and consider the PCA Φ̃. Let μ be an
invariant distribution of Φ̃. By some computations, one can prove that we necessary have μ[?] = 0, since otherwise, we would get Φ̃μ[?01] + Φ̃μ[?0] + Φ̃μ[?] <
μ[?01] + μ[?0] + μ[?], which would be in contradiction with the fact that μ is an
invariant distribution of Φ̃.
Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond
159
Thus, there is no shift-invariant and reflection-symmetric stationary distribution in which the symbol ? appears with positive probability. Let us iterate
Φ̃ from the configuration ?Z . By a coupling argument, the density Φ̃t δ?Z [?] is
decreasing with t. If it was not converging to 0, then we could extract a convergent subsequence of the Cesàro sums of Φ̃t δ?Z and obtain an invariant distribution μ of Φ̃ satisfying μ[?] > 0, which is not possible. Thus, Φ̃t δ?Z [?] −−−−→ 0,
and Φ is ergodic.
4
t→+∞
Perturbating a Nilpotent CA
Let F be a nilpotent CA. It means that there exists an integer N such that F N
is a constant function, equal to αZ for some symbol α ∈ S.
If F is not the constant function equal to αZ , then for an ε-perturbation of F
with small ε, the value p? is close to 1. Thus, Proposition 2 (and its analogous for
larger symbol sets) cannot be used to prove the ergodicity of an ε-perturbation
of F . In that case, the envelope PCA as defined in Sect. 2.1 is not an adapted
tool. Nevertheless, once again, the coupling method can be used to prove the
ergodicity.
Proposition 4. Let F be a nilpotent CA. There exists εc > 0 such for ε < εc ,
any ε-perturbation of F is ergodic.
Proof. Let ε > 0, and let Φε be an ε-perturbation of F . We prove that if ε is
small enough, we can couple all the trajectories of Φε .
Let K be a finite subset of points of Z. We consider a configuration (xk,0 )k∈Z
obtained at time t = 0, after iterating Φε from a given time in the past, using a
sequence (uk,−t )k∈Z,t≥0 of independent samples, uniformly distributed in (0, 1),
and an update function having the following property: if uk,−t > ε, then cell
k is updated according to the local rule of the deterministic CA F , while if
uk,−t ≤ ε, the value may differ. We prove that almost surely, there exists a time
T > 0 such that the evolutions from all the possible starting configurations at
time −T provide the same sequence (xk,0 )k∈K at time 0.
We define recursively the sets Ni by N0 = {0}, and Ni+1 = Ni + N =
{a + b ; a ∈ Ni , b ∈ N } for i ≥ 0, so that Nt is the neighbourhood of F t .
For k ∈ Z, and times t, i ≥ 0, we define Vi (k, −t) = {(k + , −t − i), ∈ Ni }.
It is the set of cells at time −t − i in the space-time diagram from which the
state of cell k at time −t may depend.
We also introduce W (k, −t) = 0≤i≤N −1 Vi (k, −t), where we recall that N
is such that F N is constant, equal to αZ .
We call a cell (k, −t) an error if uk,−t ≤ ε. Since F N is a constant function,
if there is no error in W (k, −t0 ), then the value xk,−t0 of cell k at time −t0
does not depend on the value of the configuration at time −t0 − N (we have
xk,−t0 = α in all cases).
For k ∈ Z, t ≥ 0, let us define the set:
∅ if there is no error in W (k, −t),
E(k, −t) =
VN (k, −t) if there is at least one error in W (k, −t).
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I. Marcovici
We define recursively the sets (Ai )i≥0 by A0 = K × {0}, and:
E(k, −t) for i ≥ 0.
Ai+1 = E(Ai ) =
(k,−t)∈Ai
Note that if (k, −t) ∈ Ai , then t = iN . We have the following property: if Ai
is empty, then if we iterate Φε from time −iN to time 0, using the samples
(uk,−t )k∈Z,0≤t<iN , the values (xk,0 )k∈K obtained on K at time 0 do not depend
on the choice of the configuration (xk,−iN )k∈Z ∈ S Z from which we start at time
−iN .
In the figure below, errors are represented by red dots. Blue domains represent
cells that are known to be in state α (there are no errors affecting them in the
last N time steps). Black domains represent cells for which we need further
information in the past to determine their state.
Time
A0 = K × {0}
0
A1
−N
−2N
−3N
A2
A3 = ∅
Let us prove that if ε is small enough, then almost surely, there exists an
integer after which all the sets Ai are empty.
We set mi = Card Ni . Let (, −t) be an error, with t = iN +j, 0 ≤ j ≤ N −1.
We have (, −t) ∈ W (k, −iN ) if and only if k ∈ −Nj . Thus, the number of points
(k, −iN ) such that (, −t) is an error of W (k, −iN ) is bounded by mj ≤ mN −1 .
If there is an error in W (k, −iN ), then Card E(k, −iN ) = mN . It follows that a
given error has a contribution of at most L = mN −1 mN points to Ai+1 .
Let M = m0 + m1 + . . . + mN −1 . We
have Card W (k, −t) = M for any
k ∈ Z, t ≥ 0. The number of points of k∈Ai W (k, −iN ) is thus smaller than
(Card Ai ) × M , and each point is an error with probability ε, independently.
Consequently, Card Ai+1 is bounded by the sum of (Card Ai ) × M independent
random variables, whose value is L with probability ε, and 0 with probability
1 − ε. If ε < 1/LM , a comparison with a branching process proves that there is
extinction: almost surely, the sets Ai are eventually empty. Consequently, Φε is
ergodic (note that the bound given for ε is rough and can certainly be improved).
Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond
5
161
Noisy Permutive Cellular Automata
Let F be a CA of neighbourhood N = {, + 1, . . . , r} and local function f :
S m → S, with m = r − + 1 ≥ 2. We say that F is left-permutive (resp.
right-permutive) if, for all w = w+1 · · · wr ∈ S m−1 , the mapping:
τw : S → S
a → f (aw)
(resp. f (wa))
is bijective. A CA is permutive if it is either left or right-permutive. It is bipermutive if it is both left and right-permutive. For example, if S = Zn and a, b, c ∈ Zn ,
the affine CA defined by f (x, y) = ax + by + c is left-permutive (resp. rightpermutive) if a (resp. b) is invertible in Zn .
Let F be a permutive CA. Using the bijections τw one can prove that F
is surjective. For deterministic CA, surjectivity is equivalent to preserving the
uniform distribution λ on S Z (that is, the product of the uniform distribution
on S). Next proposition shows that when adding a noise preserving λ, the PCA
indeed converges to λ. The proof below is adapted from a work of Vasilyev [2,4].
Proposition 5. Let F be a permutive CA, and let R be a positive noise preserving the uniform distribution. The noisy version Φ of F with noise R is ergodic,
and its unique invariant distribution is the uniform distribution λ.
Proof. We will prove that for any N ∈ N, and any initial distribution μ on S Z ,
the marginal distribution of Φt μ on K = {−N, . . . , N } converges exponentially
to the uniform Bernoulli distribution on S K , that we denote by λK . Precisely, we
will prove that for any N ∈ N, there exists θ < 1 such that for any distribution
, we have: ∀t ≥ 0, ||(Φt μ)|K − λK ||1 ≤ 2θt , where for u : S K → R,
μ on S Z
||u||1 = x∈S K |u(x)|.
Let us first assume that F is left-permutive and that N = {0, 1, . . . , r}, and
let w ∈ S r . By permutivity of F , we have a bijection
σw : S K −→ S K
x −→ f (xw),
where we still denote by f the map from S K∪{N +1,...,N +r} to S K induced by the
local function of the CA F .
So, when fixing the word w as a boundary condition on the right of K, the
noisy CA Φ maps a word x ∈ S K to a random word ZK distributed according
to a product distribution with marginal distribution R(yk ) at site k ∈ K, where
y = σw (x). From a given x ∈ S K , we denote by P
w (x, z) the probability for ZK
to be equal to some z ∈ S K , so that: Pw (x, z) = k∈K Ryk ,zk .
Recall that λK is the uniform distribution on S K . The map σw being bijective, it preserves λK . By assumption, the noise R also preserves the uniform
distribution, so that we obtain Pw λK = λK .
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I. Marcovici
For any w ∈ S r , the transition matrix Pw is positive. Therefore, there exists
θw < 1 such that for any probability distributions ν, ν on S K , we have
||Pw ν − Pw ν ||1 ≤ θw ||ν − ν ||1 ,
the above inequality being true in particular for θw = 1 − εw , where εw =
min{Pw (i, j) ; i, j ∈ S}.
Let us set θ = max{θw ; w ∈ S r }. It follows that for any sequence (wt )t≥0 of
words of S r , we have: ||Pwt−1 . . . Pw1 Pw0 ν − Pwt−1 . . . Pw1 Pw0 ν ||1 ≤ θt ||ν − ν ||1 .
In particular, for ν = λK , we obtain that for any distribution ν on S K and any
sequence (wt )t≥0 of words of S r , ||Pwt−1 . . . Pw1 Pw0 ν−λK ||1 ≤ θt ||ν−λK ||1 ≤ 2θt .
Time
t=3
V3 ∼ Pw2 Pw1 Pw0 ν
w3
t=2
V2 ∼ Pw1 Pw0 ν
w2
t=1
V 1 ∼ Pw 0 ν
w1
t=0
V0 ∼ ν
w0
−N
N
N +r
Let now μ be a distribution on S Z . When iterating Φ, it induces a random
sequence of words (Wt )t≥0 on {N + 1, . . . , N + r}. Using the above inequality,
we get:
∀t ≥ 0, ||(Φt μ)|K − λK ||1 ≤
max
w0 ,...,wt−1 ∈S r
||Pwt−1 . . . Pw1 Pw0 μ|K − λK ||1 ≤ 2θt .
If the neighbourhood of F is not of the form N = {0, 1, . . . , r}, then there
exists s ∈ Z such that F ◦ σ s is a left-permutive CA having a neighbourhood
of that form. The noisy version of F ◦ σ s is Φ ◦ σ s , and the previous inequality
provides: ||((Φ ◦ σ s )t μ )|K − λK ||1 ≤ 2θt , for any distribution μ . In particular,
for μ = σ −st μ, since Φ and σ commute, we obtain: ||(Φt μ)|K − λK ||1 ≤ 2θt ,
which ends the proof. The right-permutive case is analogous.
In a collaboration still in progress with S. Taati and M. Sablik, we investigate
the ergodicity of more general noisy surjective CA.
Concerning elementary PCA, it is still a challenging open question whether
they are ergodic as soon as θij ∈ (0, 1) for all i, j ∈ S.
Acknowledgments. The author thanks warmly S. Taati for the joint work on Sects. 4
and 5 and for his careful reading, and M. Sablik for fruitful discussions. This article is
also based on a collaboration with J.B. Martin (Sect. 3) and on a previous work with
A. Bušić and J. Mairesse (Sect. 2).
Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond
163
References
1. Bušić, A., Mairesse, J., Marcovici, I.: Probabilistic cellular automata, invariant measures, and perfect sampling. Adv. Appl. Probab. 45(4), 960–980 (2013)
2. Dobrushin, R.L., Kryukov, V.I., Toom, A.L.: Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. Nonlinear science. Manchester University Press,
Manchester (1990)
3. Holroyd, A.E., Marcovici, I., Martin, J.B.: Percolation games, probabilistic cellular
automata, and the hard-core model (2015). http://arxiv.org/abs/1503.05614
4. Vasilyev, N.B.: Bernoulli and Markov stationary measures in discrete local interactions. In: Developments in Statistics, vol. 1, pp. 99–112. Academic Press, New York
(1978)