Random Walks in Cellular Automata

Topological defects or phase boundaries discerned in a number of one-dimensional cellular automata appear to perform random walks as well as simpler motions. We analyze their properties rigorously using probabilistic methods. This results in a complete classi cation in the partially permutive case. The paper complements [1] where the general framework of tilings and subpermutivity was introduced and non-probabilistic properties were analyzed. 2 Introduction In this paper we give a comprehensive analysis of random walks in one dimensional cellular automata. The results are rigorous and based on probabilistic methods with which we analyze closely related random walks on directed graphs. The paper is mathematically self-contained and can be read as it is. However an applications oriented readers motivation would likely be enhanced by at least a super cial acquaintance of the companion paper [1]. There we present the theory of partial permutive cellular automata including the physical motivation to signal/random walk models, the general algebraic structures, the spectrum of interaction types in the case of multiple walks as well as the subtle walks that still remain unexplained in some cellular automata. The structure of the presentation is as follows. We rst brie y list the basic de nitions and results concerning partially permutive cellular automata. Some of these can be found in more elaborated form in [1] but we believe that frequent crossreferences there would make the reading too cumbersome. After this we proceed with identifying the boundary motion and unveiling its graph theoretic formulation. The graph naturally carries a markovian random walk which in turn uniquely determines the possibly non-markovian motion of the boundary walk. The analysis then branches into two cases (sections 2.1 and 2.2) depending on essentially the strength of the subalphabet interaction. In the rst case both the degenerate case of a rectilinearly moving boundary motion and a proper random walk are characterized via the graph representation. The second case only supports random walks but their structure is more involved. The mixed case is elaborated in Section 2.3. Along the way we present results 3 that identify the types as well as parameters of the boundary motions explicitly. Some examples as well as results relevant to e.g. the well documented elementary cellular automata are also included. However the main bulk of the interface to empirical studies is in [1] where we have collected some pointers to the earlier approaches to identify topological defects in lattice models. 1. Basic de nitions Let S = f0; 1; : : : ; jSj 1g be a nite alphabet i.e. the set of symbols and X = SZ and X(1=2) = SZ+1=2 be the sets of con gurations. Equipped with the product topology they become compact metric spaces homeomorphic with a Cantor set. Let be the left shift by one on a sequence in X [ X(1=2): (x)j = xj+1 8j: A two-block map f(xj; xj+1) de nes the new value of the coordinate xj+1=2. De nition 1.1.: One-dimensional cellular automaton (c.a.) is a dynamical system on X [X(1=2) de ned by a two-block map which commutes with the left shift.The global map from X to X(1=2) or from X(1=2) to X de ned by requiring F (x)j+1=2 = f(xj; xj+1) for all j 2 Z or Z+ 1=2 is continuous and conversely any such continuous map that commutes with the shift is induced by a block map (argued as in [5]). The block map f is also called the rule of the automaton. Our de nition is super cially di erent from the usual de nition of a c.a. However as shown in [1] (De nition 1.3.) there is a simple way via substitutions or tilings to generate from an arbitrary n-block map a two-block map on a larger alphabet. Since this is particularly useful in analyzing permutivity we 4 present the de nition most natural to the subsequent analysis. This by no means restricts the applicability of the results { almost all random walks arising in one-dimensional cellular automata (with a rule of any block length) are still covered. The binary operation (multiplication) represented by the two-block map is conveniently expressed in the form of a Cayley table (see e.g. Figure 1.). The following sets are of paramount importance to this paper. De nition 1.2.: A set Sr S is a right-invariant subalphabet if f(r; Sr) = Sr; 8r 2 Sr i.e. f(r; ) is right permutive on Sr for each r 2 Sr. Left-invariant subalphabets are de ned in the obvious symmetric way. One usually wants to consider maximal such subalphabets i.e. ones that can not be augmented by any element from the complement without loss of the permutivity property. If this set is the full alphabet the c.a. is (left/right)permutive ([5]). If a non-trivial maximal invariant subalphabet exists we call the c.a. partially permutive. These c.a. are much more abundant than the permutive ones. For example in [5] and in [7] permutive rules are considered in detail and a number of important results are established. Two key ones for our analysis are distilled in the following theorem. Recall that the Bernoulli measure B(1=jSj; : : : ; 1=jSj) is the product measure with uniform weights 1=jSj on symbols. Theorem 1.1.: Permutive cellular automata are onto and preserve the uniform Bernoulli measure B(1=jSj; : : : ; 1=jSj) (on X and X(1=2) appropriately). A set of con gurations is generated by a subalphabet S0 if all its elements have their coordinates in this set. 5 Corollary 1.1.: The action of a partially permutive c.a. on the set of con gurations generated by a permutive subalphabet S0 is permutive and preserves the B(1=jS0j; : : : ; 1=jS0j)-measure. 2. Characterization of the boundary motion From this on we only consider the strictly subpermutive case. Since the full alphabet is rarely referred to we use the symbol S for a subalphabet. When two di erent permutive subalphabets exist for a c.a. we have two di erent permutive actions on con gurations generated by them separately. Let the subalphabets be S and T . The natural question to ask then is what happens under the c.a. iteration if these two "phases" are mixed i.e. x 2 X consists of blocks from these subalphabets. In this paper we con ne the analysis to the basic case of two semi-in nite blocks generated from the subalphabets. A compact notation for the set of all such con gurations is ST . In order to preserve this set-up under the iteration of the rule we furthermore require that the interaction between the subalphabets is closed i.e. that for all s 2 S and t 2 T f(s; t) belongs to one of the two subalphabets. De nition 2.1.: Given two subalphabets S and T let A = S \ T be the set of ambiguous symbols. If it is nonempty it is by itself an invariant subalphabet. Ambiguous symbols are receding i.e. f(s; a) 2 S n A for all s 2 S n A; a 2 A and T identically. The con gurations in the set ST (j) = fskg j sk 2 S 8k j; sk 2 T 8k > j and sj ; sj+1 = 2 A are said to have a boundary point at j + 1=2. 6 Note that if A is empty i.e. the subalphabets are disjoint then every con guration from ST is unambiguously in some ST j . If A 6= ; then any con guration of the form SAT where A is a nite block of symbols from A is eventually reduced to the form ST . Therefore the de nition above applies again and we de ne the location of the boundary point in between these instances by interpolating. 2.1 The unambiguous interaction We now proceed to characterize the underlying graphs that determine the motion of the boundary point. In this section we restrict to the case where f(s; t) is unambiguous whenever at least one of s and t is unambiguous. The general case is analyzed in the sections 2.2 and 2.3. Let S = fs1; s2; : : : ; sng and T = ft1; t2; : : : ; tmg be two permutive subalphabets that generate the set of con gurations ST for a two-block map f . Consider the set N of all boundary pairs (i.e. pairs that can be seen around the boundary point) (s; t), s 2 S and t 2 T . If jAj = k then N has exactly nm k2 elements. Since now f(s; t) belongs to either S n A or T n A the c.a. action induces a directed graph G on the node set N . These nodes are called type I. The fan-out i.e. the number of possible successors of a given node is n or m depending on whether f(s; t) 2 T (t dominates) or f(s; t) 2 S (s dominates). The fan-in is unrestricted i.e. between zero and nm k2. The graph can have self-loops but no parallels and in general it is just weakly connected. In Figure 1. we have the Cayley table of a simple c.a. together with the graph. The permutive subalphabets are S = f1; 2g and T = f3; 4g For simplicity we have chosen the rule to be symmetric but given the ordering ST we are really 7 only interested in the shaded elements in the framed square. At the nodes of the graph the pair (s; t) is on top of f(s; t). Figure 1. here Note that the ambiguous symbols can upto bookkeeping be treated exactly as the unambiguous symbols. The location of a boundary point is de ned by interpolation for all con gurations in a given evolution starting from any element in ST : Hence whenever the boundary pair is of the form (s; a), s 2 S nA; a 2 A we know that the symbol a should be counted to belong to T ( (a; t) analogously; a is in S). The node set naturally splits in two subsets. We call Ntr the set of transient nodes if for a node n 2 Ntr either (i) there exists a transition from it such that after that it is impossible to re-enter n or (ii) a node of type (i) can be reached in a nite number of steps from n. The complement of Ntr is the set of recurrent nodes Nrec. In the forthcoming analysis all transition probabilities on the edges will be positive so to obtain the equilibrium characteristics of the boundary motion it will su ce to restrict ourselves to the set Nrec. In a moment we will investigate under which conditions this set is strongly connected. Transitions on G result in a walk fXjgj 0 on the graph which in turn uniquely determines the motion of the boundary point. Depending on the edge chosen the boundary point either jumps to the right or to the left by 1=2. Let this increment function be (n). By keeping track on the partial sum Si =Pij=1 (Xj) we will be able to locate the boundary at the ith period. The walk on the graph is markovian and the Si-process is stationary but in general non-markovian. 8 We will now present a lemma that explains how the successor node is selected. Lemma 2.1.1.: Suppose that at the ith iterate of the c.a. starting from a con guration in ST the boundary point is at 1=2. The past of the boundary motion is then determined by the block from i=2 to i=2+1 endpoints included. Let the boundary pair be (s; t) and f(s; t) = s0. Then the successor node for the graph walk is (s0; t0) where t0 is uniquely determined by the past and present of the boundary motion and the entry at i=2 + 2 in the initial con guration. The left jumps are determined analogously. Proof: From the space-time evolution this result becomes obvious. In Figure 2. the past (backward cone of (s; t)) generated by f i=2; : : : ; i=2 + 1g is the large triangle around the 1=2-line. By the permutivity of the 2-block map given this past the entry at the initial con guration at i=2 + 2 determines the entry after one iterate at (i+3)=2: But this argument can obviously be iterated i+1 times (the entries under arrow in the gure) and therefore given the past and present of (s; t) the entry at i=2 + 2 uniquely determines t0: Figure 2. here For a generic initial condition on ST the transition probabilities on the graph can be easily determined. By the Corollary 1.1. we know that the appropriate Bernoulli product measures are the invariant distributions on con gurations generated by S and T . The genericity is in the sense of these measures. Let Z be the set of non-positive integers. 9 Proposition 2.1.1.: Suppose that on ST (0) we have the product measure which is uniform i.e. B(1=m; : : : ; 1=m) on TZ+ and B(1=n; : : : ; 1=n) on SZ . Then at each node of G the transition probabilities are uniform. Proof: Suppose that (s; t) is the boundary pair and f(s; t) = s0. By the Lemma 2.1 we know that given the past of the boundary pair (s; t) the follower node (s0; t0) is determined permutively by a single entry in the positive part of the initial con guration because t0 is. But these symbols are B(1=n; : : : ; 1=n)distributed. From this on we restrict ourselves to the generic case i.e. assume the initial condition to be distributed as indicated in the Proposition above. Call this measure the natural measure. Note that in view of the proof of Lemma 2.1.1. our initial assumption on the permutivity of S and T can be weakened. When ordered ST we only need S to be left-permutive and T to be right-permutive. We now establish the dichotomy of the boundary motions. De nition 2.1.1.: A signal is a boundary motion that eventually moves monotonically to either right or left with maximum speed i.e. 1=2 at each iterate. This motion is obviously statistically degenerate i.e. has drift equal to 1=2 and vanishing dispersion (variance). De nition 2.1.2: Suppose X is a markovian random walk with uniform transition probabilities on the graph G. If the jump sequence f 1=2g that it generates is not asymptotically deterministic (almost surely) the boundary point is said to perform a random walk. 10 Remark: In many cases the boundary process is also markovian and then this de nition agrees with the usual de nition of a random walk. Due to the generating mechanism we however feel that it is appropriate to call all boundary motions of the second type random walks. Theorem 2.1.1.: The motion of a boundary point starting from an initial con guration distributed according to the natural measure is either a signal or a random walk. Proof: We will show that if the boundary motion travels at a speed strictly less than 1=2 then the 1=2-sequence describing its motion must be non-deterministic. Let si1 2 S appearing in a boundary pair be dominating i.e. f(si1 ; T ) 2 S nA. This now generates a set of chains: si2 = f(si1 ; T1); si3 = f(si2 ; T2); : : : ; sik = f(sik 1 ; Tk 1); f(sik ; T ) 2 T s0i2 = f(si1 ; T 0 1); s0i3 = f(s0i2 ; T 0 2); : : : ; s0ik = f(s0ik 1 ; T 0 k 1); f(s0ik ; T ) 2 T etc. where Ti T; T1[T 0 1[T 00 1 [ = T etc. Here we just list all the sequences generated by di erent choices of t's in the boundary pair upto a dominating t. All these chains have to be nite to obtain a speed less than 1=2. Moreover the sequences fs:i1 ; : : : ; s:ikg need to be of equal length since otherwise we would have at some iterate ambiguity whether the motion turns (depending on which fs:i1 ; : : : ; s:ikg-block we follow). So let us consider the rst block. Suppose sik is located at (j; i): Let sl be its left descendant at (j 1=2; i + 1): Now we have f(sl; t) 2 S for example when l = i1 for all t 2 T: On the other hand f(sl; t) 2 T for sl = ik for all t 2 T . But from the Lemma 2.1 we know that the entry sl is determined permutively from the initial condition. Hence if the speed is less 11 than 1=2 we are bound to have a random choice between a left and a right jump. The proof of the Theorem immediately implies the following result which is useful in nding the type of dynamics directly from the Cayley table. It generalizes the earlier notion of a dominant symbol. Corollary 2.1.1.: The dominant chain condition, 9S0 S such that 8s0 2 S0; f(s0; T ) 2 S0, (and its symmetric counterpart for T 0-dominance) is a necessary and su cient condition for a boundary motion not to be a random walk. Remark: The c.a. in Figure 1. has a dominant symbol t = 4 i.e. a dominant chain T 0 = f4g. After a nite transient the c.a. exhibits a left propagating signal. The motion type that the random walk on a given graph can generate need not be unique. The uniqueness is related to the graph topology in the following fashion. Theorem 2.1.2.: If the random walk Xi restricted to the recurrent part of the graph generates a boundary random walk the recurrent part must be strongly connected. Since the random walk on a strongly connected graph uniquely determines the statistical properties of the boundary motion Theorem 2.1.2 immediately implies a co-existence result. Corollary 2.1.2.: A signal and a boundary random walk or two di erent boundary random walks can not be generated from the same strongly connected graph. 12 The existence of a signal is a consequence of the existence of a closed (no transitions out) subgraph in G the nodes of which generates only left or only right jumps. Since several such subgraphs may exist in a weakly connected graph multiple signals may exist and in particular propagating to either direction. If however the subgraph is all of G i.e. one of the subalphabets is dominant only one signal exists. Proof of the Theorem 2.1.2.: Pick two recurrent nodes (s; t) and (s0; t0). Form the follower sets F and F 0 of both i.e. sets of nodes that can be reached from them in any number of steps. Their elements are recurrent nodes since a follower of a recurrent node is recurrent. If the nodeset N is thought as an n m array minus a k k corner its subsets F and F 0 both contain rows and columns of full length. This is because the existence of a boundary random walk generated from the set of recurrent nodes guarantees the absence of dominant chains and hence both left and right jumps are bound to happen starting from either one of (s; t) or (s0; t0). But by the geometry of F and F 0 they intersect and from any element in the intersection the starting points can be reached by recurrence. Hence (s; t) and (s0; t0) communicate. Since the transition probabilities given by Proposition 2.1.1. are positive for all edges the strong connectedness implies that the random walk on the recurrent part is irreducible and the nodes are positively recurrent. If the transition probability matrix is denoted by P then the equilibrium distribution, , on the nodes is the solution of P = : From this we get the characterization of the parameters of the random walk by a simple application of the Ergodic Theorem. Let N+ Nrec be the subset of nodes with right jumps i.e. (N+) = +1=2: 13 Theorem 2.1.3.: Suppose that we have a c.a. on ST with an unambiguous boundary action. Let the initial distribution be according to the natural measure. If the resulting boundary motion Si is a random walk then its expected spatial shift in unit time i.e. the drift is d 4 = lim I!1 1 I I Xi=1 (Xi) = X n2Nrec (n) (n) = (N+) 12 and the unit squared variation equals to 2 4 = lim I!1 1 I I Xi=1 ( (Xi) d)2 = X n2Nrec ( (n) d)2 (n) = 1 4 d2: Note that is of course compatible with our earlier result on the drift and variance of a signal in the case of a strongly connected graph on Nrec: Example 2.1.1.: Suppose that we have the subalphabets S = f1; 2g and T = f3; 4g and a rule on f1; 2; 3; 4g represented by a Cayley table in Figure 3. Note that it is only slightly di erent from that of Figure 1. But the corresponding graph is now strongly connected, each transition has probability 1=2 and the equilibrium distribution is uniform. The boundary walk generated is markovian since f (Xi)g is an independent sequence. By the Theorem the walk has zero drift and variance 1=4. Figure 3. here 2.2 The ambiguous interaction If the interaction between the (intersecting) subalpahabets can also result in an ambiguous symbol the graphs described so far will not su ce. However the 14 extended graphs are still simple enough to be explicitly analyzed. Before getting into that we characterize the ambiguous interaction and the second node type. Let S, T and A; jAj = k; be as before and let the set of ambiguous elements be non-trivial: 1 k < minfn;mg. Note that the excluded case of A coinciding with one of the subalphabets is clear { the assumption that elements of A are receding implies that there will be a signal. Moreover let there be a boundary pair (s; t) 2M = (S nA) (T nA) be such that f(s; t) 2 A and call it type II. Figure 4. illustrates the evolution of such a pair. The shaded line indicates the boundary motion according to our de nition in the beginning of Section 2. Figure 4. here Recall that the geometric distribution with parameter p 2 (0; 1) , Geom(p), assigns the probability pi(1 p) to i 2 f0; 1; 2; : : :g: The symbol stands for distribution. Lemma 2.2.1.: Suppose that the initial con guration on ST (0) is distributed according to the natural measure. At some iterate i we have at (j; i) a boundary point (s; t) such that f(s; t) 2 A i.e. type II. Let the next boundary pair (s0; t0) such that both s0 and t0 are non-ambiguous be located at (j+ ; i+ ). Then the displacement (R L)=2 and the holding time R+L+2 where R and L are independent random variables and R Geom(k=m) and L Geom(k=n). Moreover (s0; t0) is distributed uniformly over the allowed pairs. Proof: By Lemma 2.1.1. we know that given the past of (s; t) the entries at time i+1 at locations j 1 are determined permutively from two entries in the initial con guration. So the events of obtaining an element in A at these locations 15 have probabilities k=n and k=m on the left and on the right respectively. But once j 1 is determined we can iterate the same argument for j 2 and so on. Therefore L, the number of ambiguous symbols to the left of the (ambiguous) symbol at j; i+1 before the rst non-ambiguous symbol, is distributed according to Geom(k=n): R is treated analogously. Moreover L and R and the fact that elements of A are receding determine uniquely the jump and delay . The fan-in of a type II node is unrestricted and the fan-out equals to (n k)(m k). By the Lemma they in fact map onto M . An important special case is an automaton for which all interactions between S and T result in an ambiguous symbol. So let us suppose that all interaction onM is of type II. Then the graph restricted toM is strongly connected (in one step i.e. the directed graph is complete). Moreover the complement N nM is clearly transient so Nrec = M . Apart from a possible nite initial transient of jumps on N nM the following result pins down the resulting motion. Theorem 2.2.1.: Let the subalphabets have the cardinalities jSj = n, jT j = m and jAj = k, 1 k < minfn;mg: Suppose that f(s; t) 2 A whenever neither s nor t is ambiguous. Let the random variables and be as in the Lemma. Then the boundary motion performs a random walk with independent and identically distributed holding times of length and i.i.d. increments . The drift of the motion is d = E( ) E( ) = k(n m) 2nm (n+m)k and the unit variance equals to 2 = k 4 (2nm k(n+m)) m(n k) m k + n(m k) n k : 16 In particular in the symmetric case n = m the drift vanishes and the dispersion reduces to k=(4n 4k). Proof: We rst observe that in computing and we can restrict to the case where s0 = s and t0 = t in the Lemma 2.2. But this means that all the nodes of the the generating graph have same ( ; )-distributions associated with them. Moreover since the pairs ( ; ) at di erent nodes are independent we only need to consider the case of a graph with a single node and a loop. This generates a boundary random walk with independent increments delayed by iterates. Now the delays f igi 0 form a recurrent renewal process hence an application of the Renewal Theorem to the cumulative jumps fSigi 0 yields the drift d = lim t!1 1t I(t) Xj=1 (Xj) = E( ) E( ) where I(t) is the usual counting function i.e. the number of delay periods upto time t. But by the independence of L and R the expectations are easily calculated to be E( ) = k(n m) 2(n k)(m k) E( ) = 2nm nk mk (n k)(m k) : Since 2( ) = 2(L) + 2(R) =4 the unit variance formula follows via a similar argument. Remarks: 1. By choosing n = m = 2 and k = 1 this result covers a number of random walks arising e.g. in the context of elementary c.a. In particular the walks in the Rules 18 (treated in [2]) and 22 (see [1]) are just special cases of the Theorem above. 17 2. It is possible to reduce the ambiguous case to the unambiguous one via an extension. If each of the ambiguous symbols is duplicated and assigned to one of the subalphabets the new subalphabets are disjoint. On the enlargened alphabet one can then de ne a new c.a. dynamically identical to the original one (i.e. possessing statistically identical boundary motion). 2.3 The mixed case The previous results make the general case of unrestricted f(s; t) now accessible. We still consider a directed graph G with nm k2 nodes of the two types described and the random walk Xi on it. Let again M = (S nA) (T nA): Since o -M entries do not form a closed subgraph (symbols in A are receding) the novel case to be treated arises when M contains nodes of both type. We recall that the graph restricted to type II nodes is strongly connected and from any node in this set we map onto M . So in terms of recurrence the critical question is how do the follower sets of the nodes of type I lie in M . If their union is in the complement of type II nodes then Nrec is of type I and we reduce to the case treated in Section 2.1 (this phenomenon is analogous to the dominant chain case in Section 2.1). Note also that the size of such type I invariant set is bounded from below by the cardinality of A (just because (s; a) has to be in the set if s is for all a 2 A). Hence the larger A is the harder it is to type I to con ne the action on itself. Let NI and NII be a partition of Nrec into type I and II nodes and let N+ I NI be the subset of nodes with right jumps. Assume that the graph on Nrec is strongly connected and denote again by P and is the transition 18 probability matrix and equilibrium distribution on Nrec: De ne the expected displacement and visit time at a node by 4 = X n2NrecE ( (n)) (n) = (N+ I ) 12 + 1 2 +E ( jNII) (NII) and V 4 = X n2NrecE ( (n)) (n) = (NI) + E ( jNII) (NII ) : These can be readily evaluated using Theorem 2.2.1. Theorem 2.3.1.: Let a c.a. act on ST with the natural measure. Suppose that the graph on Nrec is strongly connected and NII 6= ;: Then the boundary motion generated is a stationary random walk with drift d = =V and variance 2 = 1 V ( 1 2 + 2 2 N+ I + E ( )2jNII 12 + 2! (NII)) : Remark: For NII = ; and NII =M we have the unambiguous and ambiguous case respectively so the novelty here appears when ; 6= NII 6= M: NII = ; is excluded from the Theorem since then (and only then) degeneracy can occur. Proof: By the assumptions on the existence and communication of the type I and II nodes we know that a unique random walk prevails and is stationary. For the drift we write 1t I(t) Xj=1 (Xj) = I(t) t 8<: 1 I(t) I(t) Xj=1 (Xj)9=; where I(t) counts the number of nodes visited by time t: The graph walk Xj is positive recurrent on Nrec so by the Renewal Theorem I(t)=t! 1=V : Moreover the the random variable is integrable (on NI it is bounded and on NII it is 19 the di erence between geometric random variables) so by the Ergodic Theorem the limit I !1 of the remaining Cesaro average equals to : The asymptotic squared variation is obtained via an analogous argument and some manipulation starting from the expression 1=tPI(t) j=1 (Xj) 2 : All the expressions in the Theorem can be explicitly computed. Perhaps an application is in order to illustrate this. Example 2.3.1.: Consider the simplest case where nodes of both type can coexist in the recurrent component. Let S = f0; 1g; T = f0; 2; 3g so A = f0g and N consists of ve nodes. We have a choice in terms of which node in M is of which type and which ones in its complement are transient. Of these six c.a. two have a dominant symbol i.e. they generate a signal. The remaining four yield akin random walks and are all analyzed in a similar fashion. In Figure 5. we have illustrated one of these (if the shaded element in the Cayley table would equal to 3 a signal-case would result). The two rightmost elements of the graph are transient. Clearly Nrec = f(1); (2); (3)g, N+ I = NI = f(1); (3)g and NII = f(2)g: Figure 5. here The transition matrix and equilibrium visit probabilities are P = 0B@ 1 3 1 3 1 3 0 12 1 2 13 1 3 1 3 1CA and = 15 ; 2 5 ; 25 : The ( ; )-pair associated with the nodes (1) and (3) is (1=2; 1). At (2) it has the distribution indicated in Lemma 2.2.1. The expectation E( ; ) equals to 20 ( 1=4; 7=2) by the formulas in the proof of Theorem 2.2.1. Hence by Theorem2.3.1. a boundary random walk is generated and with some computation onends that the unit drift equals to 1=10 and variance approximately to 0:205:The reader is to judge whether it is obvious that the walk drifts so slowly to theright although the only sure transitions are to this direction (at type I nodes).When the type II is recurrent no co-existence result of the type discussed inSection 2.1 is possible. However even though the random walk that Theorem2.3.1. speci es is unique it can be rather bizarre and actually look like twodistinctly di erent boundary motions intertwined. Our nal example showshow to \design" c.a. like this.Example 2.3.2.: Suppose that n = m k = 1 and that there are unambiguousspecial symbols ~s and ~t such that f(~s; t) = ~s except for t = ~t. Also let f(s; ~t) = ~tfor all s except for s = s1. Let all the rest of the nodes in M = (S nA) (T nA)be of type II. Now the Theorem applies but by design the sets NI and NIIcommunicate only with di culty i.e. jumps between them are rare. They alsogenerate very di erent motions. The type I motion consists of long monotonesequences of jumps of the same sign and size (1/2) so that the motion is a\piecewise signal". The right propagating boundary point eventually switchesinto a left propagating one whereas this ultimately (upon visit to (s1; ~t) givesbirth to a type II symmetric random walk (by Theorem 2.2.1. since n = m).This in turn is longlived since type II covers nearly all ofM but it is not immortalbecause type II nodes are onto this set.Clearly this design can be widely varied by changing S or T , by introducingbias to either type, by extending A (and hence NI) etc. and still generate pairs21 of very distinct and persistent motions. The key design principle is to partitionthe graph on the recurrent nodes into two almost disconnected parts supportedby NI and NII each generating a distinctly di erent motion.The physical phenomenon this example seems to suggests is that ofmetasta-bility. Considering the amount of freedom in our example this behavior canindeed be widespread in all but the simplest one-dimensional cellular automata.References[1] Eloranta, K. 1993 Partially permutive cellular automata, Nonlinearity, thisissue.[2] Eloranta, K., Nummelin, E. 1992 The kink of the elementary cellular au-tomaton Rule 18 performs a random walk, Journal of Statistical Physics,69, No.: 5/6.[3] Feller, W. 1966 An Introduction to Probability Theory and Its Applications,Vol. II, Wiley.[4] Harary, F. 1969 Graph Theory, Addison-Wesley.[5] Hedlund, G.A. 1969 Endomorphisms and automorphisms of the shift dy-namical system, Math. Sys. Th. 3, 320-75.[6] Petersen, K.E. 1983 Ergodic Theory, Cambridge Univ. 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