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Ergodic properties of color records
Physica 138A (1986) 183-193 North-Holland, Amsterdam
ERGODIC PROPERTIES OF COLOR RECORDS M. KEANE and W.Th.F. DEN HOLLANDER Department of Mathematics, Delft Universig of Technology, The Netherlands
Julianalaan 132, 2628 BL Delft,
In a series of papers culminating in the dissertation of the second author under the direction of P.W Kasteleyn, several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black-white colored lattice, and this under the sole assumption that the coloring is stationary, ergodic and independent of the walking. In this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk, k 2 0, of successive times needed for the walker to go from the kth to the (k + 1)st black point. As an example of the type of results obtained, we show that if a certain “induced” color record process is strongly mixing in the ergodic theoretic sense, then the average (n,) tends to the inverse of the density of black points as k tends to infinity, and, similarly, that the distribution of n, tends to a limit that can be explicitly calculated in terms of the distribution of n,. For a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing. Our main goal, however, is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records.
1. Introduction Consider an infinite lattice together with two independent probabilistic structures: - a stochastic coloring assigning either of the colors black or white to each point of the lattice, - a random walk on the points of the lattice. The infinite sequence of colors of the points visited by the walker is a (discrete time) stochastic process: the color record process. Many aspects of this process have received attention in a series of recent papers by P.W. Kasteleyn and the second author. For a recent summary and references we recommend the expository article by P.W. Kasteleyn’). On page 27.1-7 of ref. 1 the following question is raised. Let nk be the number of steps of the walker between the kth and the (k + 1)st hit of a black point. For which random walks and stochastic colorings does the so-called mean run length ( ntk) converge as k tends to infinity? It turns out that this convergence and similar limit properties, such as the 0378-4371/86/$03.50 @J Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
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convergence in distribution of nk or the asymptotic independence of color patterns seen by the walker around black points visited at times far apart, are intimately related to the strong mixing property for a certain induced transformation of the dynamical system determined by the probabilistic structures described above. In section 3 we formulate these ideas, show how they can be used, and distill out a general mathematical object of interest for further study. In section 2, however, we first formulate a theorem which implies the strong mixing property envisaged (and hence the convergence of ( nk)) and prove this theorem for a special case. Although it seems to be difficult to extend our proof in section 2 to a large class of random walks and stochastic colorings, we expect that the theorem is true in large generality. For this reason, as well as for convenience of exposition, we have chosen to restrict our presentation to the case of transient random walks on the integers having positive pausing probability and Bernoulli colorings, and to comment on possible generalizations as we proceed.
2. Asymptotic
independence
of local sceneries
Let w,,, n 2 0, be the position at time n of a random walk on the integers 2, starting at the origin at time zero, and let c(z), z E Z, be a stochastic coloring of Z with the colors black and white. P denotes the probability measure describing walk and coloring. We assume the following: 1) o
c(w, + z) = s(z) for each z E F. The event “s is perceived
at time n” is denoted by [s],.
ERGODIC PROPERTIES OF COLOR RECORDS
Proposition.
185
For all local sceneries s and t,
@ Nslll f-l[tl,) = N~lcJP([tlcJ Proof.
For each z E 2, let t + z be the translate by z of the local scenery t, i.e.
(t + z)(z’ + z) = t(z’)
(z’ E F) ,
and set (Y”(Z) := P(w, = z)
(n*O,zEZ),
P(z) : = P( [s10 f-l [t + zlO)
(z E 2) .
By conditioning on w, = z and using assumption (4), the probability of perceiving s at time zero and t at time n can be expressed in terms of p and (Y,:
fvs1l.lf-lit],) = c a,(; M(z) * ZEZ
Assumption
(1) (or rather the fact that p < 1) implies that
tm_a”(z) = 0 for each z E Z, and from assumption
(2) we obtain
I$!% P(z)= fY[slo)P([t]lJ). Since CrEZ (Y,,(Z)= 1 for all n, the proposition
follows.
W
Remarks.
1) A similar proof works for any strongly mixing color distribution (even a non-stationary one with the appropriate limit) and any random walk, except the one which stays put forever ( p = 1). 2) If the color distribution is only ergodic, then p(z) converges only in the sense of Cesaro (by the ergodic theorem) and the proposition may fail to be true. H. Berbee’) has shown that for the proposition to be true it suffices that the color distribution be totally ergodic and that the random walk satisfy a suitable aperiodicity assumption. 3) The result generalizes easily to arbitrary groups and hence also to arbitrary lattices. 4) Assumption (3) was not used, but will be needed later.
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M. KEANE AND W.Th.F. DEN HOLLANDER
The main result in this section is a version of the preceding proposition with a different time scale, viz. one in which time is counted according to the number of visits to black points. We denote by Tk the (random) time at which the random walk visits a black point for the kth time, k 5 1. It is easy to show that Tk < m with probability 1 for all k L 1. Theorem.
For all local sceneries s and t such that t does not color the origin,
F,mm Wslo ” MTJ = ~([~loN[tl,) . Proof The proof will be divided into several parts. 1) It suffices to show that if sb and S, are two local sceneries defined on the same set F and giving the same color to every point of F except for one point f E F, with ~~(2) black and s,(f) white, then for any t not coloring the origin,
E,mm ((1 - ~)~([~,I,n itIrk) - q~t[~,l, f-7[tiTk)l = 0. To see this, simply divide the bracketed
expression
(*I
by
(1 - q)fY[dl) = q~([%vlo) . This yields
so that if the statement in the theorem is valid for s,, and t, then it is also valid for S, and t, and conversely. If we define S to be the local scenery on A{2} with the same colors as sb and s,, then for all k
and hence convergence for S to P( [t],,) also implies convergence for si, and S, to the same value. Thus, by induction, it suffices to show that the theorem is valid for one particular local scenery s (and all t). But if s is the local scenery with F = (0) and s(0) black, then the sequence [tlTk conditioned on [s], (i.e. on the origin being black) is stationary (see also section 3), and thus
for all k 2 1, since t does not color the origin. Hence the convergence and t follows from (*).
for all s
ERGODIC
PROPERTIES OF COLOR RECORDS
187
2) Let sb and s, be as above, and let t be arbitrary. Since the random walk is transient, with probability 1 it visits each point of 2, and in particular the point i at which sb and s, differ, only a finite number of times. Denote by V, the event that the walk visits i exactly 1 times, 13 0. We now claim that it suffices to prove that
Ei_ {p([s&
n [tlTk n v,) - p([~& n [&+,
n v,)) = 0
(**)
for each 1 a 0. To see this, note that if 5 is visited 1 times and the color off is white, then by changing this color to black and leaving everything else the same, the walker is at the same point of Z at time Tk+[ as he was at time Tk without the color change, for k sufficiently large. Hence
jim~~~(bwln PITkn v,) -
(1 - w(b,l
and it follows that (*) may be rewritten
n [tlTk+ln w =0,
as
But (**) shows that the individual terms in this sum tend to zero, and since the Ith term is bounded in absolute value by P(y), a summable sequence, it is clear that (**) implies (*). 3) The most elegant method to prove (* *) seems to be a coupling argument. Fix a coloring c of Z and a point 5 E Z. Suppose that we can construct two coupled random walks w1 and w2 (on a suitable probability space) with the following properties: i) The distributions of both w1 and w2 are equal to the distribution of the original w. ii) If Vi denotes the event that wi visits f exactly I times (i = 1,2,1 a 0), then V: = VF for all 12 0. iii) lim,,, P(wkl # w:;+,lc) = 0, where P denotes the probability measure describing w1 and w2 (depending on c) and Ti is the kth visit time of wi to a black point. It follows from (i) and (ii) that
lmlTk ” w
- NflTk+l” WI
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M. KEANE AND W.Th.F. DEN HOLLANDER
and (iii) thus implies
Integration over those c which realize [sJ,, then yields (**). 4) We now give a construction of w1 and VV*which works for a set of colorings of probability 1. Both walkers proceed together along the same path, according to the probabilistic rules for the given random walk, until the first black point #Z is reached. Then they are “uncoupled” and perform independently a number 0, 1, . . . of “pauses”, each having a geometric distribution with parameter p = P(w, = 0). After this, they leave together in unison again until reaching the next black point Z.5, where the same procedure is repeated, and so forth. However, if after an uncoupling the total number of black visits by w2 is exactly one more than the total number of black visits by wl, then they are “coupled eternally” and continue in unison forever. It should be clear that this procedure yields the properties (i) and (ii) above. Property (iii) will follow if w1 and w* are “coupled eternally” at some point in the procedure with probability 1. But since the difference in the number of black visits by w2 and by w1 just after each uncoupling (and after the last visit to 5) is itself a random walk on Z with a jump distribution equal to the independent difference of two geometric distributions with parameter p, and since this random walk is recurrent, property (iii) is valid whenever the walk visits an infinite number of black points. To complete the proof, we note that for a set of colorings of probability 1 the walker will, with probability 1, visit an infinite number of black points by Fubini’s theorem, since for almost all walk paths (viz. all which visit an infinite number of different points of Z) an infinite number of points in the path will be colored black. The latter is of course the same as the statement that Tk < CQ n with probability 1 for all k 2 1. Remarks.
1) A similar proof works for any group and hence for any lattice. 2) The condition p > 0 can be relaxed to the requirement that with positive probability the walk returns to the origin at some finite time. For this case the coupling has to be modified, and instead of allowing w1 and w* to perform independent “pauses” at black points we now allow them to make independent “loops” of a given type, starting from black points and running over white points only. It is easily shown that with probability 1 infinitely many black points are hit which allow the given loop. 3) The ideas in the proof do not seem to be sufficient to treat recurrent
ERGODIC PROPERTIES OF COLOR RECORDS
189
random walks, such as simple random walk on 2. Note that the coupling speed is in this case roughly equal to the visit speed of a given point 5. 4) It also seems to be difficult to treat other color distributions, such as strongly mixing ones, although we expect the theorem to be true in large generality. For a counterexample involving a deterministic walk and a strongly mixing color distribution, see example 2 of section 3. For color distributions which are only ergodic it is easy to find counterexamples even for nondeterministic walks.
3. Ergodic theoretic formulations A dynamical system is a probability space together with an automorphism of this space (i.e. an invertible measure preserving mapping from the space to itself). Stationary stochastic processes and dynamical systems are intimately related to each other, and we assume in the sequel that the reader possesses some intuition for this relationship. We begin by describing the dynamical systems related to the processes discussed in section 2. Consider the sequence space J = {j = ( ji)iEZ: ji E 2 for each i E Z} together with the “natural” a-algebra 9 on J, generated by the finite-dimensional cylinder sets. We think of j E J as a sequence of jumps made by a random walk and therefore denote by h a product probability measure on (J, .$) arising from a given jump distribution on Z. If a:J+J
is the left shift on J, defined by a(j)=j’
with
ji = ji+i
(iE
Z),
then (J, 9, A, cr) is a dynamical system, and we can describe the random walk with the given jump distribution by the following sequence of random variables: n-1
w,=o, Next, let
w” =
F.
n-1 ii
=
F. t”ii)O Cna ‘1.
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M. KEANE AND W.Th.F. DEN HOLLANDER
c = {c = (Ci)&
ci is black or white for each i E Z} ,
and provide C with its natural g-algebra %‘.An element of C is a coloring of 2, and if y is a probability measure on C which is invariant under the left shift 7:
c-c,
defined by r(c) = c’
with
c; = ci+r
(iEZ),
then we have a stationary stochastic coloring of Z, as well as another dynamical system (C, %, y, 7). We now put these two systems together to describe what is seen by the walker while moving through the lattice. To that end we define X:=JxC,
2?':=$x%,
p:=hxy,
so that (X, 2, p) is the product of the walk probability probability space (so walk and coloring are independent), be the transformation
space and the color and we define T to
T:X-X
given by
T(j,c) : = (u(j),
&(c))
for each x = (j, c) E X. It is easily seen that p is T-invariant, so that (X, 2, p, T) is a dynamical system. This system is commonly called a skew product in ergodic theory. Now any event concerning the random walk and the stochastic coloring corresponds to a subset of X, namely, those points x E X for which the event is realized. For instance, let s be a local scenery and n a non-negative integer. Then it should be clear from the definitions above that the events [s],, considered as subsets of X, are related to [s& by successive applications of the mapping T:
[s], = T-“[s],
(n 20)
.
Hence the validity of the proposition
of section 2 amounts to the asymptotic
ERGODIC PROPERTIES OF COLOR RECORDS
191
independence of the sets [s10 and T-“[t],, for all local sceneries s and t as n + 03. From this it is easy to show that for any sets A E 22’and A’ E 28, !iim p(A fl T-“A’)
= p(A)p(A')
.
Thus we have:
Conclusion 1. Asymptotic independence of local sceneries (as formulated in the proposition of section 2) is equivalent to strong mixing of (X, %‘, p, T). In a similar manner the theorem elusion. Define
of section 2 leads to an analogous con-
Y:={x=(j,c)EX:c,isblack}, 9:={AE
fZ:AcY},
v(B):=s
(BE%?).
Then the probability space (Y, 93, V) has the intuitive interpretation ditioning on the origin being black. Let
of con-
S:Y+Y be the transformation S(y) := TfCY’(y)
induced by T on Y: (y E
Y) ,
where t(y):=min{nsl:
T”YE Y}.
From the general theory of induced transformations it follows (and this is also easy to verify directly) that S(y) is well defined for v-almost every y E Y and that (Y, 93, V, S) is a dynamical system. The dynamical system (Y, 93, V, S) inherits some of the properties of (X, 8?, CL,T). For example, S is ergodic if and only if T is ergodic, and the entropies of S and Tare related by the factor p(Y). However, strong mixing is not hereditary. (See example 2 below.) As above, if t is a local scenery not coloring the origin, then
M. KEANE AND W.Th.F. DEN HOLLANDER
192
[flTkfl y=
O[tl,
since S effectuates
c-lY)
7
the move to the next black point. This brings us to:
Conclusion
2. Asymptotic independence of local sceneries on the black time scale (as formulated in the theorem of section 2) is equivalent to strong mixing of (Y, 53, V, S). In particular, asymptotic independence of any two events (in 93) on the black time scale is guaranteed if we can prove the theorem in section 2. Thus, for example, if nk is the kth run length as defined in the introduction, and m is a positive integer, then it easily follows from the theorem that E_i P([n, > m]) = P([n, > ml]0 is black) = P([nO = m],;
)
where the second term is independent of k and the second equality is a consequence of the stationarity and ergodicity of the color record process’). This in turn implies that lim,,, ( nk) = q-l, provided ( nk) is uniformly bounded in k, as is the case for instance when (n,) < ~0 (see ref. 1). More generally, it is plain from the above discussion that the study of mixing properties for induced transformations is of interest for at least one application. We now formulate a definition and a general mathematical question which we deem to be worthy of further study. Let (X, 2, p, T) be a strongly mixing dynamical system, and let A be a subset of X with p(A) > 0. Denote by (Y, 93, V, S) the induced system on A. Definition.
T is kasteleyn mixing* with respect to A if the induced (Y, 93, V, S) is strongly mixing. More in particular, if
system
x = (0, l}” and A={xEX:x,=l) (as in the color record process with 0 as white and 1 as black), then we simply call the process kasteleyn mixing, omitting the set A. General question.
* “kasteleyn” castellan.
Which strongly mixing O-l processes are kasteleyn mixing?
is the traditional Dutch word for barkeeper,
with English cognates chatelain and
ERGODIC
PROPERTIES
OF COLOR
RECORDS
193
Examples.
1) If T is a Bernoulli scheme (i.e. p is a product measure on X), then S is also (isomorphic to) a Bernoulli scheme, and hence T is kasteleyn mixing. More generally, whenever 1 is a renewal state, T is kasteleyn mixing. 2) Not all strongly mixing O-l processes are kasteleyn mixing. In particular, suppose the stationary measure p on X is obtained by choosing 0 or 1 independently, but writing two l’s instead of one every time a 1 is chosen. Then T is strongly mixing, but T is not kasteleyn mixing because the “even” l’s can be distinguished from the “odd” l’s on the time scale of the l’s, given enough scenery around the origin. This example, moreover, can be interpreted as the color record process of a deterministic walk with a strongly mixing color distribution obtained by randomly placing pairs of black points on 2. 3) The form of the set A is important, since any dynamical system (X, $?, CL,T), except the one-point system, contains a set B E 2 with p(B) > 0 and with B rl T-‘B = 0. If we set A = B fl T-‘B, then the induced transformation on A has - 1 as an eigenvalue and is not strongly mixing3). 4) If (X, % P, T) comes from simple random walk in Zd and Bernoulli coloring, then the theorem in section 2 shows that T is kasteleyn mixing (for black at the origin) if d 2 3. For d = 1 or d = 2 the random walk is recurrent and we do not know whether kasteleyn mixing holds or not. (See remark 3 after the theorem of section 2.) 5) Almost all random walks on more general groups are transient4), so that in these cases our theorem will be applicable. Finally, we remark that Kalikow5) has shown that in the situation of example 4, with d = 1, T is not loosely Bernoulli and is thus in some sense ill-behaved. Nevertheless, we do feel that the process may be kasteleyn mixing. Acknowledgement
We would like to author. In particular, along the lines of exposition in section
thank H. Kesten for useful discussions with the second he independently suggested a proof of the theorem much the one we had obtained, and this has improved our 2.
References 1) P.W.Kasteleyn, Bull. I.S.I. 45 (1985) 27.1-l. 2) H.C.P. Berbee, oral communication with the second author. H.C.P. Berbee (to appear in Israel J. Math). 3) S. Kakutani, oral communicati& with the first author (before 1970). 4) Y. Guivarc’h, M. Keane and B. Roynette, Lecture Notes in Mathematics 624 (Springer, 1977). 5) S. Kalikow, Annals of Mathematics 115 (1982) 393. Literature 6) F. Spitzer, Principles of Random Walk, 2nd edition (Springer, New York, 1976). 7) U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6 (de Gruyter, 1985).
Berlin,
Berlin,
ERGODIC PROPERTIES OF COLOR RECORDS M. KEANE and W.Th.F. DEN HOLLANDER Department of Mathematics, Delft Universig of Technology, The Netherlands
Julianalaan 132, 2628 BL Delft,
In a series of papers culminating in the dissertation of the second author under the direction of P.W Kasteleyn, several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black-white colored lattice, and this under the sole assumption that the coloring is stationary, ergodic and independent of the walking. In this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk, k 2 0, of successive times needed for the walker to go from the kth to the (k + 1)st black point. As an example of the type of results obtained, we show that if a certain “induced” color record process is strongly mixing in the ergodic theoretic sense, then the average (n,) tends to the inverse of the density of black points as k tends to infinity, and, similarly, that the distribution of n, tends to a limit that can be explicitly calculated in terms of the distribution of n,. For a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing. Our main goal, however, is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records.
1. Introduction Consider an infinite lattice together with two independent probabilistic structures: - a stochastic coloring assigning either of the colors black or white to each point of the lattice, - a random walk on the points of the lattice. The infinite sequence of colors of the points visited by the walker is a (discrete time) stochastic process: the color record process. Many aspects of this process have received attention in a series of recent papers by P.W. Kasteleyn and the second author. For a recent summary and references we recommend the expository article by P.W. Kasteleyn’). On page 27.1-7 of ref. 1 the following question is raised. Let nk be the number of steps of the walker between the kth and the (k + 1)st hit of a black point. For which random walks and stochastic colorings does the so-called mean run length ( ntk) converge as k tends to infinity? It turns out that this convergence and similar limit properties, such as the 0378-4371/86/$03.50 @J Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
M. KEANE AND W.Th.F. DEN HOLLANDER
184
convergence in distribution of nk or the asymptotic independence of color patterns seen by the walker around black points visited at times far apart, are intimately related to the strong mixing property for a certain induced transformation of the dynamical system determined by the probabilistic structures described above. In section 3 we formulate these ideas, show how they can be used, and distill out a general mathematical object of interest for further study. In section 2, however, we first formulate a theorem which implies the strong mixing property envisaged (and hence the convergence of ( nk)) and prove this theorem for a special case. Although it seems to be difficult to extend our proof in section 2 to a large class of random walks and stochastic colorings, we expect that the theorem is true in large generality. For this reason, as well as for convenience of exposition, we have chosen to restrict our presentation to the case of transient random walks on the integers having positive pausing probability and Bernoulli colorings, and to comment on possible generalizations as we proceed.
2. Asymptotic
independence
of local sceneries
Let w,,, n 2 0, be the position at time n of a random walk on the integers 2, starting at the origin at time zero, and let c(z), z E Z, be a stochastic coloring of Z with the colors black and white. P denotes the probability measure describing walk and coloring. We assume the following: 1) o
c(w, + z) = s(z) for each z E F. The event “s is perceived
at time n” is denoted by [s],.
ERGODIC PROPERTIES OF COLOR RECORDS
Proposition.
185
For all local sceneries s and t,
@ Nslll f-l[tl,) = N~lcJP([tlcJ Proof.
For each z E 2, let t + z be the translate by z of the local scenery t, i.e.
(t + z)(z’ + z) = t(z’)
(z’ E F) ,
and set (Y”(Z) := P(w, = z)
(n*O,zEZ),
P(z) : = P( [s10 f-l [t + zlO)
(z E 2) .
By conditioning on w, = z and using assumption (4), the probability of perceiving s at time zero and t at time n can be expressed in terms of p and (Y,:
fvs1l.lf-lit],) = c a,(; M(z) * ZEZ
Assumption
(1) (or rather the fact that p < 1) implies that
tm_a”(z) = 0 for each z E Z, and from assumption
(2) we obtain
I$!% P(z)= fY[slo)P([t]lJ). Since CrEZ (Y,,(Z)= 1 for all n, the proposition
follows.
W
Remarks.
1) A similar proof works for any strongly mixing color distribution (even a non-stationary one with the appropriate limit) and any random walk, except the one which stays put forever ( p = 1). 2) If the color distribution is only ergodic, then p(z) converges only in the sense of Cesaro (by the ergodic theorem) and the proposition may fail to be true. H. Berbee’) has shown that for the proposition to be true it suffices that the color distribution be totally ergodic and that the random walk satisfy a suitable aperiodicity assumption. 3) The result generalizes easily to arbitrary groups and hence also to arbitrary lattices. 4) Assumption (3) was not used, but will be needed later.
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M. KEANE AND W.Th.F. DEN HOLLANDER
The main result in this section is a version of the preceding proposition with a different time scale, viz. one in which time is counted according to the number of visits to black points. We denote by Tk the (random) time at which the random walk visits a black point for the kth time, k 5 1. It is easy to show that Tk < m with probability 1 for all k L 1. Theorem.
For all local sceneries s and t such that t does not color the origin,
F,mm Wslo ” MTJ = ~([~loN[tl,) . Proof The proof will be divided into several parts. 1) It suffices to show that if sb and S, are two local sceneries defined on the same set F and giving the same color to every point of F except for one point f E F, with ~~(2) black and s,(f) white, then for any t not coloring the origin,
E,mm ((1 - ~)~([~,I,n itIrk) - q~t[~,l, f-7[tiTk)l = 0. To see this, simply divide the bracketed
expression
(*I
by
(1 - q)fY[dl) = q~([%vlo) . This yields
so that if the statement in the theorem is valid for s,, and t, then it is also valid for S, and t, and conversely. If we define S to be the local scenery on A{2} with the same colors as sb and s,, then for all k
and hence convergence for S to P( [t],,) also implies convergence for si, and S, to the same value. Thus, by induction, it suffices to show that the theorem is valid for one particular local scenery s (and all t). But if s is the local scenery with F = (0) and s(0) black, then the sequence [tlTk conditioned on [s], (i.e. on the origin being black) is stationary (see also section 3), and thus
for all k 2 1, since t does not color the origin. Hence the convergence and t follows from (*).
for all s
ERGODIC
PROPERTIES OF COLOR RECORDS
187
2) Let sb and s, be as above, and let t be arbitrary. Since the random walk is transient, with probability 1 it visits each point of 2, and in particular the point i at which sb and s, differ, only a finite number of times. Denote by V, the event that the walk visits i exactly 1 times, 13 0. We now claim that it suffices to prove that
Ei_ {p([s&
n [tlTk n v,) - p([~& n [&+,
n v,)) = 0
(**)
for each 1 a 0. To see this, note that if 5 is visited 1 times and the color off is white, then by changing this color to black and leaving everything else the same, the walker is at the same point of Z at time Tk+[ as he was at time Tk without the color change, for k sufficiently large. Hence
jim~~~(bwln PITkn v,) -
(1 - w(b,l
and it follows that (*) may be rewritten
n [tlTk+ln w =0,
as
But (**) shows that the individual terms in this sum tend to zero, and since the Ith term is bounded in absolute value by P(y), a summable sequence, it is clear that (**) implies (*). 3) The most elegant method to prove (* *) seems to be a coupling argument. Fix a coloring c of Z and a point 5 E Z. Suppose that we can construct two coupled random walks w1 and w2 (on a suitable probability space) with the following properties: i) The distributions of both w1 and w2 are equal to the distribution of the original w. ii) If Vi denotes the event that wi visits f exactly I times (i = 1,2,1 a 0), then V: = VF for all 12 0. iii) lim,,, P(wkl # w:;+,lc) = 0, where P denotes the probability measure describing w1 and w2 (depending on c) and Ti is the kth visit time of wi to a black point. It follows from (i) and (ii) that
lmlTk ” w
- NflTk+l” WI
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M. KEANE AND W.Th.F. DEN HOLLANDER
and (iii) thus implies
Integration over those c which realize [sJ,, then yields (**). 4) We now give a construction of w1 and VV*which works for a set of colorings of probability 1. Both walkers proceed together along the same path, according to the probabilistic rules for the given random walk, until the first black point #Z is reached. Then they are “uncoupled” and perform independently a number 0, 1, . . . of “pauses”, each having a geometric distribution with parameter p = P(w, = 0). After this, they leave together in unison again until reaching the next black point Z.5, where the same procedure is repeated, and so forth. However, if after an uncoupling the total number of black visits by w2 is exactly one more than the total number of black visits by wl, then they are “coupled eternally” and continue in unison forever. It should be clear that this procedure yields the properties (i) and (ii) above. Property (iii) will follow if w1 and w* are “coupled eternally” at some point in the procedure with probability 1. But since the difference in the number of black visits by w2 and by w1 just after each uncoupling (and after the last visit to 5) is itself a random walk on Z with a jump distribution equal to the independent difference of two geometric distributions with parameter p, and since this random walk is recurrent, property (iii) is valid whenever the walk visits an infinite number of black points. To complete the proof, we note that for a set of colorings of probability 1 the walker will, with probability 1, visit an infinite number of black points by Fubini’s theorem, since for almost all walk paths (viz. all which visit an infinite number of different points of Z) an infinite number of points in the path will be colored black. The latter is of course the same as the statement that Tk < CQ n with probability 1 for all k 2 1. Remarks.
1) A similar proof works for any group and hence for any lattice. 2) The condition p > 0 can be relaxed to the requirement that with positive probability the walk returns to the origin at some finite time. For this case the coupling has to be modified, and instead of allowing w1 and w* to perform independent “pauses” at black points we now allow them to make independent “loops” of a given type, starting from black points and running over white points only. It is easily shown that with probability 1 infinitely many black points are hit which allow the given loop. 3) The ideas in the proof do not seem to be sufficient to treat recurrent
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random walks, such as simple random walk on 2. Note that the coupling speed is in this case roughly equal to the visit speed of a given point 5. 4) It also seems to be difficult to treat other color distributions, such as strongly mixing ones, although we expect the theorem to be true in large generality. For a counterexample involving a deterministic walk and a strongly mixing color distribution, see example 2 of section 3. For color distributions which are only ergodic it is easy to find counterexamples even for nondeterministic walks.
3. Ergodic theoretic formulations A dynamical system is a probability space together with an automorphism of this space (i.e. an invertible measure preserving mapping from the space to itself). Stationary stochastic processes and dynamical systems are intimately related to each other, and we assume in the sequel that the reader possesses some intuition for this relationship. We begin by describing the dynamical systems related to the processes discussed in section 2. Consider the sequence space J = {j = ( ji)iEZ: ji E 2 for each i E Z} together with the “natural” a-algebra 9 on J, generated by the finite-dimensional cylinder sets. We think of j E J as a sequence of jumps made by a random walk and therefore denote by h a product probability measure on (J, .$) arising from a given jump distribution on Z. If a:J+J
is the left shift on J, defined by a(j)=j’
with
ji = ji+i
(iE
Z),
then (J, 9, A, cr) is a dynamical system, and we can describe the random walk with the given jump distribution by the following sequence of random variables: n-1
w,=o, Next, let
w” =
F.
n-1 ii
=
F. t”ii)O Cna ‘1.
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M. KEANE AND W.Th.F. DEN HOLLANDER
c = {c = (Ci)&
ci is black or white for each i E Z} ,
and provide C with its natural g-algebra %‘.An element of C is a coloring of 2, and if y is a probability measure on C which is invariant under the left shift 7:
c-c,
defined by r(c) = c’
with
c; = ci+r
(iEZ),
then we have a stationary stochastic coloring of Z, as well as another dynamical system (C, %, y, 7). We now put these two systems together to describe what is seen by the walker while moving through the lattice. To that end we define X:=JxC,
2?':=$x%,
p:=hxy,
so that (X, 2, p) is the product of the walk probability probability space (so walk and coloring are independent), be the transformation
space and the color and we define T to
T:X-X
given by
T(j,c) : = (u(j),
&(c))
for each x = (j, c) E X. It is easily seen that p is T-invariant, so that (X, 2, p, T) is a dynamical system. This system is commonly called a skew product in ergodic theory. Now any event concerning the random walk and the stochastic coloring corresponds to a subset of X, namely, those points x E X for which the event is realized. For instance, let s be a local scenery and n a non-negative integer. Then it should be clear from the definitions above that the events [s],, considered as subsets of X, are related to [s& by successive applications of the mapping T:
[s], = T-“[s],
(n 20)
.
Hence the validity of the proposition
of section 2 amounts to the asymptotic
ERGODIC PROPERTIES OF COLOR RECORDS
191
independence of the sets [s10 and T-“[t],, for all local sceneries s and t as n + 03. From this it is easy to show that for any sets A E 22’and A’ E 28, !iim p(A fl T-“A’)
= p(A)p(A')
.
Thus we have:
Conclusion 1. Asymptotic independence of local sceneries (as formulated in the proposition of section 2) is equivalent to strong mixing of (X, %‘, p, T). In a similar manner the theorem elusion. Define
of section 2 leads to an analogous con-
Y:={x=(j,c)EX:c,isblack}, 9:={AE
fZ:AcY},
v(B):=s
(BE%?).
Then the probability space (Y, 93, V) has the intuitive interpretation ditioning on the origin being black. Let
of con-
S:Y+Y be the transformation S(y) := TfCY’(y)
induced by T on Y: (y E
Y) ,
where t(y):=min{nsl:
T”YE Y}.
From the general theory of induced transformations it follows (and this is also easy to verify directly) that S(y) is well defined for v-almost every y E Y and that (Y, 93, V, S) is a dynamical system. The dynamical system (Y, 93, V, S) inherits some of the properties of (X, 8?, CL,T). For example, S is ergodic if and only if T is ergodic, and the entropies of S and Tare related by the factor p(Y). However, strong mixing is not hereditary. (See example 2 below.) As above, if t is a local scenery not coloring the origin, then
M. KEANE AND W.Th.F. DEN HOLLANDER
192
[flTkfl y=
O[tl,
since S effectuates
c-lY)
7
the move to the next black point. This brings us to:
Conclusion
2. Asymptotic independence of local sceneries on the black time scale (as formulated in the theorem of section 2) is equivalent to strong mixing of (Y, 53, V, S). In particular, asymptotic independence of any two events (in 93) on the black time scale is guaranteed if we can prove the theorem in section 2. Thus, for example, if nk is the kth run length as defined in the introduction, and m is a positive integer, then it easily follows from the theorem that E_i P([n, > m]) = P([n, > ml]0 is black) = P([nO = m],;
)
where the second term is independent of k and the second equality is a consequence of the stationarity and ergodicity of the color record process’). This in turn implies that lim,,, ( nk) = q-l, provided ( nk) is uniformly bounded in k, as is the case for instance when (n,) < ~0 (see ref. 1). More generally, it is plain from the above discussion that the study of mixing properties for induced transformations is of interest for at least one application. We now formulate a definition and a general mathematical question which we deem to be worthy of further study. Let (X, 2, p, T) be a strongly mixing dynamical system, and let A be a subset of X with p(A) > 0. Denote by (Y, 93, V, S) the induced system on A. Definition.
T is kasteleyn mixing* with respect to A if the induced (Y, 93, V, S) is strongly mixing. More in particular, if
system
x = (0, l}” and A={xEX:x,=l) (as in the color record process with 0 as white and 1 as black), then we simply call the process kasteleyn mixing, omitting the set A. General question.
* “kasteleyn” castellan.
Which strongly mixing O-l processes are kasteleyn mixing?
is the traditional Dutch word for barkeeper,
with English cognates chatelain and
ERGODIC
PROPERTIES
OF COLOR
RECORDS
193
Examples.
1) If T is a Bernoulli scheme (i.e. p is a product measure on X), then S is also (isomorphic to) a Bernoulli scheme, and hence T is kasteleyn mixing. More generally, whenever 1 is a renewal state, T is kasteleyn mixing. 2) Not all strongly mixing O-l processes are kasteleyn mixing. In particular, suppose the stationary measure p on X is obtained by choosing 0 or 1 independently, but writing two l’s instead of one every time a 1 is chosen. Then T is strongly mixing, but T is not kasteleyn mixing because the “even” l’s can be distinguished from the “odd” l’s on the time scale of the l’s, given enough scenery around the origin. This example, moreover, can be interpreted as the color record process of a deterministic walk with a strongly mixing color distribution obtained by randomly placing pairs of black points on 2. 3) The form of the set A is important, since any dynamical system (X, $?, CL,T), except the one-point system, contains a set B E 2 with p(B) > 0 and with B rl T-‘B = 0. If we set A = B fl T-‘B, then the induced transformation on A has - 1 as an eigenvalue and is not strongly mixing3). 4) If (X, % P, T) comes from simple random walk in Zd and Bernoulli coloring, then the theorem in section 2 shows that T is kasteleyn mixing (for black at the origin) if d 2 3. For d = 1 or d = 2 the random walk is recurrent and we do not know whether kasteleyn mixing holds or not. (See remark 3 after the theorem of section 2.) 5) Almost all random walks on more general groups are transient4), so that in these cases our theorem will be applicable. Finally, we remark that Kalikow5) has shown that in the situation of example 4, with d = 1, T is not loosely Bernoulli and is thus in some sense ill-behaved. Nevertheless, we do feel that the process may be kasteleyn mixing. Acknowledgement
We would like to author. In particular, along the lines of exposition in section
thank H. Kesten for useful discussions with the second he independently suggested a proof of the theorem much the one we had obtained, and this has improved our 2.
References 1) P.W.Kasteleyn, Bull. I.S.I. 45 (1985) 27.1-l. 2) H.C.P. Berbee, oral communication with the second author. H.C.P. Berbee (to appear in Israel J. Math). 3) S. Kakutani, oral communicati& with the first author (before 1970). 4) Y. Guivarc’h, M. Keane and B. Roynette, Lecture Notes in Mathematics 624 (Springer, 1977). 5) S. Kalikow, Annals of Mathematics 115 (1982) 393. Literature 6) F. Spitzer, Principles of Random Walk, 2nd edition (Springer, New York, 1976). 7) U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6 (de Gruyter, 1985).
Berlin,
Berlin,