Large deviation principle for one-dimensional
random walk in dynamic random environment:
attractive spin-flips and simple symmetric exclusion
Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In  we proved a law of large numbers for dynamic random environments satisfying a space-time mixing property called cone-mixing. If an attractive spin-flip system has a finite average coupling time at the origin for two copies starting from the all-occupied and the all-vacant configuration, respectively, then it is cone-mixing.
In the present paper we prove a large deviation principle for the empirical speed of the random walk, both quenched and annealed, and exhibit some properties of the associated rate functions. Under an exponential space-time mixing condition for the spin-flip system, which is stronger than cone-mixing, the two rate functions have a unique zero, i.e., the slow-down phenomenon known to be possible in a static random environment does not survive in a fast mixing dynamic random environment. In contrast, we show that for the simple symmetric exclusion dynamics, which is not cone-mixing (and which is not a spin-flip system either), slow-down does occur.
MSC 2000. Primary 60H25, 82C44; Secondary 60F10, 35B40.
Key words and phrases. Dynamic random environment, random walk, quenched vs. annealed large deviation principle, slow-down.
Invited paper to appear in the 15-th anniversary celebration issue of Markov Processes and Related Fields.
1 Introduction and main results
1.1 Random walk in dynamic random environment: attractive spin-flips
denote a one-dimensional spin-flip system, i.e., a Markov process on state space with generator given by
where is any cylinder function on , is the local rate to flip the spin at site in the configuration , and is the configuration obtained from by flipping the spin at site . We think of () as meaning that site is occupied (vacant) at time . We assume that is shift-invariant, i.e., for all and ,
where , , and also that is attractive, i.e., if , then, for all ,
For more on shift-invariant attractive spin-flip systems, see , Chapter III. Examples are the (ferromagnetic) Stochastic Ising Model, the Voter Model, the Majority Vote Process and the Contact Process.
We assume that
For , we write to denote the law of starting from , which is a probability measure on path space , the space of càdlàg paths in . We further write
to denote the law of when is drawn from . We further assume that
Conditional on , let
be the random walk with local transition rates
In words, on occupied sites the random walk jumps to the right at rate and to the left at rate , while at vacant sites it does the opposite. Note that, by (1.10), on occupied sites the drift is positive, while on vacant sites it is negative. Also note that the sum of the jump rates is and is independent of . For , we write to denote the law of starting from conditional on , and
to denote the law of averaged over . We refer to as the quenched law and to as the annealed law.
1.2 Large deviation principles
In  we proved that if is cone-mixing, then satisfies a law of large numbers (LLN), i.e., there exists a such that
All attractive spin-flip systems for which the coupling time at the origin, starting from the configurations and , has finite mean are cone-mixing. Theorems 1.1–1.2 below state that satisfies both an annealed and a quenched large deviation principle (LDP); the interval in (1.15) and (1.18) can be either open, closed or half open and half closed.
The interpretation of (1.13) is that is a measure for the maximal dependence of the transition rates on the states of single sites, while is a measure for the minimal rate at which the states of single sites change. See , Section I.4, for examples. In  we showed that if then is cone-mixing.
1.3 Random walk in dynamic random environment: simple symmetric exclusion
It is natural to ask whether in a dynamic random environment the rate functions always have a unique zero. The answer is no. In this section we show that when is the simple symmetric exclusion process in equilibrium with an arbitrary density of occupied sites , then for any the probability that is near the origin decays slower than exponential in . Thus, slow-down is possible not only in a static random environment (see Section 1.4), but also in a dynamic random environment, provided it is not fast mixing. Indeed, the simple symmetric exclusion process is not even cone-mixing.
The one-dimensional simple symmetric exclusion process
is the Markov process on state space with generator given by
where is any cylinder function on , the sum runs over unordered neighboring pairs of sites in , and is the configuration obtained from by interchanging the states at sites and . We will asume that starts from the Bernoulli product measure with density , i.e., at time each site is occupied with probability and vacant with probability . This measure, which we denote by , is an equilibrium for the dynamics (see , Theorem VIII.1.44).
Conditional on , the random walk
Since the simple symmetric exclusion process is not cone-mixing (the space-time mixing property assumed in ), we do not have the LLN. Since it is not an attractive spin-flip system either, we also do not have the LDP. We plan to address these issues in future work. Our main result here is the following.
For all ,
Literature. Random walk in static random environment has been an intensive research area since the early 1970’s. One-dimensional models are well understood. In particular, recurrence vs. transience criteria, laws of large numbers and central theorems have been derived, as well as quenched and annealed large deviation principles. In higher dimensions a lot is known as well, but some important questions still remain open. For an overview of these results, we refer the reader to [33, 34] and . See the homepage of Firas Rassoul-Agha [www.math.utah.edu/firas/Research] for an up-to-date list of references.
For random walk in dynamic random environment the state of the art is rather more modest, even in one dimension. Early work was done in , which considers a one-dimensional environment consisting of spins flipping independently between and , and a walk that at integer times jumps left or right according to the spin it sees at that time. A necessary and sufficient criterion for recurrence is derived, as well as a law of large numbers.
Three classes of models have been studied in the literature so far:
The focus of these references is: transience vs. recurrence [24, 21], central limit theorem [9, 7, 12, 8, 13, 14, 4, 25, 15], law of large numbers and central limit theorem , decay of correlations in space and time , convergence of the law of the environment as seen from the walk , large deviations [22, 32]. In classes (1) and (2) the random environment is uncorrelated in time, respectively, in space. In  we moved away from this restriction by proving a law of large numbers for a class of dynamic random environments correlated in space and time, satisfying a space-time mixing condition called cone-mixing. We showed that a large class of uniquely ergodic attractive spin-flip systems falls into this class.
Consider a static random environment with law , the Bernoulli product measure with density , and a random walk with transition rates (compare with (1.9))
where . In  it is shown that is recurrent when and transient to the right when . In the transient case both ballistic and non-ballistic behavior occur, i.e., for -a.e. , and
and, for ,
Attractive spin flips. The analogues of (1.15) and (1.18) in the static random environment (with no restriction on the interval in the annealed case) were proved in  (quenched) and  (quenched and annealed). Both and are zero on the interval and are strictly positive outside (“slow-down phenomenon”). For the same symmetry property as in (1.19) holds. Moreover, an explicit formula for is known in terms of random continued fractions.
We do not have explicit expressions for and in the dynamic random environment. Even the characterization of their zero sets remains open, although under the stronger assumptions that and we know that both have a unique zero at .
Theorems 1.1–1.2 can be generalized beyond spin-flip systems, i.e., systems where more than one site can flip state at a time. We will see in Sections 2–3 that what really matters is that the system has positive correlations in space and time. As shown in , this holds for monotone systems (see , Definition II.2.3) if and only if all transitions are such that they make the configuration either larger or smaller in the partial order induced by inclusion.
Simple symmetric exclusion. What Theorem 1.3 says is that, for all choices of the parameters, the annealed rate function (if it exists) is zero at , and so there is a slow-down phenomenon similar to what happens in the static random environment. We will see in Section 4 that this slow-down comes from the fact that the simple symmetric exclusion process suffers “traffic jams”, i.e., long strings of occupied and vacant sites have an appreciable probability to survive for a long time.
To test the validity of the LLN for the simple symmetric exclusion process, we performed a simulation the outcome of which is drawn in Figs. 1–2. For each point in these figures, we drew initial configurations according to the Bernoulli product measure with density , and from each of these configurations ran a discrete-time exclusion process with parallel updating for steps. Given the latter, we ran a discrete-time random walk for steps, both in the static environment (ignoring the updating) and in the dynamic environment (respecting the updating), and afterwards averaged the displacement of the walk over the initial configurations. The probability to jump to the right was taken to be on an occupied site and on a vacant site, where replaces in the continuous-time model. In Figs. 1–2, the speeds resulting from these simulations are plotted as a function of for , respectively, as a function of for . In each figure we plot four curves: (1) the theoretical speed in the static case (as described by (1.30)); (2) the simulated speed in the static case; (3) the simulated speed in the dynamic case; (4) the speed for the average environment, i.e., . The order in which these curves appear in the figures is from bottom to top.
Fig. 1 shows that, in the static case with fixed, as increases the speed first goes up (because there are more occupied than vacant sites), and then goes down (because the vancant sites become more efficient to act as a barrier). In the dynamic case, however, the speed is an increasing function of : the vacant site are not frozen but move around and make way for the walk. It is clear from Fig. 2 that the only value of for which there is a zero speed in the dynamic case is , for which the random walk is recurrent. Thus, the simulation suggests that there is no (!) non-ballistic behavior in the transient case. In view of Theorem 1.3, this in turn suggests that the annealed rate function (if it exists) has zero set .
In both pictures the two curves at the bottom should coincide. Indeed, they almost coincide, except for values of the parameters that are close to the transition between ballistic and non-ballistic behavior, for which fluctuations are to be expected. Note that the simulated speed in the dynamic environment lies inbetween the speed for the static environment and the speed for the average environment. We may think of the latter as corresponding to a simple symmetric exclusion process running at rate , respectively, rather than at rate as in (1.24).
2 Proof of Theorem 1.1
In Section 2.1 we prove three lemmas for the probability that the empirical speed is above a given threshold. These lemmas will be used in Section 2.2 to prove Theorems 1.1(a–b). In Section 2.3 we prove Theorems 1.1(c).
2.1 Three lemmas
For all ,
For and , let denote the operator acting on as
Fix , and let be the non-negative grid of width . For any , we have
The first inequality holds because two copies of the random walk running on the same realization of the random environment can be coupled so that they remain ordered. The second inequality uses that
are non-decreasing and that the law of an attractive spin-flip system has the FKG-property in space-time (see , Corollary II.2.12). Let
Then it follows from (2.3) that is subadditive along , i.e., for all . Since for all , it thefore follows that
Because takes values in , the restriction can be removed. This proves the claim for . The claim easily extends to , because the transition rates of the random walk are bounded away from and uniformly in (recall (1.9)).
is non-decreasing and convex on .
for and .
2.2 Annealed LDP
Clearly, depends on , and . Write
to exhibit this dependence. So far we have not used the restriction in (1.10). By noting that is equal in distribution to when and are swapped and is replaced by , the image of under reflection in the origin (recall (1.9)), we see that the upward annealed LDP proved in Section 2.1 also yields a downward annealed LDP
whose qualitative shape is given in Fig. 4. Note that
because , the speed in the LLN proved in , must lie in the zero set of both and .
Our task is to turn the upward and downward annealed LDP’s into the annealed LDP of Theorem 1.1.
for all closed intervals such that either or .
We distinguish three cases.
(1) , : Let . Then, because is continuous,
(2) , : Same as for (1) with replacing .
2.3 Unique zero of when
In  we showed that if and , then a proof of the LLN can be given that is based on a perturbation argument for the generator of the environment process
i.e., the random environment as seen relative to the random walk. In particular, it is shown that is uniquely ergodic with equilibrium . This leads to a series expansion for in powers of , with coefficients that are functions of and and that are computable via a recursive scheme. The speed in the LLN is given by
with , where denotes expectation over ( is the fraction of time spends on occupied sites).
Let be an attractive spin-flip system with . If , then the rate function in (2.18) has a unique zero at .
It suffices to show that
To that end, put . Then, by (2.21), . Let
be the time spends on occupied sites up to time , and define
Conditional on , behaves like a homogeneous random walk with speed in . Therefore the second term in the r.h.s. of (2.25) vanishes exponentially fast in . In , Lemma 3.4, Eq. (3.26) and Eq. (3.36), we proved that
for some , where denotes the semigroup associated with the environment process , and denotes the triple norm of . As shown in , (2.26) implies a Gaussian concentration bound for additive functionals, namely,
for some , uniformly in , with and . By picking , , we get
for some . Therefore also the first term in the r.h.s. of (2.25) vanishes exponentially fast in .
3 Proof of Theorem 1.2
In Section 3.1 we prove three lemmas for the probability that the empirical speed equals a given value. These lemmas will be used in Section 3.2 to prove Theorems 1.2(a–b). In Section 3.3 we prove Theorem 1.2(c). Theorem 1.2(d) follows from Theorem 1.1(c) because .
3.1 Three lemmas
For all ,
Fix , and recall that is the non-negative grid of width . For any , we have
where . Let
exists, is finite -a.s, and is -invariant for every . Moreover, since is ergodic under space-time shifts (recall (1.5) and (1.7)), this limit is constant -a.s. Because the transition rates of the random walk are bounded away from and uniformly in (recall (1.9)), the restriction may be removed after is replaced by in (3.4). This proves the claim for . By the boundedness of the transition rates, the claim easily extends to .
is convex on .
for and .
Same as Lemma 2.3.
3.2 Quenched LDP
We are now ready to prove the quenched LDP.
For -a.e. , the family of probability measures , , satisfies the LDP with rate and with deterministic rate function .
3.3 A quenched symmetry relation
For all , the rate function in Theorem 3.4 satisfies the symmetry relation
We first consider a discrete-time random walk, i.e., a random walk that observes the random environment and jumps at integer times. Afterwards we will extend the argument to the continuous-time random walk defined in (1.8–1.10).
1. Path probabilities. Let
be the random walk with transition probablities
where w.l.o.g. . For an oriented edge , , write to denote the reverse edge. Let denote the probability for the walk to jump along the edge at time . Note that in the static random environment these probabilities are time-independent, i.e., for all .
We will be interested in -step paths with and for a given . Write to denote the time-reversed path, i.e., . Let denote the number of times the edge is crossed by , and write , , to denote the successive times at which the edge is crossed. Let denote the set of edges in the path , and the subset of forward edges, i.e., edges of the form . Then we have
Given a realization of , the probability that the walk follows the path equals