1 Introduction

Random Walks in Random Environments

L. V. Bogachev, University of Leeds, Leeds, UK

© 2006 Elsevier Ltd. All rights reserved.

1

1 Introduction

Random walks provide a simple conventional model to describe various transport processes, for example propagation of heat or diffusion of matter through a medium (for a general reference see, e.g., Hughes (1995)). However, in many practical cases the medium where the system evolves is highly irregular, due to factors such as defects, impurities, fluctuations etc. It is natural to model such irregularities as random environment, treating the observable sample as a statistical realization of an ensemble, obtained by choosing the local characteristics of the motion (e.g., transport coefficients and driving fields) at random, according to a certain probability distribution.

In the random walks context, such models are referred to as Random Walks in Random Environments (RWRE). This is a relatively new chapter in applied probability and physics of disordered systems initiated in the 1970s. Early interest in RWRE models was motivated by some problems in biology, crystallography and metal physics, but later applications have spread through numerous areas (see review papers by Alexander et al. (1981), Bouchaud and Georges (1990), and a comprehensive monograph by Hughes (1996)). After 30 years of extensive work, RWRE remain a very active area of research, which has been a rich source of hard and challenging questions and has already led to many surprising discoveries, such as subdiffusive behavior, trapping effects, localization, etc. It is fair to say that the RWRE paradigm has become firmly established in physics of random media, and its models, ideas, methods, results, and general effects have become an indispensable part of the standard tool kit of a mathematical physicist.

One of the central problems in random media theory is to establish conditions ensuring homogenization, whereby a given stochastic system evolving in a random medium can be adequately described, on some spatial-temporal scale, using a suitable effective system in a homogeneous (non-random) medium. In particular, such systems would exhibit classical diffusive behavior with effective drift and diffusion coefficient. Such an approximation, called effective medium approximation (EMA), may be expected to be successful for systems exposed to a relatively small disorder of the environment. However, in certain circumstances EMA may fail due to atypical environment configurations (“large deviations”) leading to various anomalous effects. For instance, with small but positive probability a realization of the environment may create “traps” that would hold the particle for anomalously long time, resulting in the subdiffusive behavior, with the mean square displacement growing slower than linearly in time.

RWRE models have been studied by various non-rigorous methods including Monte Carlo simulations, series expansions, and the renormalization group techniques (see more details in the above references), but only few models have been analyzed rigorously, especially in dimensions greater than one. The situation is much more satisfactory in the one-dimensional case, where the mathematical theory has matured and the RWRE dynamics has been understood fairly well.

The goal of this article is to give a brief introduction to the beautiful area of RWRE. The principal model to be discussed is a random walk with nearest-neighbor jumps in independent identically distributed (i.i.d.) random environment in one dimension, although we shall also comment on some generalizations. The focus is on rigorous results; however, heuristics will be used freely to motivate the ideas and explain the approaches and proofs. In a few cases, sketches of the proofs have been included, which should help appreciate the flavor of the results and methods.

1.1 Ordinary Random Walks: A Reminder

To put our exposition in perspective, let us give a brief account of a few basic concepts and facts for ordinary random walks, that is, evolving in a non-random environment (see further details in Hughes 1995). In such models, space is modelled using a suitable graph, e.g., a -dimensional integer lattice , while time may be discrete or continuous. The latter distinction is not essential, and in this article we will mostly focus on the discrete-time case. The random mechanism of spatial motion is then determined by the given transition probabilities (probabilities of jumps) at each site of the graph. In the lattice case, it is usually assumed that the walk is translation invariant, so that at each step distribution of jumps is the same, with no regard to the current location of the walk.

In one dimension (), the simple (nearest-neighbor) random walk may move one step to the right or to the left at a time, with some probabilities and , respectively. An important assumption is that only the current location of the walk determines the random motion mechanism, whereas the past history is not relevant. In terms of probability theory, such a process is referred to as Markov chain. Thus, assuming that the walk starts at the origin, its position after steps can be represented as the sum of consecutive displacements, , where are independent random variables with the same distribution ,  .

The strong law of large numbers (LLN) states that almost surely (i.e., with probability )

(1)

where denotes expectation (mean value) with respect to . This result shows that the random walk moves with the asymptotic average velocity close to . It follows that if then the process , with probability 1, will ultimately drift to infinity (more precisely, if and if ). In particular, in this case the random walk may return to the origin (and in fact visit any site on ) only finitely many times. Such behavior is called transient. However, in the symmetric case (i.e., ) the average velocity vanishes, so the above argument fails. In this case the walk behavior appears to be more complicated, as it makes increasingly large excursions both to the right and to the left, so that ,   (-a.s.). This implies that a symmetric random walk in one dimension is recurrent, in that it visits the origin (and indeed any site on ) infinitely often. Moreover, it can be shown to be null-recurrent, which means that the expected time to return to the origin is infinite. That is to say, return to the origin is guaranteed, but it takes very long until this happens.

Fluctuations of the random walk can be characterized further via the central limit theorem (CLT), which amounts to saying that the distribution of is asymptotically normal, with mean and variance :

(2)

These results can be extended to more general walks in one dimension, and also to higher dimensions. For instance, the criterion of recurrence for a general one-dimensional random walk is that it is unbiased, . In the two-dimensional case, in addition one needs . In higher dimensions, any random walk (which does not reduce to lower dimension) is transient.

1.2 Random Environments and Random Walks

The definition of an RWRE involves two ingredients: (i) the environment, which is randomly chosen but remains fixed throughout the time evolution, and (ii) the random walk, whose transition probabilities are determined by the environment. The set of environments (sample space) is denoted by , and we use to denote the probability distribution on this space. For each , we define the random walk in the environment as the (time-homogeneous) Markov chain on with certain (random) transition probabilities

(3)

The probability measure that determines the distribution of the random walk in a given environment is referred to as the quenched law. We often use a subindex to indicate the initial position of the walk, so that e.g. .

By averaging the quenched probability further, with respect to the environment distribution, we obtain the annealed measure , which determines the probability law of the RWRE:

(4)

Expectation with respect to the annealed measure will be denoted by .

Equation (4) implies that if some property of the RWRE holds almost surely (a.s.) with respect to the quenched law for almost all environments (i.e., for all such that ), then this property is also true with probability under the annealed law .

Note that the random walk is a Markov chain only conditionally on the fixed environment (i.e., with respect to ), but the Markov property fails under the annealed measure . This is because the past history cannot be neglected, as it tells what information about the medium must be taken into account when averaging with respect to environment. That is to say, the walk learns more about the environment by taking more steps. (This idea motivates the method of “environment viewed from the particle”, see Section 7 below.)

The simplest model is the nearest-neighbor one-dimensional walk, with transition probabilities

where and () are random variables on the probability space . That is to say, given the environment , the random walk currently at point will make a one-unit step to the right, with probability , or to the left, with probability . Here the environment is determined by the sequence of random variables . For the most of the article, we assume that the random probabilities are independent and identically distributed (i.i.d.), which is referred to as i.i.d. environment. Some extensions to more general environments will be mentioned briefly in Section 9. The study of RWRE is simplified under the following natural condition called (uniform) ellipticity:

(5)

which will be frequently assumed in the sequel.

2 Transience and Recurrence

In this section, we discuss a criterion for the RWRE to be transient or recurrent. The following theorem is due to Solomon (1975).

Theorem 1.

Set , , and .

(i) If then is transient (-a.s.); moreover, if then , while if then (-a.s.).

(ii) If then is recurrent (-a.s.); moreover,

Let us sketch the proof. Consider the hitting times and denote by the quenched first-passage probability from to :

Starting from the first step of the walk may be either to the right or to the left, hence by the Markov property the return probability can be decomposed as

(6)

To evaluate , for set

which is the probability to reach prior to , starting from . Clearly,

(7)

Decomposition with respect to the first step yields the difference equation

(8)

with the boundary conditions

(9)

Using , eqn (8) can be rewritten as

whence by iterations

(10)

Summing over and using the boundary conditions (9) we obtain

(11)

(if , the product over is interpreted as ). In view of eqn (7) it follows that if and only if the right-hand side of eqn (11) tends to , that is,

(12)

Note that the random variables are i.i.d., hence by the strong LLN

That is, the general term of the series (12) for large behaves like , hence for the condition (12) holds true (and so ), whereas for it fails (and so ).

By interchanging the roles of and , we also have if and if . From eqn (6) it then follows that in both cases , i.e. the random walk is transient.

In the critical case, , by a general result from probability theory, for infinitely many (-a.s.), and so the series in eqn (12) diverges. Hence, and, similarly, , so by eqn (6) , i.e. the random walk is recurrent.

It may be surprising that the critical parameter appears in the form , as it is probably more natural to expect, by analogy with the ordinary random walk, that the RWRE criterion would be based on the mean drift, . In the next section we will see that the sign of may be misleading.

A canonical model of RWRE is specified by the assumption that the random variables take only two values, and , with probabilities

(13)

where ,  . Here , and it is easy to see that, e.g., if , or , . The recurrent region where splits into two lines, and . Note that the first case is degenerate and amounts to the ordinary symmetric random walk, while the second one (except where ) corresponds to Sinai’s problem (see Section 6). A “phase diagram” for this model, showing various limiting regimes as a function of the parameters , , is presented in Figure 1.

Figure 1: Phase diagram for the canonical model, eqn (13). In the regions where or , the RWRE is transient to or , respectively. The recurrent case, , arises when or . The asymptotic velocity is given by eqn (14). Adapted from Hughes B.D. (1996) Random Walks and Random Environments. Volume 2: Random Environments, Ch. 6, p. 391. Clarendon Press, Oxford, by permission of Oxford University Press.

3 Asymptotic Velocity

In the transient case the walk escapes to infinity, and it is reasonable to ask at what speed. For a non-random environment, , the answer is given by the LLN, eqn (1). For the simple RWRE, the asymptotic velocity was obtained by Solomon (1975). Note that by Jensen’s inequality, .

Theorem 2.

The limit    exists(-a.s.) and is given by

(14)

Thus, the RWRE has a well-defined non-zeroasymptotic velocity except when . For instance, in the canonical example eqn (13) (see Figure 1) the criterion for the velocity to be positive amounts to the condition that both and lie on the same side of point .

The key idea of the proof is to analyze the hitting times first, deducing results for the walk later. More specifically, set , which is the time to hit after hitting (providing that ). If and then . Note that in fixed environment the random variables are independent, since the quenched random walk “forgets” its past. Although there is no independence with respect to the annealed probability measure , one can show that, due to the i.i.d. property of the environment, the sequence is ergodic and therefore satisfies the LLN:

In turn, this implies

(15)

(the clue is to note that ).

To compute the mean value , observe that

(16)

where is the indicator of event and , are, respectively, the times to get from to and then from to . Taking expectations in a fixed environment , we obtain

(17)

and so

(18)

Note that is a function of and hence is independent of . Averaging eqn (18) over the environment and using yields

(19)

and by eqn (15) “half” of eqn (14) follows. The other half, in terms of , can be obtained by interchanging the roles of and , whereby is replaced with .

Let us make a few remarks concerning Theorems 1 and 2. First of all, note that by Jensen’s inequality , with a strict inequality whenever is non-degenerate. Therefore, it may be possible that, with -probability , but (see Figure 1). This is quite unusual as compared to the ordinary random walk (see Section 1.1), and indicates some kind of slowdown in the transient case.

Furthermore, by Jensen’s inequality

so eqn (14) implies that if then

and the inequality is strict if is genuinely random (i.e., does not reduce to a constant). Hence, the asymptotic velocity is less than the mean drift , which is yet another evidence of slowdown. What is even more surprising is that it is possible to have but , so that -a.s.  (although with velocity ). Indeed, following Sznitman (2004) suppose that

with . Then if , hence . On the other hand,

if is sufficiently small.

4 Critical Exponent, Excursions and Traps

Extending the previous analysis of the hitting times, one can obtain useful information about the limit distribution of (and hence ). To appreciate this, note that from the recursion (16) it follows

and, similarly to eqn (17),

Taking here expectation , one can deduce that if and only if . Therefore, it is natural to expect that the root of the equation

(20)

plays the role of a critical exponent responsible for the growth rate (and hence, for the type of the limit distribution) of the sum . In particular, by analogy with sums of i.i.d. random variables one can expect that if then is asymptotically normal, with the standard scaling , while for the limit law of is stable (with index ) under scaling .

Alternatively, eqn (20) can be obtained from consideration of excursions of the random walk. Let be the left excursion time from site , that is the time to return to after moving to the left at the first step. If , then (-a.s.). Fixing an environment , let be the quenched mean duration of the excursion and observe that , where is the time to get back to after stepping to .

As a matter of fact, this representation and eqn (19) imply that the annealed mean duration of the left excursion, , is given by

(21)

Note that in the latter case (and bearing in mind ), the random walk starting from will eventually drift to , thus making only a finite number of visits to , but the expected number of such visits is infinite.

In fact, our goal here is to characterize the distribution of under the law . To this end, observe that the excursion involves at least two steps (the first and the last ones) and, possibly, several left excursions from , each with mean time . Therefore,

(22)

By the translation invariance of the environment, the random variables and have the same distribution. Furthermore, similarly to recursion (22), we have . This implies that is a function of with only, and hence and are independent random variables. Introducing the Laplace transform and conditioning on , from eqn (22) we get the equation

(23)

Suppose that

then eqn (23) amounts to

Expanding the product on the right, one can see that a solution with is possible only if , in which case

We have already obtained this result in eqn (21).

The case is possible if , which is exactly eqn (20). Returning to , one expects a slow decay of the distribution tail,

In particular, in this case the annealed mean duration of the left excursion appears to be infinite.

Although the above considerations point to the critical parameter , eqn (20), which may be expected to determine the slowdown scale, they provide little explanation of a mechanism of the slowdown phenomenon. Heuristically, it is natural to attribute the slowdown effects to the presence of traps in the environment, which may be thought of as regions that are easy to enter but hard to leave. In the one-dimensional case, such a trap would occur, for example, between two long series of successive sites where the probabilities are fairly large (on the left) and small (on the right).

Remarkably, traps can be characterized quantitatively with regard to the properties of the random environment, by linking them to certain large deviation effects (see Sznitman (2002, 2004)). The key role in this analysis is played by the function ,  . Suppose that (so that by Theorem 1 the RWRE tends to , -a.s.) and also that and (so that by Theorem 2, ). The latter means that and , and since is a smooth strictly convex function and , it follows that there is the second root , so that , i.e., (cf. eqn (20)).

Let us estimate the probability to have a trap in where the RWRE will spend anomalously long time. Using eqn (11), observe that

where as . However, due to large deviations may exceed level with probability

where is the Legendre transform of . We can optimize this estimate by assuming that and minimizing the ratio . Note that can be expressed via the inverse Legendre transform, , and it is easy to see that if then , so is the second (positive) root of .

The “left” probability is estimated in a similar fashion, and one can deduce that for some constants , and any , for large

That is to say, this is a bound on the probability to see a trap centered at , of size , which will retain the RWRE for at least time . It can be shown that, typically, there will be many such traps both in and , which will essentially prevent the RWRE from moving at distance from the origin before time . In particular, it follows that for any , so recalling that , we have indeed a sublinear growth of . This result is more informative as compared to Theorem 2 (the case ), and it clarifies the role of traps (see more details in Sznitman (2004)). The non-trivial behavior of the RWRE on the precise growth scale, , is characterized in the next section.

5 Limit Distributions

Considerations in Section 4 suggest that the exponent , defined as the solution of eqn (20), characterizes environments in terms of duration of left excursions. These heuristic arguments are confirmed by a limit theorem by Kesten et al. (1975), which specifies the slowdown scale. We state here the most striking part of their result. Denote ; by an arithmetic distribution one means a probability law on concentrated on the set of points of the form , , ,

Theorem 3.

Assume that and the distribution of is non-arithmetic (excluding a possible atom at ). Suppose that the root of equation (20) is such that and . Then

where is the distribution function of a stable law with index , concentrated on .

General information on stable laws can be found in many probability books; we only mention here that the Laplace transform of a stable distribution on with index has the form .

Kesten et al. (1975) also consider the case . Note that for , we have , so by eqn (14). For example, if then, as expected (see Section 4),

Let us describe an elegant idea of the proof based on a suitable renewal structure. (i) Let () be the number of left excursions starting from up to time , and note that . Since the walk is transient to , the sum is finite (-a.s.) and so does not affect the limit. (ii) Observe that if the environment is fixed then the conditional distribution of , given , is the same as the distribution of the sum of i.i.d. random variables , each with geometric distribution (). Therefore, the sum (read from right to left) can be represented as , where is a branching process (in random environment ) with one immigrant at each step and the geometric offspring distribution with parameter for each particle present at time . (iii) Consider the successive “regeneration” times , at which the process vanishes. The partial sums form an i.i.d. sequence, and the proof amounts to showing that the sum of has a stable limit of index . (iv) Finally, the distribution of can be approximated using (cf. eqn (11)), which is the quenched mean number of total progeny of the immigrant at time . Using Kesten’s renewal theorem, it can be checked that as , so is in the domain of attraction of a stable law with index , and the result follows.

Let us emphasize the significance of the regeneration times . Returning to the original random walk, one can see that these are times at which the RWRE hits a new “record” on its way to , never to backtrack again. The same idea plays a crucial role in the analysis of the RWRE in higher dimensions (see Sections 10.1, 10.2 below).

Finally, note that the condition allows , so the distribution of may have an atom at (and hence at ). In view of eqn (20), no atom is possible at . The restriction for the distribution of to be non-arithmetic is important. This will be illustrated in Section 8 where we discuss the model of random diodes.

6 Sinai’s Localization

The results discussed in Section 5 indicate that the less transient the RWRE is (i.e., the critical exponent decreasing to zero), the slower it moves. Sinai (1982) proved a remarkable theorem showing that for the recurrent RWRE (i.e., with ), the slowdown effect is exhibited in a striking way.

Theorem 4.

Suppose that the environment is i.i.d. and elliptic, eqn (5), and assume that , with . Denote , . Then there exists a function of the random environment such that for any

(24)

Moreover, has a limit distribution:

(25)

and thus also the distribution of under converges to the same distribution .

Sinai’s theorem shows that in the recurrent case, the RWRE considered on the spatial scale becomes localized near some random point (depending on the environment only). This phenomenon, frequently referred to as Sinai’s localization, indicates an extremely strong slowdown of the motion as compared with the ordinary diffusive behavior.

Following Révész (1990), let us explain heuristically why is measured on the scale . Rewrite eqn (11) as

(26)

where is defined in (12). By the central limit theorem, the typical size of for large is of order of , and so eqn (26) yields

This suggests that the walk started at site will make about visits to the origin before reaching level . Therefore, the first passage to site takes at least time . In other words, one may expect that a typical displacement after steps will be of order of (cf. eqn (24)). This argument also indicates, in the spirit of the trapping mechanism of slowdown discussed at the end of Section 4, that there is typically a trap of size , which retains the RWRE until time .

It has been shown (independently by H. Kesten and A.O. Golosov) that the limit in (25) coincides with the distribution of a certain functional of the standard Brownian motion, with the density function

7 Environment Viewed from the Particle

This important technique, dating back to Kozlov and Molchanov (1984), has proved to be quite efficient in the study of random motions in random media. The basic idea is to focus on the evolution of the environment viewed from the current position of the walk.

Let be the shift operator acting on the space of environments as follows:

Consider the process

which describes the state of the environment from the point of view of an observer moving along with the random walk . One can show that is a Markov chain (with respect to both and ), with the transition kernel

(27)

and the respective initial law or (here is the Dirac measure, i.e., unit mass at ).

This fact as it stands may not seem to be of any practical use, since the state space of this Markov chain is very complex. However, the great advantage is that one can find an explicit invariant probability for the kernel (i.e., such that ), which is absolutely continuous with respect to .

More specifically, assume that and set , where (cf. eqn (14))

(28)

Using independence of , we note

hence is a probability measure on . Furthermore, for any bounded measurable function on we have

(29)

By eqn (28),

and similarly

So from eqn (29) we obtain

which proves the invariance of .

To illustrate the environment method, let us sketch the proof of Solomon’s result on the asymptotic velocity (see Theorem 2 in Section 3). Set . Noting that , define

Due to the Markov property, the process is a martingale with respect to the natural filtration and the law ,

and it has bounded jumps, . By general results, this implies (-a.s.).

On the other hand, by Birkhoff’s ergodic theorem

The last integral is easily evaluated to yield

and the first part of the formula (14) follows.

The case can be handled using a comparison argument (Sznitman 2004). Observe that if for all then for the corresponding random walks we have (-a.s.). We now define a suitable dominating random medium by setting (for )

Then if is large enough, so by the first part of the theorem, -a.s.,

(30)

Note that is a continuous function of with values in , so there exists such that attains the value . Passing to the limit in eqn (30) as , we obtain (-a.s.). Similarly, we get the reverse inequality, which proves the second part of the theorem.

A more prominent advantage of the environment method is that it naturally leads to statements of CLT type. A key step is to find a function