# A Comparison of Random Walks in Dependent Random Environments

TL;DR: Comparing random walks in various dependent environments, it is demonstrated that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.

Abstract: Although the theoretical behavior of one-dimensional random walks in random environments is well understood, the actual evaluation of various characteristics of such processes has received relatively little attention. This paper develops new methodology for the exact computation of the drift in such models. Focusing on random walks in dependent random environments, including $k$-dependent and moving average environments, we show how the drift can be characterized and found using Perron-Frobenius theory. We compare random walks in various dependent environments and show that their drift behavior can differ signicantly.

## Summary (2 min read)

### 1 Introduction

- Random walks in environments are well-known mathematical models for motion through disorganized media.
- They generalize ordinary random walks whereby the transition probabilities from any position are determined by the random state of the environment at that position.
- From an applied and computational point of view significant gaps in their understanding remain.
- Exact drift computations and comparisons (as oposed to comparisons using simulation) between dependent random environments seem to be entirely missing from the literature.
- In Section 3 the authors prove explicit results for the drift for each of these models, and compare their behaviors.

### 2.1 General theory

- (2.1) The theoretical behavior of {Xn} is well understood, as set out in the seminal work of Solomon [14].
- In particular, Theorems 2.1 and 2.2 below completely describe the transience/recurrence behavior and the Law of Large Numbers behavior of {Xn}.
- The authors follow the notation of Alili [1] and first give the key quantities that appear in these theorems.
- These follow directly from the stationarity of U.

### 3 Evaluating the drift

- And then further specify the transience/recurrence and drift results to the Markov environment, the 2-dependent environment, and the moving average environments.the authors.
- The authors omit a separate derivation for the i.i.d. environment, which can be viewed as a special case of the Markovian environment, see Remark 3.1.

### 3.1 General solution for swap models

- Consider now the RWRE swap model with a random environment generated by a Markov chain {Yi, i ∈ Z}, as specified in Section 2.2.
- The authors summarize these findings in the following theorem.

### 3.2.1 Comparison with the iid environment

- Substitution into the expression for V (here in the case of positive drift only, see (3.7)) and rewriting yields V = (2p−.
- This enables us not only to immediately quickly obtain the drift for the iid case (take ̺ = 0), but also to study the dependence of the drift V on ̺.
- Figures 2 illustrates various aspects of the difference between iid and Markov cases.
- Clearly, compared to the iid case (for the same value of α), the Markov case with positive correlation coefficient has lower drift, but also a lower ‘cutoff value’ of p at which the drift becomes zero.
- For negative correlation coefficients the authors see a higher cutoff value, but not all values of α are possible (since they should have a < 1).

### 3.3 2-dependent environment

- Unfortunately, the eigenvalues of PD are now the roots of a 4-degree polynomial, which are hard to find explicitly.
- Using Perron–Frobenius theory and the implicit function theorem it is possible to prove the following lemma, which has the same structure as in the Markovian case.
- Now, moving σ from 1 to any other positive value, λ0(σ) must continue to play the role of the Perron–Frobenius eigenvalue; i.e., none of the other λi(σ) can at some point take over this role.
- Including the transience/recurrence result from the first part of this section, and including the cases with negative drift, the authors obtain the following analogon to Proposition 3.1.

### 3.4 Moving average environment

- The proof is similar to that of Lemma 3.2; the authors only give an outline, leaving details for the reader to verify.
- The cutoff value for p is now easily found as (1+σcutoff) −1, which can be numerically evaluated.
- It is interesting to note that the cutoff points (where V becomes 0) are significantly lower in the moving average case than the iid case, using the same α, while at the same time the maximal drift that can be achieved is higher for the moving average case than for the iid case.
- This is different behavior from the Markovian case; see also Figure 2.

### 4 Conclusions

- Random walks in random environments can exhibit interesting and unusual behavior due to the trapping phenomenon.
- The dependency structure of the random environment can significantly affect the drift of the process.
- For the wellknown swap RWRE model, this approach allows for easy computation of drift, as well as explicit conditions under which the drift is positive, negative, or zero.
- The cutoff values where the drift becomes zero, are determined via Perron–Frobenius theory.
- Various generalizations of the above environments can be considered in the same (swap model) framework, and can be analyzed along the same lines, e.g., replacing iid by Markovian {Ûi} in the moving average model, or taking moving averages of more than 3 neighboring states.

Did you find this useful? Give us your feedback

...read more

##### Citations

28 citations

##### References

3,877 citations

[...]

3,253 citations

2,478 citations

1,708 citations

1,560 citations