Slow movement of a random walk on the range of a random walk in the presence of an external field
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Abstract
In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (\(d\ge 5\)). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a nontrivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cuttimes to relate the associated biased random walk to a onedimensional random walk in a random environment in Sinai’s regime. Via this approach, a corresponding aging result is also proved.
Keywords
Biased random walk Range of random walk Sinai’s walk Localisation AgingMathematics Subject Classification
60K37 60K35 60G501 Introduction
In studying random walks in random environments, there is a particular focus at the moment on understanding the effect of an external field. Indeed, some quite remarkable results have been proved in this area. For instance, whereas adding a deterministic unidirectional bias to the random walk on the integer lattice \(\mathbb Z ^d\) results in ballistic escape, the same has been shown not to hold for supercritical percolation clusters. Instead, the random environment arising in the percolation model creates traps which become stronger as the bias is increased, so that when the bias is set above a certain critical value, the speed of the biased random walk is zero [5, 9, 16]. This phenomenon, which has also been observed for the biased random walks on supercritical Galton–Watson trees [4, 14] and a onedimensional percolation model [1], is of physical significance, as it helps to explain how a particle could in some circumstances actually move more slowly when the strength of an external field, such as gravity, is greater [3].
For percolation on the integer lattice close to criticality, physicists have identified two potential trapping mechanisms for the associated biased random walk: ‘trapping in branches’ and ‘traps along the backbone’ [3]. More concretely, in high dimensions the incipient infinite cluster for bond percolation on the integer lattice is believed to be formed of a single infinite path—the backbone, to which a collection of ‘branches’ or ‘dangling ends’ is attached. If the dangling end is aligned with the bias, then the random walk will find it easy to enter this section of the graph, but very difficult to escape. Similarly, there will be sections of the backbone that flow with and sections that flow against the bias, and this will mean the random walk will prefer to spend time in certain locations along it.
Given that rigourous results for the incipient infinite cluster for critical bond percolation in \(\mathbb Z ^d\) are currently rather limited, exploring the biased random walks on it directly is likely to be difficult. Nonetheless, the above heuristics motivate a number of interesting, but more tractable research problems, one of which will be the focus of this article. In particular, to investigate the effect of ‘traps along the backbone’, it makes sense to initially study how the presence of an external field affects a random walk on a random path. A natural choice for such a path is the one generated by a simple random walk on \(\mathbb Z ^d\), and it is for this reason that we pursue here a study of the biased random walk on this object.
Theorem 1.1
Remark 1.2
The characterisation of \(L_\beta \) that will be given in the proof of Theorem 1.1 readily yields that the distribution of \(L_\beta \log \beta \) is independent of \(\beta \). (This fact can also seen on Fig. 2 below, which provides a sketch of the localisation point of \(X\).) Thus, as the bias is increased, the biased random walk will be found closer to the origin.
To show that the unbiased random walk \(X\) on the graph \(\mathcal G \) in dimensions \(d\ge 5\) is diffusive, it was exploited in [7] that the point process of cuttimes of \(S\), i.e. those times where the past and future paths do not intersect (of which there are infinite), is stationary. In particular, this observation allowed \(\mathcal G \) to be decomposed at cutpoints into a stationary chain of finite graphs, effectively reducing the problem into a onedimensional one. (Note that the same techniques are no longer applicable when \(d\le 4\), as there are no longer an infinite number of cuttimes for the twosided random walk path.) This idea will again prove useful when proving Theorem 1.1, with the difference being that now it must be taken into account how the bias affects each of the graphs in the chain. Since the orientations of the graphs in the chain are random, it turns out that the onedimensional model it is relevant to compare to is a random walk in a random environment in the socalled Sinai regime. It is now wellknown that, because of the large traps that arise, a random walk in a random environment in Sinai’s regime escapes at a rate \((\log n)^2\) [15]. This will also be true for \(X\) with respect to the graph distance, but taking into account that \(S\) satisfies a diffusive scaling, we arrive at the \(\log n\) scaling of the above result.
Another phenomenon occurring in Sinai’s regime is aging—that is, the existence of correlations in the system over long time scales. One way of formalising this is to show that the asymptotic probability that the locations of the process on two different time scales are close does not decay to zero. Providing such a result for the biased random walk on the range of a random walk is the purpose of the following corollary. Note that the two time scales considered in this aging result, \(n\) and \(n^h\), where \(h> 1\), are the same as those for which the analogous result is known to hold for the onedimensional random walk in a random environment in Sinai’s regime (see [8], and also [17, Section 2.5]).
Corollary 1.3
The main difficulty in pursuing the line of reasoning outlined above is that the underlying simple random walk \(S\) has loops, and so it is necessary to estimate how much time the biased random walk \(X\) spends in these. If we start from a random path that is nonself intersecting, then there is not such a problem and, as long as the first coordinate of the random path still converges to a Brownian motion, verifying that a biased random walk exhibits a localisation phenomenon is much more straightforward. Thus, as a warm up to proving Theorem 1.1, we start by considering biased random walks on nonself intersecting paths. As a particular example, we are able to prove the following annealed scaling limit for the biased random walk on the range of a twosided looperased random walk in high dimensions (see the end of Sect. 2 for precise definitions). It would also be possible to derive an aging result corresponding to Corollary 1.3 for this model, but since the proof would be identical (actually, slightly simpler), we choose not to present such a conclusion here.
Theorem 1.4
This article contains only two further sections. In Sect. 2 we explain the relationship between the biased random walk on a random path and a random walk in a onedimensional random environment, and prove Theorem 1.4. In Sect. 3, we adapt the argument in order to prove Theorem 1.1 and Corollary 1.3.
2 Biased random walk on a selfavoiding random path
The aim of this section is to describe how a biased random walk on a selfavoiding random path can be expressed as a random walk in a onedimensional random environment. As we will demonstrate, this enables us to transfer results proved for the latter model to the former. To illustrate this, we will apply our techniques to the biased random walk on the range of the twosided looperased random walk in dimensions \(d\ge 5\).
The potential is of particular relevance when understanding the behaviour of the random walk in a random environment in the Sinai regime. In particular, by applying the fact that the potential converges to a Brownian motion, it is possible to describe where the large traps in the environment appear, and thus where the random walk prefers to spend time. Hence, at least when \(S\) satisfies a scaling result that incorporates a functional invariance principle in the first coordinate (and the increments of \(S^{(1)}\) are bounded), it is possible to use the relationship between \(S^{(1)}\) and \(R\) derived above to obtain the behaviour of the biased random walk on the random path.
Proposition 2.1
Proof
3 Biased random walk on the range of simple random walk
 (i)
Firstly, we need to check that the associated potential has Brownian scaling (this is an assumption in [17]). To do this, we use the connection between random walks and electrical networks, scaling and continuity properties of \(S\), and the distribution of the cuttimes of \(S\) (see Lemma 3.1).
 (ii)
Secondly, the random environment that arises is nonelliptic (i.e. its transition probabilities are not uniformly bounded from below). Since ellipticity is assumed in [17], we need to be careful about controlling the effect of small transition probabilities (see (14) and the proof of Lemma 3.2).
 (iii)
Thirdly, to ensure that the behaviour of the jump chain \(J\) suitably wellapproximates the behaviour of the original biased random walk \(X\), we are required to provide an estimate for the time \(X\) spends in loops of the graph (see Lemma 3.3). Note also that, it is in order to have the flexibility to accommodate the time in loops into the proof of the main localisation result that we prove in Lemma 3.2 a slightly sharper estimate for \(J\) than is necessary in the onedimensional case.
Lemma 3.1
Proof

\(b(n)=b_\delta (n)\),

any refinement \((a,b,c)\) of \((a_\delta (n),b_\delta (n),c_\delta (n))\) with \(b \ne b(n)\) has depth \(<(1\delta )\log n\),

\(\min _{m\in [a_\delta (n),c_\delta (n)]\backslash [b(n)\delta (\log n)^2,b(n)+\delta (\log n)^2]}(R_mR_{b(n)})>\delta ^3\log n\),

\(a_\delta (n)+c_\delta (n)\le K(\log n)^2\),

\(\sup _{m\le K(\log n)^2+1}\left[\log (T_{m+1}T_m)+\log \beta \sup _{T_m\le k\le T_{m+1}}\leftC_m^{(1)}S_k^{(1)}\right\right] \le {\delta ^4}\log n\).
The following lemma outlines some first properties of the jump process \(J\) defined at (8).
Lemma 3.2
Proof
We now provide an upper estimate for the growth of hitting times.
Lemma 3.3
Proof
Proof of Theorem 1.1
As in the proof of [17, Theorem 2.5.3], the proof strategy will be to show that \(X\) hits \(C_{b(n)}\) before time \(n\) and then stays there for a sufficient amount of time. For the majority of the proof, we will assume that \(A(n,K,\delta )\) holds, with \(\delta \) small and \(n\ge n_0(K,\delta )\).
To complete the article, we derive the aging result of Corollary 1.3.
Proof of Corollary 1.3
Notes
Acknowledgments
The author would like to thank two anonymous referees for suggesting some improvements to the presentation of this article, one of whom also encouraged the inclusion of the aging result that appears as Corollary 1.3.
Open Access
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