Biased random walk on critical Galton–Watson trees conditioned to survive
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DOI: 10.1007/s004400120462z
Abstract
We consider the biased random walk on a critical Galton–Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain onedimensional trapping models with slowlyvarying tails. Indeed, in each of these two settings, we establish closelyrelated functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.
Mathematics Subject Classification
Primary 60K37 Secondary 60F17 60G70 60J801 Introduction
Biased random walks in inhomogeneous environments are a natural setting to witness trapping phenomena. In the case of supercritical Galton–Watson trees with leaves (see [6, 22, 28]) or the supercritical percolation cluster on \(\mathbb Z ^d\) (see [17]), for example, it has been observed that deadends found in the environment can, for suitably strong biases, create a subballistic regime that is characteristic of trapping. More specifically, for both of these models, the distribution of the time spent in individual traps has polynomial tail decay, and this places them in the same universality class as the onedimensional heavytailed trapping models considered in [33]. Indeed, although the case of a deterministically biased random walk on a Galton–Watson tree with leaves is slightly complicated by a certain lattice effect, which means it can not be rescaled properly [6], in the case of randomly biased random walks on such structures, it was shown in [22] (see also the related article [8]) that precisely the same limiting behaviour as the onedimensional models of [15] and [33] occurs. Moreover, there is evidence presented in [17] that suggests the biased random walk on a supercritical percolation cluster also has the same limiting behaviour. The universality class that connects these models was previously investigated in [3, 4] and [5], and is characterised by limiting stable subordinators and aging properties.
The aim of this paper is to investigate biased random walks on critical structures. To this end, we choose to study the biased random walk on a critical Galton–Watson tree conditioned to survive. With the underlying environment having radically different properties from its supercritical counterpart, we would expect different limiting behaviour, with more extreme trapping phenomena, to arise. It is further natural to believe that some of the properties of the biased random walk on the incipient infinite cluster for critical percolation on \(\mathbb Z ^d\), at least in high dimensions, would be similar to the ones proved in our context, as is observed to be the case for the unbiased random walk (compare, for instance, the results of [1] and [25]). Nevertheless, our current understanding of the geometry of this object is not sufficient to extend our results easily, and so we do not pursue this inquiry here. In particular, we anticipate that, as indicated by physicists in [2], for percolation close to criticality there is likely to be an additional trapping mechanism that occurs due to spatial considerations, which means that, even without taking the effect of deadends into account, it is more likely for the biased random walk to be found in certain regions of individual paths than others (see [9] for a preliminary study in this direction).
Our main model—the biased random walk on critical Galton–Watson trees conditioned to survive—is presented in the next section, along with a summary of the results we are able to prove for it. This is followed in Sect. 1.2 with an introduction to a onedimensional trapping model in which the trapping time distributions have slowlyvarying tails. This latter model, which is of interest in its own right, is of particular relevance for us, as it allows us to comprehensively characterise the universality class into which the Galton–Watson trees we consider fall. Furthermore, the arguments we apply for the onedimensional model provide a useful template for the more complicated tree framework.
1.1 Biased random walk on critical Galton–Watson trees
Theorem 1.1
It is interesting to observe that this result is extremely explicit compared to its supercritical counterparts. Indeed, notwithstanding the fact the latticeeffect that was the source of somewhat complicated behaviour in [6] does not occur in the critical setting, the above scaling limit clearly describes the \(\beta \)dependence of the relevant slowdown effect. Note that, unlike in the supercritical case where there is a ballistic phase, this slowdown effect occurs for any nontrivial bias parameter, i.e. for any \(\beta >1\). Furthermore, we remark that the dependence on \(\alpha \) is natural: as \(\alpha \) decreases and the leaves get thicker (in the sense that tree’s Hausdorff dimension of \(\alpha /(\alpha 1)\) increases, see [12, 21]), the biased random walk moves more slowly away from its start point.
As suggested by comparing Theorem 1.1 with (1.3), the critical Galton–Watson tree case is closely linked with a sum of independent and identicallydistributed random variables where \(\bar{F}\) is asymptotically equivalent to \(\ln \beta /(\alpha 1)\ln x\). Although the logarithmic rate of decay is relatively easy to guess, finding the correct constant is slightly subtle, particularly for \(\alpha \ne 2\). This is because, unlike in the supercritical case and the critical case with \(\alpha =2\), when \(\alpha \ne 2\) it can happen that there are multiple deep traps emanating from a single backbone vertex. As a result, we have to take special care which of these have actually been visited when determining the time spent there, meaning that the random variable which actually has the \(\ln \beta /(\alpha 1)\ln x\) tail behaviour is not environment measurable (see Lemma 3.11). To highlight the importance of this consideration, which is also relevant albeit in a simpler way for \(\alpha =2\), in Theorem 3.14 we show that the constant that appears differs by a factor \(\alpha \) when \(\Delta _n\) is replaced by its quenched mean \(E^\mathcal{T ^*}_\rho \Delta _n\).
Corollary 1.2
Remark 1.3
Since the height of the leaves in which the random walk can be found at time \(e^{n}\) (see the localisation result of Lemma 4.5) will typically be of order \(n\), some further argument will be necessary to deduce a limit result for the graph distance \(d_\mathcal{T ^*}(\rho ,X_n)\) itself.
Another characteristic property that we are able to show is that the random walk also exhibits extremal aging.
Theorem 1.4
Although regular aging has previously been observed for random walks in random environments in the subballistic regime on \(\mathbb Z \) (see [14]), as far as we know, this is the first example of a random walk in random environment where extremal aging has been proved. As already hinted at, this kind of behaviour, as well as that demonstrated in Theorem 1.1 and Corollary 1.2, places the biased random walk on a critical Galton–Watson tree conditioned to survive in a different universality class to that of the supercritical structures discussed previously. In the class of critical Galton–Watson trees we have instead the spin glass models considered in [7] and [20], and the trap models with slowlyvarying tails we introduce in the next section.
1.2 Onedimensional directed trap model with slowlyvarying tails
Theorem 1.5
Similarly to [23, Remark 2.4], we note that the proof of the above result may be significantly simplified in the case when \(\bar{F}\) decays logarithmically. The reason for this is that, in the logarithmic case, the hitting time \(\Delta _{n}\) is very wellapproximated by the maximum holding time within the first \(n\) vertices, and so the functional scaling limit for \((\Delta _n)_{n\ge 0}\) can be readily obtained from a simple study of the maximum holding time process. For general slowly varying functions, the same approximation does not provide tight enough control on \(\Delta _{n}\) to apply this argument, and so a more sophisticated approach is required.
As a simple corollary of Theorem 1.5, it is also possible to obtain a scaling result for the process \(X\) itself. The definition of \(m^{1}\) should be recalled from (1.9). We similarly define the rightcontinuous inverse \(\bar{F}^{1}\) of \(\bar{F}\), only with \(>\) replaced by \(<\).
Corollary 1.6
Remark 1.7
(ii) In a number of places in the proofs of Theorem 1.5 and Corollary 1.6, we are slightly cavalier about assuming that \(\bar{F}(\bar{F}^{1}(x))=x\) for \(x\in (0,1)\). This is, of course, only true in general when \(\bar{F}\) is continuous. In the case when this condition is not satisfied, however, we can easily overcome the difficulties that arise by replacing \(\bar{F}\) with any nonincreasing continuous function \(\bar{G}\) that satisfies \(\bar{G}(0)=1\) and \(\bar{G}(u)\sim \bar{F}(u)\) as \(u\rightarrow \infty \). For example, one could define such a \(\bar{G}\) by setting \(\bar{G}(u):=(\frac{1}{u}\int _0^u L(v)dv)^{1}\).
The extremal aging result we are able to prove in this setting is as follows.
Theorem 1.8
Remark 1.9
Note that if \(\bar{F}\) (and the functions \(\bar{F}_n\) introduced below at (2.6)) are not continuous and eventually strictly decreasing, a minor modification to the proof of the above result (cf. Remark 1.7(ii)) is needed.
1.3 Article outline and notes
The remainder of the article is organised as follows. In Sect. 2, we study the onedimensional trap model introduced in Sect. 1.2 above, proving Theorem 1.5 and Corollary 1.6. In Sect. 3, we then adapt the relevant techniques to derive Theorem 1.1 and Corollary 1.2 for the Galton–Watson tree model. The arguments of both these sections depend on the extension of the limit at (1.3) that is proved in Sect. 5. Before this, in Sect. 4, we derive the extremal aging results of Theorems 1.4 and 1.8. Finally, as noted earlier, the appendix recalls some basic facts concerning Skorohod space.
We finish the introduction with some notes about the conventions used in this article. Firstly, there are two widely used versions of the geometric distribution with a given parameter, one with support \(0,1,2,\dots \) and one with support \(1,2,3,\dots \). In the course of this work, we will use both, and hope that, even without explanation, it is clear from the context which version applies when. Secondly, there are many instances when for brevity we use a continuous variable where a discrete argument is required, in such places \(x\), say, should be read as \(\lfloor x\rfloor \). Finally, we recall that \(f\sim g\) will mean \(f(x)/g(x)\rightarrow 1\) as \(x\rightarrow \infty \).
2 Directed trap model with slowlyvarying tails
Lemma 2.1
Fix \(T\in (0,\infty )\). As \(n\rightarrow \infty \), the \(\mathbf{P}\)probability of the events \(\mathcal E _1(n,T)\) and \(\mathcal E _2(n)\) converge to one.
Proof
Lemma 2.2
Fix \(T\in (0,\infty )\). As \(n\rightarrow \infty \), the \(\mathbb P _0\)probability of the event \(\mathcal E _3(n,T)\) converges to one.
Lemma 2.3
Fix \(T\in (0,\infty )\). As \(n\rightarrow \infty \), the \(\mathbb P _0\)probability of the event \(\mathcal E _4(n,T)\) converges to one.
Proof
Lemma 2.4
Proof
Lemma 2.5
Proof
We are now in a position to prove Theorem 1.5 by showing that the rescaled sums considered in the previous lemma suitably well approximate the sequence \((\Delta _n)_{n\ge 1}\).
Proof of Theorem 1.5
From this, the proof of Corollary 1.6 is relatively straightforward.
Proof of Corollary 1.6
3 Biased random walk on critical Galton–Watson trees
In this section, we explain how techniques similar to those of the previous section can be used to deduce the corresponding asymptotics for a biased random walk on a critical Galton–Watson tree conditioned to survive. Prior to proving our main results (Theorem 1.1 and Corollary 1.2), however, we proceed in the next two subsections to derive certain properties regarding the structure of the tree \(\mathcal T ^*\) and deduce some preliminary simple random walk estimates, respectively. These results establish information in the present setting that is broadly analogous to that contained in Lemmas 2.1–2.4 for the directed trap model.
3.1 Structure of the infinite tree
With this picture, it is clear how we can view \(\mathcal T ^*\) as an essentially onedimensional trap model with the backbone playing the role of \(\mathbb Z \) in the previous section. Rather than having an exponential holding time at each vertex \(\rho _i\), however, we have a random variable representing the time it takes \(X\) to leave the tree \(\mathcal T _{i}:=\{\rho _i\}\cup (\cup _{j=1,\dots ,\tilde{Z}_i1}\mathcal T _{ij})\) starting from \(\rho _i\). As will be made precise later, key to determining whether this time is likely to be large or not are the heights of the leaves connected to \(\rho _i\). For this reason, the rest of this section will be taken up with an investigation into the big, or perhaps more accurately tall, leaves of \(\mathcal T ^*\).
Lemma 3.1
Proof
It will be important for our future arguments that the sites from which big leaves emanate are not too close together, and that there are no big traps close to \(\rho \). The final lemma of this section demonstrates that the sequence of critical heights we have chosen achieves this.
Lemma 3.2
Proof
This is essentially the same as Lemma 2.1. \(\square \)
3.2 Initial random walk estimates
This section collects together some preliminary results for the biased random walk \((X_m)_{m\ge 0}\) on \(\mathcal T ^*\), regarding in particular: the amount of backtracking performed by the embedded biased random walk on the backbone; the amount of time \(X\) spends in small leaves; the amount of time \(X\) spends close to the base of big leaves; and tail estimates for the amount of time \(X\) spends deep within big leaves.
Lemma 3.3
Lemma 3.4
Proof

let the first generation size of \(\tilde{\mathcal{T }}_{n+1}\) be \(\zeta _{n+1}\),

let \(\tilde{\mathcal{T }}_n\) be the subtree founded by the \(\xi _{n+1}\)th first generation particle of \(\tilde{\mathcal{T }}_{n+1}\),

attach independent Galton–Watson trees conditioned on having height strictly less than \(n\) to the \(\xi _{n+1}1\) siblings to the left of the distinguished first generation particle,

attach independent Galton–Watson trees conditioned on height strictly less that \(n+1\) to the \(\zeta _{n+1}\xi _{n+1}\) siblings to the right of the distinguished first generation particle.
Lemma 3.5
Proof
Lemma 3.6
Proof
\(\square \)
Lemma 3.7
Proof
The lemma readily follows from the symmetry of the situation, which implies that, starting from \(\rho _i\), the biased random walk \(X\) is equally likely to visit any one of \(x_{ij},\,j\in B_i\) and \(z_i\) first. \(\square \)
Although the above lemma might seem simple, it allows us to deduce the distributional tail behaviour of the greatest height of a big leaf at a particular backbone vertex visited by the biased random walk \(X\). Note that we continue to use the notation \(q_n=\mathbf{P}(Z_n>0)\).
Lemma 3.8
Proof
Lemma 3.9
Proof
We can also prove a lower bound for the distributional tail of \(t_i\) that matches the upper bound proved above. Similarly to a proof strategy followed in [6], a key step in doing this is obtaining a concentration result to show that the time spent in a leaf visited deeply by the process \(X\) will be on the same scale as its expectation.
Lemma 3.10
Proof
Lemma 3.11
Proof
3.3 Proof of main result for critical Galton–Watson trees
The purpose of this section is to complete the proof of our main results for biased random walks on critical Galton–Watson trees (Theorem 1.1 and Corollary 1.2).
Proof of Theorem 1.1
We start the proof by claiming that the conclusion of the lemma holds when the hitting time sequence \((\Delta _{n})_{n\ge 0}\) is replaced by \((\sum _{i=0}^{n1}\tilde{t}_i)_{n\ge 0}\). By imitating the proof of Lemma 2.5 with Lemma 3.2 in place of Lemma 2.1, to verify that this is indeed the case, it will be enough to prove the same result for \((\sum _{i=0}^{n1}\tilde{t}_i^{\prime })_{n\ge 0}\), where \((\tilde{t}_i^{\prime })_{i\ge 0}\) in an independent sequence such that \(\tilde{t}_i^{\prime }\sim \tilde{t}_{1+(\ln n)^{1+\gamma }}\) for each \(i\). (Note that, because the elements of the sequence \((\tilde{t}_i)_{i\ge 0}\) are only identicallydistributed for \(i\ge 1+(\ln n)^{1+\gamma }\), we do not take \(\tilde{t}_i^{\prime }\sim \tilde{t}_i\) for each \(i\). By applying the second part of Lemma 3.2, which shows that with high probability there will be no big leaves in the interval close to \(\rho \), it is easy to adapt the argument of Lemma 2.5 to overcome this issue.) Since the tail asymptotics of Lemma 3.11 mean that the relevant functional scaling limit for \((\sum _{i=0}^{n1}\tilde{t}^{\prime }_i)_{n\ge 0}\) is an immediate application of Theorem 5.1 (with \(h_1(n)=\ln n\) and \(h_2(n)=n^{1}\)), our claim holds as desired.
Proof of Corollary 1.2
Since the proof is identical to that of Corollary 1.6, with \(\bar{F}(x)\) being taken to be a distribution function that is asymptotically equivalent to \(\ln \beta /(\alpha 1)\ln x\), we omit it. \(\square \)
3.4 Growth rate of quenched mean hitting times
The purpose of this section is to compare the growth rate of \(E^\mathcal{T ^*}_\rho \Delta _n\), that is, the quenched expectations of the hitting times \(\Delta _n\), with the growth rate of \(\Delta _n\) that was established in the previous section. Interestingly, in the result corresponding to Theorem 1.1 (see Theorem 3.14 below), an extra factor of \(\alpha \) appears, meaning that the sequence of quenched expectations grows more quickly than the hitting times themselves. This is primarily due to the fact that the quenched expectation \(E^\mathcal{T ^*}_\rho \Delta _n\) feels all the big leaves at a particular backbone vertex, whereas the hitting time \(\Delta _n\) only feels the big leaves that are deeply visited by \(X\). Indeed, the extra \(\alpha \) is most easily understood by comparing the following lemma, which describes the height of the biggest leaf at a particular backbone vertex, with Lemma 3.8, which concerns only deeply visited big leaves.
Lemma 3.12
Proof
Lemma 3.13
Proof
We are now ready to prove the main result of this section.
Theorem 3.14
Proof
Remark 3.15
For comparison, recall the directed trap model of Sect. 2, but, so as to avoid having to consider the time that the biased random walk \(X\) spends at negative integers, replace \(\mathbb Z \) by the halfline \(\mathbb Z _+\). As in Theorem 1.5, we have that \((n^{1}L(\Delta _{nt}))_{t\ge 0}\) converges in distribution under the annealed law \(\mathbb P _0\) to \((m(t))_{t\ge 0}\). For the corresponding quenched expectation, similarly to (3.23), we have that \(\sum _{i=0}^{n1}\tau _i\le E^\tau _0(\Delta _n)\le \frac{\beta +1}{\beta 1}\sum _{i=0}^{n1}\tau _i\). Thus, again applying [23, Theorem 2.1] (or Theorem 5.1 below), it is possible to check that \((n^{1}L(E^\tau _0(\Delta _{nt})))_{t\ge 0}\) converges in distribution under \(\mathbf{P}\) to \((m(t))_{t\ge 0}\). In particular, in contrast to the critical Galton–Watson tree case, the asymptotic behaviour of \(E^\tau _0(\Delta _n)\) and \(\Delta _n\) are identical. This is because, although certain big leaves will be avoided by certain realisations of the biased random walker in the tree setting, the geometry of the graph \(\mathbb Z _+\) forces \(X\), when travelling from \(0\) and \(n\), to visit all the traps in between on every realisation.
4 Extremal aging
In this section, we will prove Theorem 1.4 and Theorem 1.8, which state that the biased random walk on critical Galton–Watson tree conditioned to survive and the onedimensional trap model, respectively, experience extremal aging. The phenomenon we describe for these models is similar to what happens in the trapping models considered by Onur Gun in his PhD thesis [20] and to results observed for spin glasses in [7].
4.1 Extremal aging for the onedimensional trap model
We start by considering the onedimensional trap model introduced in Sect. 1.2, with the goal of this section being to prove Theorem 1.8. The reason for proving this result before its counterpart for trees is that the simpler argument it requires will be instructive when it comes to tackling the more challenging tree case in the subsequent section.
Lemma 4.1
We now establish the relevant localisation result for \(X\).
Lemma 4.2
Proof
Combining Lemmas 4.1 and 4.2, we readily obtain Theorem 1.8.
4.2 Extremal aging for the critical Galton–Watson tree model
Lemma 4.3
Proof
Before proceeding to prove the analogue of Lemma 4.2 in the tree setting—see Lemma 4.5 below, we prove a preliminary estimate which rules out the possibility that any leaves have heights that are close to any particular level on the appropriate scale.
Lemma 4.4
Proof
Lemma 4.5
Proof
Putting Lemmas 4.3 and 4.5 together, we obtain Theorem 1.4.
5 A limit theorem for sums of independent random variables with slowly varying tail probability
In this section, we derive the limit theorem for sums of independent random variables with slowly varying tail probability that was applied in the proofs of Lemma 2.5 and Theorem 1.1. The result we prove here is a generalisation of [23, Theorem 2.1].
Theorem 5.1
Remark 5.2
(i) Note that, similarly to Remark 1.9, if \(\bar{F}_n\) and \(\bar{F}\) are not continuous and eventually strictly decreasing, a minor modification to the proof of the above result (cf. Remark 1.7(ii)) is needed.
(ii) The same conclusion holds if on the lefthand side of (5.2) we replace \(L\) by \(L_n(x)=1/\bar{F}_n(x)\).
We are now ready to present the key lemma needed to establish Theorem 5.1. (This corresponds to [23, Lemmas 2.2 and 2.3].) In its statement, we use the notation \(\phi _n(x):=F_n^{1}(F_*(x))\), and we also define \(\phi (x):=F^{1}(F_*(x))\) for its proof.
Lemma 5.3
 (i)For every \(\lambda >0\) and \(T>0\), as \(n\rightarrow \infty \),$$\begin{aligned} \sup _{0\le x\le T}\left\frac{1}{n} L\left(\lambda \phi _n\left(n^2x^2\right)\right)x\right\rightarrow 0. \end{aligned}$$(5.4)
 (ii)For each \(\delta >0,\,T>\delta \), there exist random constants \(K_1,K_2>0\) and \(n_0\) such that, for every \(t\in [\delta ,T]\) and \(n\ge n_0\),almostsurely.$$\begin{aligned} K_1\phi _n\left(n^2m_n(t)^2\right)\le \sum _{i\le nt}\phi _n\left(n^2\eta _{n,i}\right)\le K_2\phi _n\left(n^2m_n(t)^2\right)\!, \end{aligned}$$(5.5)
Proof
Proof of Theorem 5.1
Acknowledgments
Part of this work was completed while D.C. was undertaking a three month JSPS Postdoctoral Fellowship at the Research Institute for Mathematical Sciences, Kyoto University, during which time he was most generously hosted by T.K. A.F. would like to thank Gérard Ben Arous for suggesting that these models would exhibit extremal aging.
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