Random walk on barely supercritical branching random walk
RRANDOM WALK ON BARELY SUPERCRITICALBRANCHING RANDOM WALK
REMCO VAN DER HOFSTAD, TIM HULSHOF, AND JAN NAGEL
Abstract.
Let T be a supercritical Galton-Watson tree with a bounded offspringdistribution that has mean µ >
1, conditioned to survive. Let ϕ T be a randomembedding of T into Z d according to a simple random walk step distribution. Let T p be percolation on T with parameter p , and let p c = µ − be the critical percolationparameter. We consider a random walk ( X n ) n ≥ on T p and investigate the behaviorof the embedded process ϕ T p ( X n ) as n → ∞ and simultaneously, T p becomes critical,that is, p = p n (cid:38) p c . We show that when we scale time by n/ ( p n − p c ) and spaceby (cid:112) ( p n − p c ) /n , the process ( ϕ T p ( X n )) n ≥ converges to a d -dimensional Brownianmotion. We argue that this scaling can be seen as an interpolation between the scalingof random walk on a static random tree and the anomalous scaling of processes incritical random environments. MSC 2010.
Primary: 60K37, 82C41. Secondary: 60F17, 60K40.
Key words and phrases.
Branching random walk, random walk indexed by a tree, percolation,scaling limit, supercriticality. Introduction
Percolation has long stood as a simple and tractable model for random media inphysics. Starting with the infinite integer lattice Z d with nearest-neighbor edges, eachedge is kept independently with probability p ∈ (0 , phase transition : there exists a p c ( Z d ) ∈ (0 ,
1) such that when p > p c the model has aninfinite connected component or cluster, while it does not when p < p c . The behaviorfor p close to and at the critical point p c , moreover, has many remarkable features.Random walk on percolation clusters of the lattice Z d at or near the critical pointhas been a central model in both physics and modern probability ever since de Gennesproposed it more than forty years ago [dG76] (see [BAF16] for an overview). Computersimulations and non-rigorous studies suggest that the model, which de Gennes dubbed“the ant in the labyrinth”, has many intriguing features, such as the observation thatrandom walk on large critical clusters exhibits anomalous diffusion. This fact hassince been rigorously verified in two dimensions [Kes86] and in high dimensions [KN09].Yet many questions still remain unanswered. For instance, almost all rigorous resultswe have are either for the two-dimensional lattice or when the dimension is “high Date : March 14, 2019. a r X i v : . [ m a t h . P R ] M a r REMCO VAN DER HOFSTAD, TIM HULSHOF, AND JAN NAGEL enough” (depending on the problem this is typically above either six or ten dimensions[HS90, FvdH17]). Dimensions three to six are terra incognita. And even in highdimensions, where the picture is perhaps most complete, there are still many big openproblems. For instance, it is currently unknown what the scaling limit of random walkon large critical clusters is (although there are good conjectures [HS00, Sla02], on whichmuch progress has been made recently [BACF16a, BACF16b]).One subject that is now, after a long line of research, rather well understood forpercolation in any dimension, is random walk on the infinite supercritical percolationcluster, i.e., when p > p c . When the temporal and spatial scaling factors are n and 1 / √ n , respectively, it has a Brownian scaling limit, just as for random walk on Z d [KV86, DMFGW85, DMFGW89, SS04, BB07, MP07]. That is, writing ( ˜ X n ) n ≥ fora random walk on an infinite percolation cluster C ∞ at supercriticality, one obtains ascaling limit of the form (cid:0) n − / ˜ X (cid:98) tn (cid:99) (cid:1) t ≥ −−−→ n →∞ (˜ σ ( p ) B t ) t ≥ , (1.1)where ( B t ) t ≥ is a standard d -dimensional Brownian motion. The diffusion constant˜ σ ( p ), which describes the typical fluctuations of this Brownian motion, depends on p (and the dimension d ). Remarkably, it is at this time not even known whether ˜ σ ( p ) ismonotonically increasing for p > p c . There is, however, clear evidence that the abovescaling limit cannot hold for (an infinite version of) critical clusters, because there it isknown that ( n − / | ˜ X n | ) n ≥ is a tight sequence of random variables [HvdHH14]. Thissuggests that either ˜ σ ( p ) → p (cid:38) p c , or that there exists a discontinuity for ˜ σ ( p ).Following heuristic arguments (see [Bis11], Problem 4.21, or [HvdH17], Chapter 14), itis conjectured that ˜ σ ( p ) ∼ ( p − p c ) for p slightly above p c in high dimensions.In the current paper we investigate this barely supercritical regime for a different modelthat is believed to be in the same universality class as the high-dimensional setting,namely for random walk on a branching random walk (defined below). The barelysupercritical regime here refers to the case where the underlying branching processbecomes critical, which we can also parametrize in a natural way as p approaching p c . In Theorem 1.2, we show that this model indeed has σ ( p ) ∼ ( p − p c ), the scalingconjectured for the high-dimensional percolation case. Furthermore, the underlying treestructure allows for a finer analysis of the process, and as the main result in Theorem1.1, we establish a scaling limit of the random walk after a large number of steps when p (cid:38) p c and the underlying branching process thus becomes more and more critical.This requires a bounds that have a high level of uniformity in p , which is one of themain challenges in this paper. We show that the Brownian scaling remains observablein the limit where we apply a temporal and spatial scaling with factors n/ ( p − p c ) and (cid:112) ( p − p c ) /n , respectively. In a way, this scaling interpolates between the diffusiverescaling by n − / of the strictly supercritical case, and the subdiffusive rescaling by n − / of the critical case, by letting p n tend to p c slower or faster, respectively. To ourknowledge this is the first result of such a scaling limit on a sequence of increasinglycritical graphs, interpolating between the supercritical and the critical regime. ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 3
Random walk on a randomly embedded random tree.
Before we proceedwith the main results, let us define the model.
Percolation on Galton-Watson trees.
Let T be a Galton-Watson tree rooted at (cid:37) with law P , and let ξ be the random number of offspring of the root. Suppose thatthe tree is supercritical, i.e., µ := E [ ξ ] >
1. We write ∆ for the maximal number ofchildren that a single individual in the (unpercolated) tree can have, i.e.,∆ := sup { n : P ( ξ = n ) > } . We consider percolation on the edges of the tree and let T p denote the connectedcomponent of the root (cid:37) in T , when each edge is deleted with probability 1 − p ∈ (0 , T . Then T p is again a Galton-Watson tree,whose distribution we denote by P p . The root of a tree T chosen according to P p nowhas ξ p offspring, such that, conditioned on { ξ = n } , ξ p is a binomial random variablewith parameters n and p .The percolated tree has mean number of offspring pµ and is supercritical if and onlyif p > p c := 1 /µ . In this setting, denote by ¯ P p = P p ( · | |T | = ∞ ) the distribution of thepercolated tree, conditioned on non-extinction. Given a tree T rooted at (cid:37) , let ( X n ) n ≥ be a simple random walk on T started at (cid:37) . Given v ∈ T , we write P vT for the law of( X n ) n ≥ with X = v , and write P T if v = (cid:37) . Given a realization of a Galton-Watsontree T , we call P T the quenched law of the random walk, and we call P p := ¯ P p × P T the annealed law of the random walk on the tree (writing P vp := ¯ P p × P v T , with theconvention that P vT = δ { ( X n ) n ≥ =( v ) n ≥ } if v / ∈ T ). Spatial embedding: branching random walk.
We embed a Galton-Watson tree T into Z d by means of a branching random walk, which we will define now: Let D denotea non-degenerate probability distribution on Z d . Given a tree T rooted at (cid:37) , set Z ( (cid:37) ) = 0and assign to each vertex v (cid:54) = (cid:37) of T an independent random variable Z ( v ) with law D . For any v ∈ T there exists a unique path (cid:37) = v , v , . . . , v m = v . The branchingrandom walk embedding ϕ = ϕ T is defined such that ϕ ( v ) := Z ( v ) + · · · + Z ( v m ). Ifwe write ( Y n ) n ≥ for a random walk on Z d with step distribution D started at 0, thentwo vertices v, w ∈ T with the unique path v = v , v , . . . , v k = w between them aremapped such that ( ϕ ( v ) , . . . , ϕ ( v k )) has the same law as the translation of ( Y , . . . , Y k )by ϕ ( v ), i.e., the marginal of the branching random walk embedding of any simple pathin T is a random walk path. This type of branching random walk is also sometimesreferred to in the literature as “random walk indexed by a tree”. Random walk on barely supercritical BRW.
The process that we consider inthis paper is that of ( ϕ T ( X i )) ≤ i ≤ a n under P p n as p n (cid:38) p c , i.e., the first a n steps of arandom walk on a barely supercritical infinite tree, embedded by a branching randomwalk ϕ into Z d . Note that this is a different process than if we were to consider a simple random walk on thesubgraph of Z d traced out by the branching random walk, as is for instance the topic of [BACF16a](for a different kind of trees): random walk on the trace is, in our setting, a vacuous complication,because T is supercritical, and thus grows at an exponential rate, while Z d has polynomial growth, sothat ϕ T ( T ) = Z d P p -almost surely. REMCO VAN DER HOFSTAD, TIM HULSHOF, AND JAN NAGEL
The main result of this paper is the following scaling limit. In its statement, weregard (cid:0) ϕ ( X (cid:98) tn (cid:99) ) (cid:1) t ≥ as a random element in the space D R d [0 , ∞ ), endowed with theSkorokhod topology (see [Bil99]) and the Borel σ -algebra. Theorem 1.1.
Consider random walk ( X m ) m ≥ a percolated Galton-Watson tree with µ > , ∆ < ∞ , and percolation parameter p n satisfying p n (cid:38) p c = 1 /µ , embeddedinto Z d with d ≥ by ϕ whose one-step distribution D satisfies (cid:80) x ∈ Z d xD ( x ) = 0 and (cid:80) x ∈ Z d e c (cid:107) x (cid:107) D ( x ) < ∞ for all c > . For any such sequence ( p n ) n ≥ satisfying e δ √ n ( p n − p c ) → ∞ for any δ > as n → ∞ , (cid:18)(cid:113) p n − p c n ϕ ( X (cid:98) tn ( p n − p c ) − (cid:99) ) (cid:19) t ≥ −−−→ n →∞ (cid:0) ( κ Σ) / B t (cid:1) t ≥ under P p n , with ( B t ) t ≥ a standard Brownian motion in R d , Σ the covariance matrix of D , and κ is an explicit constant given in Theorem 1.2 below. Note that the process in Theorem 1.1 is effectively a sequence pairs indexed by n :The first element of the pair is the random embedding of a tree percolated at p n suchthat, as n tends to infinity, it becomes more and more critical. The second elementof the pair is a time-rescaled random walk (cid:0) X (cid:98) tn ( p n − p c ) − (cid:99) ) (cid:1) t ≥ on this tree, that, as n tends to infinity, takes more steps per unit of time t . The increasing number of steps isneeded to see the Brownian motion in the limit, because the closer to criticality thetree is, the more steps it takes before the ballistic nature of the random walk becomesvisible. In Section 1.2 we give a heuristic explanation for why this is the correct order,and also discuss the implications of observing the random walk on different time scales.In Section 1.4 we explain the assumptions of the theorem.One of the main features of the convergence in Theorem 1.1 is that it, in some sense, interpolates between a random walk in a strictly supercritical and critical environment.Indeed, for p = p n > p c fixed, the scaling is the same as in (1.1) (and also in (1.2)below), while in the barely supercritical regime where p n (cid:38) p c , the additional factors( p n − p c ) bring the scaling close to that on a critical tree. We discuss the latter point inmore detail in Section 1.3.To understand the problem of random walk on a barely supercritical BRW, and togive a clearer context to the main results, let us first discuss the behavior of randomwalk on a strictly supercritical BRW. Random walk on supercritical BRW.
Consider for the moment random walkon BRW under P p , with p > p c fixed. Given v ∈ T , let | v | denote the distance of avertex v to the root. It is shown in [LPP95] that there exists an infinite sequence of regeneration times ( τ k ) k ≥ such that | X n | < | X τ k | for any n < τ k and | X n | > | X τ k | for any n > τ k , P p -almost surely. These regeneration times decompose the trajectoryof the random walk into independent increments with good moment bounds. Since (cid:0) ϕ ( X τ k +1 ) − ϕ ( X τ k ) (cid:1) k ≥ are then independent as well, and this increment is given by thedisplacement of the embedding random walk after τ k +1 − τ k steps, standard argumentsallow us to conclude the convergence (cid:0) n − / ϕ ( X (cid:98) tn (cid:99) ) (cid:1) t ≥ −−−→ n →∞ (Σ( p ) / B t ) t ≥ (1.2) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 5 in distribution under P p , with ( B t ) t ≥ a standard d -dimensional Brownian motion andΣ( p ) denoting a covariance matrix. (The proof of Theorem 1.1 in Section 5 below alsoestablishes (1.2).) Further writing Σ for the covariance matrix associated with D , i.e.,Σ ij := (cid:80) x x i x j D ( x ), and v T for the transpose of v ∈ Z d , we can determine that thecovariance matrix Σ( p ) is given byΣ( p ) = lim n →∞ n E p [ ϕ ( X n ) ϕ ( X n ) T ] = lim n →∞ n E p [ | X n | ] Σ , (1.3)where the last equality follows by conditioning on the distance of X n from the root, sothat ϕ ( X n ) can be written as a sum of i.i.d. increments with covariance Σ.It is a well-known result [LPP95] that the effective speed v ( p ) := lim n →∞ | X n | n (1.4)exists P p -almost surely and is non-zero when p > p c . This implies that the covariancematrix in the supercritical setting is given byΣ( p ) = v ( p )Σ . (1.5)We know that v ( p ) > p > p c . It is not known what happens to v ( p )when p (cid:38) p c . The following result establishes that v ( p ) →
0, and at what rate it doesso.For k ∈ N , define the k -th factorial moment of the (unpercolated) offspring distributionas m k := E (cid:34) k − (cid:89) i =0 ( ξ − i ) (cid:35) . Theorem 1.2 (The speed of random walk on a barely supercritical tree) . Consider asimple random walk ( X n ) n ≥ with X = (cid:37) on a Galton-Watson tree with m > and m < ∞ . Then, lim p (cid:38) p c v ( p )( p − p c ) = m m =: κ. We prove this theorem in Section 11 by analyzing the Taylor expansion of theextinction probability in terms of the generating function. Note that (1.5) and Theo-rem 1.2 imply that the variance of the limit in Theorem 1.1 is given by the limit of thesupercritical variance, appropriately rescaled. That is,lim p (cid:38) p c Σ( p )( p − p c ) = κ Σ , (1.6)where the scaling in ( p − p c ) is thus same as the conjectured scaling for random walkon a high-dimensional barely supercritical infinite percolation cluster mentioned above. REMCO VAN DER HOFSTAD, TIM HULSHOF, AND JAN NAGEL
On the scaling factors in Theorem 1.1.
Heuristically, the fact that rescalingspace by (cid:112) ( p − p c ) /n and time by n/ ( p − p c ) yields a Brownian limit can be understoodas follows: The duality principle tells us that any infinite supercritical GW-tree can bedecomposed into a supercritical GW-tree with no leaves (which we call the “backbonetree”) to which, at each vertex, are attached a random number of i.i.d. subcriticalGW-trees (which we call “traps”), see Section 2 below for details.The furcations or branch points in the backbone tree are then separated by path-likesegments with lengths of order ( p − p c ) − . Simple random walk on a GW-tree with noleaves is transient because each time the walk reaches a furcation, it is more likely totake a step away from the root than towards it. If we look at the sequence of pairs ofan n step walk ( X k ) ≤ k ≤ n on the tree T p n , then if the walk sees a growing number offurcations as n tends to infinity, a limit of the form (1.2) can be expected. Together, therate at which p n tends to p c , and the length of the long line segments in the backbonetree, thus imply that we need to rescale the tree by ( p n − p c ) /n for the walk to seea linearly growing number of furcations. Since the embedding of the tree into Z d isdiffusive, this means that we need to rescale space by (cid:112) ( p n − p c ) /n .If the walk were restricted to the backbone tree, then it would take the walk order( p − p c ) − steps to visit the next furcation, because that is how long it takes for arandom walk to travel distance O (( p − p c ) − ) on a line. Between two furcations, we canexpect a tight number of large traps of size O (( p − p c ) − ), each visited O (( p − p c ) − )times. An application of electrical network theory for Markov chains tells us that thetime to exit such a large trap is of order ( p − p c ) − . The time spent in traps betweenfurcations thus accumulates to O (( p − p c ) − ). Since the above spatial scaling impliesthat the number of furcations for the walk restricted to the backbone grows linearly in n , to see a scaling limit like (1.2), we need to rescale time by a factor n/ ( p − p c ) forthe same to be true about the walk on the full tree.We can expect rather different behaviour when the number of furcation points visitedby the random walk remains bounded, i.e., in the setting where we consider the walkat a time scale a n such that a n ( p n − p c ) remains bounded. In this case, the walk willbarely observe that the BRW is supercritical. In particular, when a n ( p n − p c ) →
0, therandom walk does not observe the supercritical nature at all, and basically observes a critical
BRW conditioned to survive forever. As discussed in more detail in the nextsection, we do not expect that the limit in Theorem 1.1 holds for these time scales.1.3.
Interpolation of supercritical and critical scaling.
As mentioned, we viewthe limit in Theorem 1.1 as interpolating between the supercritical and the criticalregime. This behavior becomes more visible when a nonstandard scaling is applied tothe random walk ( X n ) n ≥ : Writing a n := n ( p n − p c ) − , an equivalent representation ofthe convergence in Theorem 1.1 then reads (cid:0) ( p n − p c ) − a − / n ϕ ( X (cid:98) ta n (cid:99) ) (cid:1) t ≥ −−−→ n →∞ (cid:0) ( κ Σ) / B t (cid:1) t ≥ . (1.7)Consider a sequence ( p n ) n ≥ converging to the critical value p c very slowly. Theprefactor ( p n − p c ) then plays a negligible role compared to the prefactor a − / n . Indeed,completely omitting ( p n − p c ) − yields exactly the same diffusive scaling factors as those ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 7 of the random walk on the supercritical percolation cluster as in (1.1), or of the randomwalk on a supercritical BRW as in (1.2). In this setting, the random walk on the BRWbehaves “almost supercritical”.Now consider a sequence ( p n ) n ≥ converging very quickly to p c . This gives rise tobehavior of the random walk that is “almost critical”. To observe this, note that inthis setting the dominant factor in the time rescaling by a n is ( p n − p c ) − and theconvergence in Theorem 1.1 may be rewritten as (cid:0) n − / a − / n ϕ ( X (cid:98) ta n (cid:99) ) (cid:1) t ≥ −−−→ n →∞ (cid:0) ( κ Σ) / B t (cid:1) t ≥ . (1.8)This time, the prefactor n − / plays a negligible role compared to a / n when n → ∞ .The rescaling we see here, of time by a n and of space by a − / n , is characteristic forprocesses on high-dimensional critical random objects, as we discuss now.A key example of such critical processes is a random walk on a critical BRW con-ditioned to survive forever. There, it is known that a − / n ϕ ( X (cid:98) ta n (cid:99) ) is tight ( [Kes86]proves for random walk on the critical Galton-Watson tree conditioned to survive that a − / n | X (cid:98) ta n (cid:99) | is tight, and tightness of the associated BRW then easily follows from thediffusive nature of the random walk embedding ϕ ).For percolation a similar result is proved in [HvdHH14] (which builds upon [BJKS08,KN09]). We expect that the scaling limit of critical BRW conditioned to survive foreveris superBrownian motion (SBM) conditioned to survive forever (or equivalently, SBMwith an immortal particle, see [Eva93]). See [Hof06] for several examples of criticalmodels that converge to this superprocess, including the incipient infinite cluster oforiented percolation. For a random walk ( Y n ) n ≥ on such a critical high-dimensionalstructure, we further expect that the limit (cid:0) a − / n Y (cid:98) ta n (cid:99) (cid:1) t ≥ −−−→ n →∞ (cid:0) √ ¯ κB IIC¯ νt (cid:1) t ≥ (1.9)holds, where ( B IIC¯ νt ) t ≥ is a Brownian motion on the trace of SBM conditioned to surviveforever (mind that this combined process has not yet been defined rigorously, as far aswe are aware).A full scaling limit for random walk on a critical Galton-Watson tree conditioned tobe infinite was recently proved by Athreya, L¨ohr, and Winter [ALW17], with a rescalingthat, after embedding the tree, exactly corresponds to the one in (1.9). Also recently,Ben-Arous, Cabezas, and Fribergh [BACF16b] proved that the same scaling limit holdsin high dimensions (with different ¯ κ and ¯ ν ) for the model where the random walk moveson the trace of the critical BRW in Z d (rather than directly on the tree, as in thecurrent paper).Croydon [Cro09] showed that a random walk ( Y m ) m ≥ on a sequence of critical trees( T n ) n ≥ conditioned to have size n , and randomly embedded by ϕ (with Σ = Id), hasthe scaling limit (cid:0) b − / n ϕ ( Y (cid:98) tb n (cid:99) ) (cid:1) t ≥ −−−→ n →∞ (cid:0) √ ¯ κB ISE¯ νt (cid:1) t ≥ , (1.10)with b n = n / and where B ISE is a Brownian motion on the
Integrated superBrownianExcursion, or ISE. The ISE is now understood to be a canonical random object. It
REMCO VAN DER HOFSTAD, TIM HULSHOF, AND JAN NAGEL can be viewed as a Brownian embedding of a continuum random tree of a fixed size(cf. [Ald91]), or as the scaling limit of high-dimensional lattice trees, cf. [DS98]. Here,the time scale b n = n / is asymptotically the minimal time scale that ensures thatthe random walk explores a non-vanishing fraction of the tree, and thus notices that itmoves on a finite structure. We expect that if we set b n (cid:28) n / instead, using the samescaling as in (1.10) will yield a limit equal to that in (1.9). This is because the randomwalk only explores a small part of the embedded tree that is close to its starting point,and this local perspective is the same as the local perspective of a critical branchingprocess conditioned to survive forever as in (1.9).It would be interesting to consider the regime when a n ( p n − p c ) converges to aconstant. We expect that when a n → ∞ but a n ( p n − p c ) →
0, the limit is the same asin (1.9). In the case where a n ( p n − p c ) converges to a positive constant, it is not clearto us what the scaling limit should be (but we expect that it will not be equal to thatin (1.9)).Heuristically, the case where a n ( p n − p c ) converges can be viewed as the settingin which the random walk sees only a finite number of furcation points of the tree.Returning to the point of view of (1.7), where the time scale is set to be a n = n ( p n − p c ) − ,which diverges as n → ∞ for any p n → p c , we see that this choice can be understood asa time scale such that the random walk sees a growing number of furcation points. Thismeans that it can “feel” the drift that arises from the fact that the barely supercriticaltree is in fact supercritical. The factor n in a n does not seem necessary for that, andwe thus expect that any diverging a n leads to a Brownian limit. In particular, we donot see the need for a restriction on p n here. The reason we do have restrictions on ( p n )in the assumptions of Theorem 1.1 is different: we need to control the displacement intraps, which requires n = a n ( p n − p c ) to grow sufficiently fast compared to ( p n − p c ) − ,as formulated in the condition that e δ √ n ( p n − p c ) → ∞ for any δ > n → ∞ .1.4. About the assumptions in Theorem 1.1. (1) We only assume that there existsa finite maximal degree ∆ to simplify the proofs below. We believe that the proof canbe modified (albeit with some lengthy computations) to admit any offspring distributionwhose generating function f ( s ) = E [ s ξ ] has derivatives that satisfy f ( n +1) ( s ) ≤ Cnf ( n ) ( s )for all n ≥ ≤ s ≤ D that (cid:80) x ∈ Z d xD ( x ) = 0 is necessary, otherwise the BRWwould have a drift, which would require a different analysis and yield a different limit.(3) The assumption that (cid:80) x ∈ Z d e c (cid:107) x (cid:107) D ( x ) < ∞ prevents the BRW from making verylarge jumps. The strength of the assumption is for techical reasons. We believe that theresult should remain valid when (cid:80) x ∈ Z d (cid:107) x (cid:107) D ( x ) < ∞ . It is known that the behavior ofBRW (and other statistical mechanical models) alters dramatically when this restrictionis relaxed further, see e.g. [Kes95, JM05, Hul15]. In particular, macroscopically longedges in the BRW embedding are not sufficiently rare that the random walk can avoidthem w.h.p. When the random walk crosses such a long edge, it will likely only spend avery short time at the other end of the edge, which should not affect the random walk’sdiffusivity, but we expect that it will affect the continuity of the limiting process. Wethus conjecture that when D is such that α = sup { a : (cid:80) x ∈ Z d (cid:107) x (cid:107) a D ( x ) < ∞} satisfies ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 9 ≤ α <
4, we may still be able to see convergence to a Brownian motion in finite-dimensional distributions, but we would not be able to achieve such convergence in theSkorokhod topology. When α < α <
2, we expect to see convergenceto an α -stable motion, but again only for the finite-dimensional distributions.(4) We require that ( p n ) n ≥ satisfies that for any δ > δ √ n ( p n − p c ) → ∞ , sothat we may apply a result of Neuman and Zheng regarding the maximal displacementof subcritical BRW [NZ17], which we crucially use to estimate the size of the trapsafter embedding into Z d . We expect the result to remain valid when thus condition isrelaxed (see Section 1.3 below).1.5. About the proofs.
The proof of Theorem 1.1 is, at its core, close to the classicalproofs of Brownian limits for random walk in random environment. In particular, weconstruct a sequence of regeneration times to find (almost) independent increments,and follow the standard approach from there.What is new about our proof is that we require all the necessary moment bounds tohold uniformly for p ∈ ( p c , p d ) for some p d > p c . To obtain such uniform bounds weneed to perform a careful analysis of the structure of the backbone tree. In particular,we need to take into account that furcations occur only on a length scale of order( p − p c ) − and the influence of the traps grow as p tends to p c .Our regeneration structure is based on that of [SZ99], but is specifically designedto account for the length rescaling of the tree, due to furcation points growing farapart. Classical regeneration times would require the random walk to never visit aparent vertex, but this probability tends to zero as the tree becomes critical. Instead,inspired by the regenerations of [GMP12], we construct a sequence of regenerationtimes ( τ k ) k ≥ that allows the random walk to backtrack a distance of order ( p − p c ) − after a regeneration time, to increase the density of the regeneration times to the rightscale. This relaxation comes at the cost of (1) having a small and localized intersectionbetween the past and future of the walk at τ k , and (2) making ( τ k +1 − τ k ) k ≥ a stationaryand 1-dependent sequence, rather than an i.i.d. sequence (but these are not seriouscomplications). From this new construction of regeneration times we are then ableto derive all the necessary moment bounds on the regeneration times and distancesuniformly for p ∈ ( p c , p d ). We stress the fact that the distribution of ( τ k +1 − τ k ) k ≥ depends sensitively on p n , and thus it might be appropriate to write ( τ ( n ) k +1 − τ ( n ) k ) k ≥ instead. We omit it to simplify notation. Recalling from (1.7) that we investigate therandom walk on a time scale a n = n ( p n − p c ) is it worthwhile however to note that n can be interpreted as the order of the number of regenerations before time a n . The factthat n → ∞ allows us to use classical laws of large numbers for the sequence ( τ k ) k ≥ .Besides controlling the regeneration structure, we also need exponentially tight controlover the displacement of the random walk in the traps (Lemma 5.1), and a bound on allmoments of the size of the trace of the random walk on the backbone tree (Lemma 5.2).Uniformity in p is again an important requirement here, since we apply these boundsto p = p n (cid:38) p c . If, rather than looking at ϕ ( X (cid:98) a n t (cid:99) ), we would consider the positionof the walker only at regeneration times, i.e., ϕ ( X τ n ), then Theorem 1.1 would holds without any assumption on p n except for p n (cid:38) p c . This follows from the first threesteps of the proof in Section 5. The restriction e δ √ n ( p n − p c ) → ∞ is needed to controlthe maximal displacement between regeneration times.The proof of Theorem 1.2 in Section 11 is a straightforward Taylor expansion of agenerating function for the effective speed of simple random walk on a supercriticalGW-tree derived in [LPP95], applied to our setting. It is, however, interesting to notethat in two seemingly far removed parts of the proof we see a term involving the thirdmoment of ξ arise, but these terms cancel perfectly, leaving us with an expression ofthe asymptotic speed that only involves the first two moments. It is unclear to us whythe third moment should drop out like this.An alternative proof of the scaling as in Theorem 1.2 may be obtained as a byproductof the regeneration structure. Although this method does not give the precise limit, itis more robust and could be applied to more general models (see Remark 4.7).1.6. The structure of the paper.
We start by stating some preliminary lemmasabout the structure of slightly supercritical trees in Section 2. We conclude that thetypical length scale of the trees is ( p − p c ) − , and introduce a rescaling by this lengththat we use throughout the paper. In Section 3, we state some further lemmas withpreliminary bounds on escape probabilities of random walk on such trees. The proofsof these lemmas can be found in Section 6.In Section 4 we define a new regeneration structure, which decomposes the trajectoryinto one-dependent increments. We state moment bounds on the regeneration distancesin Lemma 4.5 (which we prove in Section 7), and moment bounds on times betweenregenerations in Lemma 4.6 (which we prove in Section 8).In Section 5 we give the main steps of the proof of Theorem 1.1. Utilizing thedecoupling effect of the regenerations, we first prove that the limit holds if we considerthe increments of the walk between regeneration times. For this we need the momentbounds on the regeneration distances of Lemma 4.5, and also bounds on the inter-regeneration times of Lemma 4.6. Because this process does not take into accountwhere the random walk is between regeneration times, and the regeneration times bydefinition never occur in the traps of the tree, the final step of the proof is to showthat the random walk is not able to walk great distances in traps. For this we use arecent result by Neuman and Zheng [NZ17] on the maximal displacement of subcriticalBRW, which yields a bound on the maximal displacement of the traps in Lemma 5.1.The proof of Lemma 5.1 can be found in Section 9. We also need bounds on the sizeand shape of the trace of the backbone of the BRW, in Lemma 5.2 (which we prove inSection 10).Finally, Section 11 contains the expansion of the speed on the tree and thus provesTheorem 1.2. 2. Preliminaries: the shape of the tree
In this section we establish some facts about the percolated trees, such as their growthrate. We also discuss a useful decomposition of the tree.
ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 11
To simplify notation we frequently drop the subscript p . We write c and C for genericconstants whose value may change from line to line. Central bounds appearing in thea-priori estimates in this section are denoted by a i and c i . The constants a i , c, C, and c i may depend on the original offspring distribution and on d , but we stress that they areindependent of p . Since we are interested in p close to p c , we will state many resultsonly for p ∈ ( p c , p d ) for some unspecified p d > p c .Given a tree T rooted at (cid:37) and v, w ∈ T , v (cid:54) = w , we say w is a descendant of v if theunique self-avoiding path from (cid:37) to w passes through v , and we call w an ancestor of v if the path from (cid:37) to v passes through w . We write T v for the subtree of T rootedat v induced by v and all descendants of v . We say a vertex v has an infinite line ofdescent if T v is an infinite tree. We can decompose any infinite tree T into its backbonetree T Bb , the tree induced by all vertices with an infinite line of descent, and the forest T \ T Bb , which consists only of finite trees. Here, we denote by the difference T \ T ofa tree T and a subgraph T of T the graph obtained by removing from T all edges of T and all then isolated vertices. This is equivalent to removing from T all verticesthat are only part of edges in T . Note that by this definition, T \ T v is the subtree of T with all descendants of v removed, but still containing v . Given v ∈ T Bb , we call theconnected component of v in T \ T Bb the trap at v and denote it by T trap v .Lyons [Lyo92, Proposition 4.10] gives an explicit description of this decompositionfor Galton-Watson trees using the “duality principle” (cf. [AN72, Chapter 12]). For f agenerating function of an offspring distribution of a Galton-Watson tree with extinctionprobability q ∈ (0 , f ( s ) := f ((1 − q ) s + q ) − q − q and f ∗ ( s ) := f ( qs ) q . (2.1)A supercritical Galton-Watson tree T with generating function f conditioned on survivalcan be generated by sampling a tree T Bb with generating function ˆ f , and then addingto every vertex v ∈ T Bb a random number U v of edges, and to the other ends of thoseedges independent Galton-Watson trees T ∗ i , i ∈ { , . . . , U v } with generating function f ∗ . While T Bb has no leaves, the trees generated by f ∗ go extinct with probability 1.Conditionally on T Bb , the ( U v ) v ∈T Bb are independent (but not identically distributed).The marginal distribution of U v can be characterized as follows: for v ∈ T Bb write δ v = deg T Bb ( v ) − f ( n ) ( s ) for the n -th derivative of f ( s ). Then, E [ s U v ] = f ( δ v ) ( qs ) f ( δ v ) ( q ) . (2.2)Then T trap v is the random subtree consisting of v , the U v edges attached to v , and thefinite trees generated by f ∗ attached to the edges. Lemma 2.1 (Properties of the generating functions) . Consider a Galton-Watson treewith generating function f and m > and m < ∞ . There exist a p d > p c andconstants c ∈ (0 , and c > such that for any p ∈ ( p c , p d ) , f (cid:48) p (0) ≥ c , µ p = ˆ µ p = f (cid:48) p (1) = p/p c , (2.3) and, for p (cid:38) p c , q p = 1 − c ( p − p c )(1 + o (1)) ,µ ∗ p = f (cid:48) p ( q p ) = 1 − m ( p − p c )(1 + o (1)) , (2.4)ˆ f (cid:48)(cid:48) p (0) = 2 m ( p − p c )(1 + o (1)) . The proof of this lemma is standard and can be found in Section 11.
Remark 2.2 (Topology of barely supercritical tree, backbone tree and traps as p n (cid:38) p c ) . From the above lemma we can infer that, heuristically speaking, the percolated GW-tree
T ∼ ¯ P p , conditioned on survival, and its decomposition look as follows for p close to p c :Both T and T Bb grow at rate f (cid:48) p (1) = p/p c , but only a fraction p − p c of the vertices in T is contained in T Bb . By the asymptotics above, P (deg T Bb ( ρ ) ≥
3) = 1 − ˆ f (cid:48) p (0) − ˆ f (cid:48)(cid:48) p (0) = o ( p − p c ) , so we are unlikely to see any vertices in T Bb with out-degree three or more up to aheight of order ( p − p c ) − . Up to this height, the tree looks like a binary tree whereeach edge has been replaced by a path whose length is distributed as an independentgeometric random variable with a parameter of order p − p c . The trap at any vertex v consists of a random number of i.i.d. subcritical Galton-Watson trees with meanoffspring distribution µ ∗ p ≈ − c ( p − p c ), whose expected size and depth are bothknown to be of order ( p − p c ) − . The distribution of the traps, however, is such that P ( |T trap v | ≥ A ( p − p c ) − ) ≈ p − p c A e − A/ , which implies that among O (( p − p c ) − ) typicaltraps, almost all traps will be very small (or non-existent), while a tight number ofthem are macroscopically large, having a size of the order of ( p − p c ) − and depth ofthe order ( p − p c ) − . Although we do not use any of the computations of this heuristicdirectly in the proofs that follow, we do use them often as guiding principles.For a tree T with backbone tree T Bb , let G (cid:48) m ( T ) = { v ∈ T Bb : | v | = m } be thebackbone-tree vertices in generation m ≥
0. As discussed above, the relevant spatialscaling factor of the random tree will be of order p − p c , so throughout this paper we willoften consider the backbone tree only at generations that are multiples of L/ ( p − p c ),for some L ≥ L -levels of T , andfor m ∈ N write G [ m ] := G [ m ] ( T ) := G (cid:48) m (cid:98) L/ ( p − p c ) (cid:99) ( T ) (2.5)for the m -th L -level, and write G [0] := { (cid:37) } , the root of the tree. We tacitly ignore thedependency on L (and p ) in the definition of G [ m ] , the reason being that we soon fix thevalue of L . Roughly speaking, the choice of L will be such that with good probability(uniform in p − p c ), the random walk starting from a vertex v ∈ G [ m ] never hits G [ m − .For m < n , let T Bb [ m,n ] := { v ∈ T Bb : m (cid:98) L/ ( p − p c ) (cid:99) ≤ | v | ≤ n (cid:98) L/ ( p − p c ) (cid:99)} (2.6)denote the forest segment of the backbone tree between the m -th and n -th L -level. Let T [ m,n ] := (cid:91) v ∈ T Bb [ m,n ] T trap v (2.7) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 13 denote the same forest segment with all the traps attached.The next lemma shows that on the scale of a single L -level, the backbone tree T Bb looks like a non-degenerate tree: Lemma 2.3 (The size of the first L -level) . For any
L > there exists p d > andconstants < a ≤ a < and a < ∞ , such that for all p ∈ ( p c , p d ) a ≤ ¯ P p ( | G [1] ( T ) | = 1) ≤ a and ¯ E p [ | G [1] ( T ) | ] ≤ a . Proof.
By (2.1) we know that the backbone tree T Bb is a Galton-Watson tree withgenerating function ˆ f ( s ), so the probability that a vertex in the backbone tree hasexactly one child in T Bb is ˆ p = f (cid:48) ( q p ). By Lemma 2.1, f (cid:48) ( q p ) = 1 − c ( p − p c )(1 + o (1))as p (cid:38) p c . Therefore,lim p (cid:38) p c ¯ P p ( | G [1] ( T ) | = 1) = lim p (cid:38) p c (1 − c ( p − p c )(1 + o (1))) L/ ( p − p c ) = e − c L . (2.8)This proves the first part of the lemma.For the second part, recall that T p has offspring mean µ p = µp and that p c = 1 /µ , sothat ¯ E p [ | G [1] ( T ) | ] = µ (cid:98) L/ ( p − p c ) (cid:99) p = (1 + µ ( p − p c )) (cid:98) L/ ( p − p c ) (cid:99) , which is uniformly bounded as p (cid:38) p c . (cid:3) Preliminaries: escape probabilities
In this section we establish three useful bounds on the probability that the randomwalk escapes to infinity before returning to the previous L -level. Recall that ( X n ) n ≥ denotes the random walk on T .For A a set of vertices of T , let η ( A ) = inf { n ≥ X n ∈ A } be the hitting time of the set A . For m ≥
0, we denote by η m := η ( G [ m ] ( T )) the hitting time of the m th L -level. A crucial step in defining the regeneration times is the following uniform boundfor the probability that the random walk backtracks (cid:98) L/ ( p − p c ) (cid:99) steps in tree distance: Lemma 3.1 (Annealed escape probability) . There exists an L ≥ depending onlyon the original offspring distribution, such that for any L ≥ L , for any tree T with G [1] ( T ) (cid:54) = ∅ and any v ∈ G [1] ( T ) , P vp ( η = ∞ | T [0 , = T [0 , ) ≥ . We prove this lemma in Section 6.1.In what follows, fix an L such that the estimate in Lemma 3.1 holds.It would simplify our argument substantially if we could get a quenched version ofLemma 3.1. Unfortunately, such a quenched version does not hold. We do, however,have the following statement, which shows that the backtracking probabilities are smallwith a high probability, even when we additionally delete all outgoing edges at thestarting point v ∈ G [1] ( T ), so that the walker has to escape to infinity via a differentvertex in G [1] ( T ): Lemma 3.2 (Quenched indirect escape probability) . There exists a function h inde-pendent of p with h ( α ) → as α → , such that ¯ E p (cid:88) v ∈ G [1] ( T ) { P v T \T v ( η = ∞ ) <α } ≤ a + h ( α ) , with a as in Lemma 2.3. We prove this lemma in Section 6.2.For a vertex v ∈ G [ m ] ( T ) with m ≥
1, let anc ( v ) ∈ G [ m − ( T ) denote the L -ancestor of v , its ancestor in L -level m −
1, and define the number of L -siblings of v as sib ( v ) := | G [1] ( T anc ( v ) ) | − . (3.1)Note that anc ( v ) and sib ( v ) are only defined for vertices on the backbone tree, and that sib ( v ) only counts the siblings on the backbone tree. The following bound shows thatwe have a uniformly bounded probability of hitting vertices that have no L -siblings: Lemma 3.3 (Escape probability on thin parts of the backbone tree) . There exists aconstant a > such that for any tree T with G [1] ( T ) (cid:54) = ∅ and any v ∈ G [1] ( T ) , P vp ( η < η , sib ( X η ) = 0 | T [0 , = T [0 , ) ≥ a . We prove this lemma in Section 6.3.4.
The regeneration structure
Regeneration times are a classical tool to decouple the increments of random walks inrandom environments. For random walks on GW-trees, they were utilized already in thepapers [LPP95, LPP96] or [PZ08]. In our definition we follow the formulation of [SZ99].There are two main changes from the classical regeneration time structure of the abovementioned papers: First, we have to allow the random walk to backtrack a distance oforder ( p − p c ) − , similar to the construction in [GMP12, GGN17]. Second, to obtain astationary sequence even with backtracking, we have to control the environment wherethe walker regenerates. For the first point, we can rely on Lemma 3.1 to see that wehave a good probability of not backtracking too far. For the second point, we wantto ensure that a regeneration point has no siblings, such that the random walk canbacktrack only on a branch where the backbone has no furcations.Intuitively, the regeneration times are constructed as follows: We wait until therandom walker for the first time reaches a vertex with the potential for regeneration,namely a vertex in an L -level with no siblings. We call this time S and the associated L -level M . If the previous L -level M − S thefirst regeneration time τ . But suppose that the walker does revisit L -level M − R . Then S is not a regeneration time. Instead, we denote by N the highest L -level that the walker visited between S and R on the backbone tree, and we waituntil the walker first reaches an L -level with no siblings with generation greater than N + 1 (the additional generation is required to guarantee that we see a new part ofthe tree, so that we can apply annealed estimates). We call this potential regenerationtime S and the corresponding l -level M . If M − ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 15 set τ = S . If it is revisited again, then we repeat the above procedure. Because thewalk on the tree is transient, the first regeneration time τ is finite almost surely. Werepeat the entire procedure to construct the sequence ( τ k ) k ≥ of regeneration times. SeeFigure 1 for a sketch of this construction.To define the above construction formally, we start with some more notation. Let( z n ) n ≥ be an infinite path, and let θ m be the time shift on the path such that ( θ m z ) n = z m + n and such that for any function f that takes an infinite path as its argument, f ◦ θ m denotes the same function applied to the time-shifted path. Define the backtrackingtime η (cid:48) = η (cid:48) (( X n ) n ≥ ) := inf { n ≥ | X n | = | X | − (cid:98) L/ ( p − p c ) (cid:99)} . We need η (cid:48) = ∞ toregenerate. The regeneration structure that we rely on is then defined as follows: Definition 4.1.
Given a tree T and a random walk ( X n ) n ≥ on T , we define a sequenceof stopping times S ≤ R ≤ S ≤ R ≤ . . . and distances M k , N k , beginning with M := min { m ≥ sib ( X η m ) = 0 } , S := η M ,R := S + η (cid:48) ◦ θ S , (4.1) N := max { m ≥ η m < R } , and recursively, for k ≥ , M k +1 := min { m ≥ N k + 2 : sib ( X η m ) = 0 } , S k +1 := η M k +1 ,R k +1 := S k +1 + η (cid:48) ◦ θ S k +1 , (4.2) N k +1 := max { m : η m < R k +1 } . These definitions make sense until R k = ∞ for some k and we set K := inf { k : R k = ∞} , τ := S K . We call τ the first regeneration time. By Lemma 3.1,
K < ∞ almost surely so that τ is well-defined. We set τ := 0 and for k ≥ we define the subsequent regenerationtimes as τ k := τ k − + τ ◦ θ τ k − . (4.3) Finally, denote by π k := η ( anc ( X τ k )) the times when the L -ancestors of the regenerationpoints are visited for the first time, and by Λ k the L -generation at which the k -thregeneration time occurs, i.e., Λ k := | X τ k | / ( (cid:98) L/ ( p − p c ) (cid:99) ) . Remark 4.2 (Decoupling property of the regenerations) . We can make the followingobservations:(a) Up to time π k , the random walk walks on the tree up to L -generation Λ k − τ k on, the walk never visits L -generation Λ k − X n ) n ≤ π k and ( X n ) n ≥ τ k visit disjoint parts of the tree.(b) The only part of the environment that is visited by both ( X n ) n ≤ τ k and ( X n ) n ≥ τ k is the tree segment consisting of the tree rooted at X π k with all descendants of X τ k removed. Since we require sib ( X τ k ) = 0, we know that the correspondingbackbone-segment T Bb X πk \ T Bb X τk is isomorphic to a line graph of length (cid:98) L/ ( p − p c ) (cid:99) − G [1] G [2] G [3] G [4] = G [Λ ] G [5] b ̺ b S b R b π b S = τ Figure 1. An idealized sketch of the sample path of the random walk (in red) ona portion of the backbone tree (in black). The labels correspond to hitting timesof vertices, L -levels are indicated by dashed lines. The hitting time η of the first L -level is not the first potential regeneration time S , since sib ( X η ) = 1 . Observethat η is the first potential regeneration time S because X S has no L -siblings,but that the random walk then backtracks more than one L -level. Finally, η = S satisfies the condition for a regeneration time and π = η . (c) The definition above allows for the random walk to move distance at most (cid:98) L/ ( p − p c ) (cid:99) − | X τ k | − | X τ k − | are independent of the traps, we may conclude thatthe inter-regeneration distances are independent. With the exception of k = 1,they are identically distributed as well. (This observation is formalized inLemma 4.3 below.)(d) The inter-regeneration times τ k − τ k − are not independent, since two subsequenttime intervals both depend on the traps in the tree segment between X π k − and X τ k . The inter-regeneration times are, however, stationary and 1-dependent.(This is formalized in Lemma 4.4 below.)To state the above observations generally and unambiguously, we introduce the σ -fields G k := σ (cid:0) T \ T X πk , ( X n ) ≤ n<π k , π k (cid:1) and the time shift ¯ θ m defined for a set B = (cid:8) T ∈ B , ( X n ) n ≥ ∈ B (cid:9) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 17 by B ◦ ¯ θ k = (cid:8) T X πk ∈ B , ( X n − X τ k ) n ≥ τ k ∈ B (cid:9) . We say that a sequence ( Y n ) n ≥ is m -dependent, if ( Y n ) n ≤ k and ( Y n ) n ≥ k + (cid:96) are independentwhenever (cid:96) > m . The following two lemmas are standard, so we omit their proofs (seee.g. [Guo16, Lemma 17 and Proposition 18]): Lemma 4.3 (Stationarity of the tree and walk) . For any measurable set B = (cid:8) T ∈ B , ( X n ) n ≥ ∈ B (cid:9) and k ≥ , P p ( B ◦ ¯ θ k | G k ) = P vp ( B | G [1] ( T ) = { v } , η = ∞ ) . Lemma 4.4 (Stationarity and 1-dependence of the regeneration times) . Under P p , thesequence (cid:0) T X πk \ T X τk +1 , ( X n − X τ k ) τ k ≤ n<τ k +1 , τ k +1 − τ k (cid:1) k ≥ is stationary and 1-dependent. Furthermore, the marginal distribution of this sequenceis given by P p (cid:0) T X πk \ T X τk +1 ∈ B , ( X n − X τ k ) τ k ≤ n<τ k +1 ∈ B , τ k +1 − τ k ∈ B (cid:1) = P vp (cid:0) T \ T X τ ∈ B , ( X n ) ≤ n<τ ∈ B , τ ∈ B | G [1] ( T ) = { v } , η = ∞ (cid:1) . The moment bounds for the regeneration distances and the regeneration times in thetwo following lemmas are crucial ingredients for the proof of the main result:
Lemma 4.5 (Moment bounds on regeneration distances) . For any a ≥ , there existsa finite constant C a such that ( p − p c ) a E p [ | X τ | a ] ≤ C a , ( p − p c ) a E p [( | X τ | − | X τ | ) a ] ≤ C a , for any p > p c . We prove this lemma in Section 7 below. Note that we also have the trivial lowerbound | X τ | ≥ c ( p − p c ) − . Lemma 4.6 (Moment bounds on regeneration times) . There exists a constant C suchthat ( p − p c ) E p (cid:2) ( τ ) (cid:3) ≤ C, ( p − p c ) E p (cid:2) ( τ − τ ) (cid:3) ≤ C, for any p > p c . Furthermore, there exists a constant c > , such that ( p − p c ) E p [( τ − τ )] ≥ c . We prove this lemma in Section 8 below.
Remark 4.7 (Robust bounds on the effective speed near criticality) . Given the momentbounds of Lemmas 4.5 and 4.6, we may apply the Law of Large Numbers to obtain anexpression for the effective speed in terms of regeneration times, as v ( p ) = lim n →∞ | X n | n = lim n →∞ | X τ n | τ n = E p [ | X τ | − | X τ | ] E p [ τ − τ ] . (4.4) In particular, the p -independence of the moment bounds implies0 < lim inf p n (cid:38) p c ( p n − p c ) − v ( p n ) ≤ lim sup p n (cid:38) p c ( p n − p c ) − v ( p n ) < ∞ . (4.5)This gives the order of the effective speed v ( p ) close to criticality. The result ofTheorem 1.2 is stronger and shows that lim inf and lim sup in (4.5) agree, but the proofof Theorem 1.2 in Section 11 below relies on an explicit formula for the effective speedof SRW on GW-trees obtained in [LPP95], while the above arguments are quite robustagainst changes to tree or the behavior of the random walk.5. The scaling limit: proof of Theorem 1.1
In this section we prove Theorem 1.1, subject to the proofs of Theorem 1.2, thelemmas in Sections 2, 3, and 4, and subject to the proofs of two further lemmas, Lemmas5.1 and 5.2, which are stated below as we need them.The proof of Theorem 1.1 will go in four steps. In the first two, we consider thescaling limit of a simple random walk whose jumps have the same size distribution asthe increments of the random walk on the BRW at regeneration times, but with jumpsat a fixed rate. We show that under a rescaling equivalent to the one of Theorem 1.1,this process converges to a Brownian motion with diffusion ( κ Σ) / as desired. Then,we will apply a time change to have the jumps occur at random times with the samedistribution as the regeneration times and show that the difference with the processof the first step vanishes in the limit. In the final step, we show that the trajectoriesof the random walk during excurions between regeneration times also vanishes in thelimit, thus yielding the scaling limit for the process that we are after.We assume in this section some familiarity with the theory of convergence for Markovprocesses. We refer the reader who is insufficiently knowledgable about this topicto [Bil99, EK86], where all convergence-related topics of this section that are notexplicitly cited are defined and discussed.To start, let us recall that in the setting of Theorem 1.1 we consider p = p n with p n (cid:38) p c as n → ∞ . Therefore, the distribution of the random walk changes with n .We will thus indicate this dependency on n clearly in this section. First step: i.i.d. increments between regenerations.
For the first step of theproof, we define the process W (1) t,n := 1 √ n (cid:98) nt (cid:99) (cid:88) k =1 W k as a random element of the Skorokhod space D R d [0 , T ]. Without loss of generality, wetake T ∈ N and then, rescaling n linearly and using the scale invariance of the limitingBrownian motion, may restrict the proof further to the case T = 1. The increments of W (1) t,n are given by W := ϕ ( X τ ) (cid:112) ( p n − p c ) E p n [ τ ] , and W k := ϕ ( X τ k ) − ϕ ( X τ k − ) (cid:112) ( p n − p c ) E p n [ τ − τ ] for k ≥ ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 19
By Definition 4.1 and Lemma 4.3, the W k are independent and W , W , . . . areidentically distributed. Conditioned on ( X n ) n ≥ , ϕ ( X τ k ) − ϕ ( X τ k − ) is the increment of( Y n ) n ≥ (the random walk on Z d with step distribution D ) after | X τ k | − | X τ k − | steps.Thus, by the assumption in Theorem 1.1 that (cid:80) x ∈ Z d xD ( x ) = 0, we get E [ W k ] = 0 andby Lemma 4.5 and Lemma 4.6 we get E p n (cid:2) (cid:107) W k (cid:107) (cid:3) = E p n (cid:2) | X τ k | − | X τ k − | (cid:3) E p n [ (cid:107) Y (cid:107) ]( p n − p c ) E p n [ τ − τ ] ≤ C (5.1)for k ≥ E p n (cid:2) (cid:107) W (cid:107) (cid:3) = E p n [ | X τ | ] E p n [ (cid:107) Y (cid:107) ]( p n − p c ) E p n [ τ ] ≤ C. (5.2)Writing v T for the transpose of v , we also have the convergencelim n →∞ n E p n (cid:32) n (cid:88) k =1 W k (cid:33) (cid:32) n (cid:88) k =1 W k (cid:33) T = lim n →∞ E p n [ | X τ n | ] n ( p n − p c ) E p n [ τ − τ ] E p n (cid:2) Y Y T (cid:3) = lim n →∞ E p n [ | X τ | − | X τ | ]( p n − p c ) E p n [ τ − τ ] E p n (cid:2) Y Y T (cid:3) = lim n →∞ v ( p n )( p n − p c ) E p n (cid:2) Y Y T (cid:3) (5.3)= κ Σ , where in the last step we have used Theorem 1.2 and the assumption in Theorem 1.1that (cid:80) x ∈ Z d (cid:107) x (cid:107) D ( x ) < ∞ , so that Σ exists. Because E [ W k ] = 0 and (5.1) and (5.3)hold, we may now apply the Invariance Principle for triangular arrays (see [Bil99], p.147, for a one-dimensional version, which applies thanks to the Cram´er-Wold technique),and conclude the convergence (cid:16) W (1) t,n (cid:17) t ∈ [0 ,
1] d −−−→ n →∞ (cid:0) ( κ Σ) / B t (cid:1) t ∈ [0 , , (5.4)under P p n , with ( B t ) t ≥ a standard Brownian motion on R d . This concludes the firststep. Second step: The correct (random) number of jumps.
The process W (1) n :=( W (1) t,n ) t ∈ [0 , is a piecewise constant function in t , jumping exactly n times in [0 , n steps. Given n ≥
0, let k n be the integer satisfying τ k n ≤ n < τ k n +1 , (5.5)where we recall that τ = 0. Set ν n := ( p n − p c ) − k n and define W (2) t,n := W (1) t,ν n . (5.6) This process jumps ν n times and we next show that this number of jumps is asymptoti-cally equal to a n := n ( p n − p c ) − E p n [ τ − τ ] − . (5.7)Note that by Lemma 4.6, a n → ∞ . Now, by definition of k n , a n ν n = nk n E p n [ τ − τ ] ≤ τ k n +1 k n E p n [ τ − τ ] = 1 k n E p n [ τ − τ ] k n +1 (cid:88) m =1 ( τ m − τ m − ) . (5.8)Using the 1-dependence of ( τ m − τ m − ) m and the moment bound in Lemma 4.6, Cheby-shev’s inequality shows that the variance of the right hand side of (5.8) converges tozero as n → ∞ . Thus, the right hand side of (5.8) converges to 1 in distribution under P p n . Bounding n ≥ τ k n , the same argument shows that a n /ν n ≥ P p n , a n ν n d −−−→ n →∞ . (5.9)We may then apply [Bil99, Theorem 14.4] to conclude that W (2) t,n has the same limit as W (1) t,n , that is, (5.4) with W (1) t,n replaced by W (2) t,n . This concludes the second step. Third step: The correct (random) jump times.
The process W (2) n := ( W (2) t,n ) t ∈ [0 , still jumps with equal intervals, at times t = ν n , ν n , . . . ,
1. The third step of the proofis to consider instead the process W (3) n := ( W (3) t,n ) t ∈ [0 , , which has the same incrementsas W (2) n , but jumps at times t = τ /τ ν n , τ /τ ν n , . . . ,
1. That is, W (3) t,n := W (2) k/ν n ,n = ϕ ( X τ k ) (cid:112) ν n ( p n − p c ) E p n [ τ − τ ] for τ k ≤ (cid:98) tτ ν n (cid:99) < τ k +1 , so W (3) n is a random time change of W (2) n . We can bound their distance in the Skorokhodmetric d S as d S ( W (3) n , W (2) n ) ≤ max k =1 ,...,ν n (cid:12)(cid:12)(cid:12)(cid:12) τ k τ ν n − kν n (cid:12)(cid:12)(cid:12)(cid:12) ≤ max k =1 ,...,ν n (cid:12)(cid:12)(cid:12)(cid:12) τ k − E p n [ τ k − τ ] τ ν n (cid:12)(cid:12)(cid:12)(cid:12) + max k =1 ,...,ν n (cid:12)(cid:12)(cid:12)(cid:12) E p n [ τ k − τ ] τ ν n − kν n (cid:12)(cid:12)(cid:12)(cid:12) . (5.10)By Lemma 4.4 we have E p n [ τ k − τ ] = k E p n [ τ − τ ], so we may write the second term asmax k =1 ,...,ν n (cid:12)(cid:12)(cid:12)(cid:12) E p n [ τ k − τ ] τ ν n − kν n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν n E p n [ τ − τ ] τ ν n − (cid:12)(cid:12)(cid:12)(cid:12) . Using again the moment bounds of Lemma 4.6, τ ν n ν n E p n [ τ − τ ] = τ ν n E p n [ τ − τ ] + 1 ν n ν n (cid:88) k =2 τ k − τ k − E p n [ τ − τ ] d −−−→ n →∞ , (5.11)in distribution under P p n . This shows that the second term in (5.10) vanishes as n → ∞ . ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 21
We write the first term in (5.10) asmax k =1 ,...,ν n (cid:12)(cid:12)(cid:12)(cid:12) τ ν n E p n [ τ − τ ] + ( τ k − τ ) − E p n [ τ k − τ ] ν n E p n [ τ − τ ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) τ ν n ν n E p n [ τ − τ ] (cid:12)(cid:12)(cid:12)(cid:12) − . The second factor converges in distribution to 1 by (5.11). That the first factor alsoconverges to 1 follows from Lemma 4.4 combined with an Invariance Principle fortriangular arrays of 1-dependent random variables due to Chen and Romano [CR99,Theorem 2.1]. This implies that (5.10) vanishes in probability and thus (5.4) also holdswith W (1) n replaced by W (3) n . This concludes the third step. Fourth step: Adding the path between regenerations.
The fourth and lastmain step of the proof is now to transfer the convergence to W (4) t,n := ϕ ( X (cid:98) tτ νn (cid:99) ) (cid:112) ν n ( p n − p c ) E p n [ τ − τ ] , i.e., to the full (rescaled) process of random walk on a BRW, where we now alsoincorporate the fluctuations of ( X n ) n ≥ between regeneration times. Note that bydefinition of a n and ν n and by (5.9), we have1 (cid:112) ν n ( p n − p c ) E p n [ τ − τ ] = (cid:115) a n ( p n − p c ) ν n n ≤ C (cid:114) p n − p c n , so bounding d S ( W (4) n , W (3) n ) ≤ C (cid:114) p n − p c n max k =1 ,...,ν n max τ k − ≤ i<τ k (cid:107) ϕ ( X i ) − ϕ ( X τ k − ) (cid:107) , (5.12)it suffices to show that the right-hand side vanishes in probability. Again by themoment bounds of Lemma 4.6, ν n ≤ Cn with probability converging to 1, so thatwe may replace the remaining ν n in (5.12) by Cn . We need a lemma that controlsthe maximal displacement of the embedded traps. As we show in Section 9, this is adirect consequence of a precise estimate by Neuman and Zheng [NZ17] for the maximaldisplacement of subcritical branching random walks.Recall that in the decomposition of T into the backbone tree and the traps given inSection 2, for any v ∈ T Bb we have that the law of T trap v only depends on v throughdeg T ( v ), and that we have assumed in Theorem 1.1 that deg T ( v ) ≤ ∆ + 1 almost surely. Lemma 5.1 (Maximal displacement inside traps) . Under the assumptions of Theorem1.1, there exists a γ > and C < ∞ , both independent of p , such that max ≤ δ ≤ ∆ − E p (cid:34) exp (cid:32) γ √ p − p c sup w ∈T trap v (cid:107) ϕ ( w ) − ϕ ( v ) (cid:107) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) deg T trap v ( v ) = δ (cid:35) ≤ C. (5.13)We prove this lemma in Section 9 below.We also need to bound the number of visited backbone-tree vertices. For this wewrite BBT k := { X τ k − , X τ k − +1 , . . . , X τ k } ∩ T Bb , i.e., BBT k is the trace of ( X n ) τ k − ≤ n ≤ τ k restricted to the backbone tree. Lemma 5.2 (Size of and maximal distance from BBT k ) . Under the assumptions ofTheorem 1.1, for any a ≥ there exists a C a such that for any k ≥ E p [ | BBT k | a ] ≤ C a ( p − p c ) − a , (5.14) and E p [ max v ∈ BBT k (cid:107) ϕ ( v ) − ϕ ( X τ k − ) (cid:107) a ] ≤ C a ( p − p c ) − a/ . (5.15)We prove this lemma in Section 10 below.Using the estimates of Lemmas 5.1 and 5.2, we can show that the right-hand side of(5.12) vanishes. Indeed, (cid:114) p n − p c n max τ k − ≤ i<τ k (cid:107) ϕ ( X i ) − ϕ ( X τ k − ) (cid:107)≤ (cid:114) p n − p c n (cid:32) max v ∈ BBT k (cid:107) ϕ ( v ) − ϕ ( X τ k − ) (cid:107) + max v ∈ BBT k sup w ∈T trap v (cid:107) ϕ ( w ) − ϕ ( v ) (cid:107) (cid:33) . (5.16)By (5.15) with a = 3, we have for any ε > P p n (cid:32)(cid:114) p n − p c n max v ∈ BBT k (cid:107) ϕ ( v ) − ϕ ( X τ k − ) (cid:107) > ε (cid:33) ≤ ε n / C. (5.17)Using further that by the decomposition (2.1), the GW-trees of the traps are independentof the backbone tree, P p n (cid:32)(cid:114) p n − p c n max v ∈ BBT k sup w ∈T trap v (cid:107) ϕ ( w ) − ϕ ( v ) (cid:107) > ε (cid:33) ≤ E p n [ | BBT k | ] max ≤ δ ≤ ∆ − P p n (cid:32) (cid:114) p n − p c n sup w ∈T trap v (cid:107) ϕ ( w ) − ϕ ( v ) (cid:107) > ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) deg T trap v ( v ) = δ (cid:33) ≤ e − γε √ n E p n [ | BBT k | ] × max ≤ δ ≤ ∆ − E p n (cid:34) exp (cid:32) γ √ p n − p c sup w ∈T trap v (cid:107) ϕ ( w ) − ϕ ( v ) (cid:107) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) deg T trap v ( v ) = δ (cid:35) ≤ C (cid:16) ( p n − p c )e γε √ n (cid:17) − , (5.18)where the last inequality follows from (5.13) and (5.14). From (5.16), (5.17), and (5.18)we may conclude P p n (cid:32)(cid:114) p n − p c n max k =1 ,...,Cn max τ k − ≤ i<τ k (cid:107) ϕ ( X i ) − ϕ ( X k − ) (cid:107) > ε (cid:33) ≤ Cε n / + C (cid:16) ( p n − p c )e γε √ n (cid:17) − . (5.19)Recall that in Theorem 1.1 we assumed that for any δ >
0, e δ √ n ( p n − p c ) → ∞ . Underthis assumption the right-hand side converges to zero, which implies that the right-hand ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 23 side of (5.12) vanishes in probability. Therefore, the convergence (5.4) also holds with W (1) n replaced by W (4) n , that is, (cid:32) ϕ ( X (cid:98) tτ νn (cid:99) ) (cid:112) ν n E p n [ τ − τ ]( p n − p c ) (cid:33) t ∈ [0 ,
1] d −−−→ n →∞ (cid:0) ( κ Σ) / B t (cid:1) t ∈ [0 , . (5.20)This completes the fourth step.All that remains to finish the proof of Theorem 1.1 is to apply the time change t (cid:55)→ tn ( p n − p c ) − τ − ν n , which, by the scale invariance of Brownian motion, yields (cid:32)(cid:115) ( p n − p c ) τ ν n nν n E p n [ τ − τ ]( p n − p c ) ϕ ( X (cid:98) tn ( p n − p c ) − (cid:99) ) (cid:33) t ∈ [0 ,
1] d −−−→ n →∞ (cid:0) ( κ Σ) / B t (cid:1) t ∈ [0 , . (5.21)The Law of Large Numbers for τ n then allows us to replace the prefactor by (cid:112) ( p − p c ) /n ,which completes the proof. (cid:3) Proofs of the escape time estimates
In this section we prove the lemmas of Section 3. We assume in this section somefamiliarity with the theory of reversible Markov chains and electrical networks. Werefer the reader who is insufficiently knowledgable about this topic to [AF02, LP16],where all reversiblility-related topics of this section that are not explicitly cited aredefined and discussed.6.1.
Proof of Lemma 3.1.
Let T be an infinite tree and v ∈ G [1] ( T ). We have P vT ( η < ∞ ) = lim n →∞ P vT ( η < η n ) . Denote by C G ( x, A ) the effective conductance in a graph G = ( V, E ) with unit edgeweights between a vertex x ∈ V and a subset A ⊂ V , and write R G ( x, A ) := C G ( x, A ) − for the associated effective resistance. For two disjoint sets A, B and x / ∈ A, B we havethe bound P xG ( η ( A ) < η ( B )) ≤ C G ( x, A ) C G ( x, A ∪ B ) ≤ C G ( x, A ) C G ( x, B ) , (6.1)see [LP16, Exercise 2.34] or [BGP03, Fact 2]. In fact, the arguments in the latterreference even show that if it is impossible that the random walk hits both A and B during the same excursion starting from x , then P xG ( η ( A ) < η ( B )) = C G ( x, A ) C G ( x, A ∪ B ) = C G ( x, A ) C G ( x, A ) + C G ( x, B ) . (6.2)In our setting, (6.1) gives P vT ( η < η n ) ≤ C T ( v, (cid:37) ) C T ( v, G [ n ] ( T )) . (6.3) The Series Law for conductances implies that for any v ∈ G [1] ( T ) we have C T ( v, (cid:37) ) = (cid:98) L/ ( p − p c ) (cid:99) − . Rayleigh’s Monotonicity Principle, moreover, implies that for any v ∈ G [1] ( T ) we have C T ( v, G [ n ] ( T )) ≥ C T v ( v, G [ n − ( T v )) . Observe that C T v ( v, G [ n − ( T v )) (and any bound we formulate from here on) dependsonly on the subtree rooted at v , and is thus independent of the tree segment T [0 , under¯ P p . This implies P vp ( η < ∞ | T [0 , = T [0 , ) ≤ lim n →∞ p − p c L ¯ E p (cid:2) R T v ( v, G [ n − ( T v )) (cid:3) . (6.4)Now given an infinite tree T and v ∈ G [1] ( T ), we prune T v as follows: first, removeany vertex w ∈ T v that does not have an infinite line of descent to obtain T Bb v , andsecond, at every vertex w ∈ T Bb v with more than two children in T Bb v , keep the first twoin Ulam-Harris ordering and delete all other children (and their subtrees). Call theresulting tree ˜ T v . Since we only removed edges, Rayleigh’s Monotonicity Principle oncemore implies R T v ( v, G [ n ] ( T )) ≤ R ˜ T v ( v, G [ n ] ( ˜ T v )) . (6.5)Now let ˜ G [ n ] ( ˜ T v ) denote the set of vertices w ∈ ˜ T v such that deg ˜ T v ( w ) = 3 and suchthat the unique path v = v , v , . . . , v m = w in ˜ T v satisfies |{ v i : deg ˜ T v ( v i ) = 3 }| = n (i.e., there are exactly n degree-three vertices between v and w in ˜ T v ). For ¯ P -almost alltrees, ˜ G [ n ] ( ˜ T v ) is nonempty for all n . Moreover, for any T ,lim n →∞ R ˜ T v ( v, G [ n ] ( ˜ T v )) = lim n →∞ R ˜ T v ( v, ˜ G [ n ] ( ˜ T v )) , (6.6)because this limit does not depend on the choice of the sequence of exhausting subgrapsof ˜ T v (see [LP16, Exercise 2.4]).Under ¯ P p , the subtree of ˜ T v of all paths connecting v to ˜ G [ n ] ( ˜ T v ) now has a particularlyeasy structure: vertices with two children are connected by line segments (a sequenceof vertices with one child) up to the n -th “generation” of branching points. By theSeries Law, the effective resistances of these line segments are just their respectivelengths, so they are independent Geo(1 − ˆ p ) random variables (which take values on { , , . . . } ), where ˆ p is the probability that the root of a Galton-Watson tree withgenerating function ˆ f ( s ) as described in (2.1) has exactly one child. By Lemma 2.1,ˆ p = ˆ f (cid:48) (0) = f (cid:48) ( q p ) = 1 − c ( p − p c )(1 + o (1)), so the mean length of a path segmentbetween two branching points in ˜ T v is at most λ ( p − p c ) − for some λ > G [1] ( ˜ T v ) = { w } and call ˜ T (1) w , ˜ T (2) w the two subtrees of ˜ T v rooted at w . Anapplication of the Series Law and the Parallel Law gives rise to the recursion R ˜ T v ( v, ˜ G [ n ] ( ˜ T v )) = d ˜ T v ( v, w ) + (cid:32) R ˜ T (1) w ( w, ˜ G [ n − ( ˜ T (1) w )) + 1 R ˜ T (2) w ( w, ˜ G [ n − ( ˜ T (2) w )) (cid:33) − . (6.7) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 25
Now write ˜ R [ n ] := R ˜ T v ( v, ˜ G [ n ] ( ˜ T v )) for the effective resistance between v and ˜ G [ n ] ( ˜ T v ),and write ˜ R (1)[ n − and ˜ R (2)[ n − for independent copies of ˜ R [ n − . Then, under ¯ P p , we havethe following equality in distribution:˜ R [ n ] d = S + (cid:32) R (1)[ n − + 1˜ R (2)[ n − (cid:33) − , (6.8)where S ∼ Geo(1 − ˆ p ). Bounding the harmonic mean by the arithmetic mean, i.e.,using that ( x + y ) − ≤ ( x + y ) for x, y >
0, this yields the stochastic domination˜ R [ n ] (cid:52) S + ( ˜ R (1)[ n − + ˜ R (2)[ n − ) . (6.9)Iterating (6.9), we obtain ˜ R [ n ] (cid:52) n − (cid:88) i =0 − i i (cid:88) j =1 S i,j , (6.10)where ( S i,j ) i,j ≥ is an array of i.i.d. copies of S . Taking the expectation,lim n →∞ ¯ E p [ ˜ R [ n ] ] ≤ lim n →∞ n − (cid:88) i =0 − i E p [ S ] = 2 E p [ S ] ≤ λp − p c . (6.11)Combining (6.4), (6.5), (6.6) and (6.11), we obtain P vp ( η < ∞ | T [0 , = T [0 , ) ≤ p − p c L λp − p c = 2 λL , and this upper bound is smaller than when we choose L ≥ L := 3 λ . (cid:3) Proof of Lemma 3.2.
Define the event G := {| G [1] ( T ) | > } and, for α < , let F ( α ) := {| G (cid:48)(cid:98)√ αL/ ( p − p c ) (cid:99) ( T ) | = 1 } denote the event that the backbone tree does notbranch before the (cid:98)√ αL/ ( p − p c ) (cid:99) -th generation. Splitting the expectation along G and F ( α ), we bound¯ E p (cid:20) (cid:88) v ∈ G [1] ( T ) { P v T \T v ( η = ∞ ) <α } (cid:21) ≤ ¯ P p ( G c ) + ¯ E p (cid:2) F ( α ) c | G [1] | (cid:3) (6.12)+ ¯ E p G∩F ( α ) (cid:88) v ∈ G [1] ( T ) ¯ P p ( P v T \T v ( η = ∞ ) < α | T [0 , ) . By Lemma 2.3 the first term on the right-hand side is bounded from above by a , asdesired. It remains to show that the second and third term on the right-hand side aboveboth vanish as α → B k be the event that the first branching of thebackbone tree occurs in generation k , that is, | G (cid:48) | = · · · = | G (cid:48) k − | = 1, but | G (cid:48) k | >
1. Attime B k , the backbone branch splits into at most ∆ branches. Let G ( i )[1] , ≤ i ≤ ∆ be i.i.d. copies of G [1] , then we can stochastically dominate | G [1] | by | G (1)[1] | + · · · + | G (∆)[1] | ,on any of the events B k , k ≤ (cid:98)√ αL/ ( p − p c ) (cid:99) . This implies¯ E p (cid:2) F ( α ) c | G [1] | (cid:3) = (cid:98)√ αL/ ( p − p c ) (cid:99) (cid:88) k =1 ¯ E p (cid:2) B k | G [1] | (cid:3) ≤ (cid:98)√ αL/ ( p − p c ) (cid:99) (cid:88) k =1 ¯ E p (cid:34) B k ∆ (cid:88) i =1 | G ( i )[1] | (cid:35) = ∆ ¯ P p ( F ( α ) c ) ¯ E p (cid:2) | G [1] | (cid:3) , where we used the independence of B k and the | G ( i )[1] | and Wald’s identity for the lastequality. By Lemma 2.3, ¯ E [ | G [1] ( T ) | ] ≤ a and, by an argument similar to (2.8), thereexists a constant c > P p ( F ( α ) c ) ≤ C √ α when α is sufficiently close to 0,so we conclude ¯ E p (cid:2) F ( α ) c | G [1] | (cid:3) ≤ C √ αa . (6.13)Now we show that the third term on the right-hand side of (6.12) also tends to 0as α →
0. Let T ∈ G ∩ F ( α ) and denote by v the unique vertex of G (cid:48)(cid:98)√ αL/ ( p − p c ) (cid:99) ( T ).Then, for any v ∈ G [1] , P vT \ T v ( η = ∞ ) ≥ P v T \ T v ( η = ∞ ) = C T \ T v ( v , ∞ ) C T \ T v ( v , (cid:37) ) + C T \ T v ( v , ∞ )by (6.2). By the Series Law and the fact that T ∈ G ∩ F ( α ) we have C T \ T v ( v , (cid:37) ) = (cid:98)√ αL/ ( p − p c ) (cid:99) − . It follows that P v T \ T v ( η = ∞ ) < α can only hold if R T \ T v ( v , ∞ ) > − α √ α Lp − p c . This implies that for any vertex w ∈ G [1] \ { v } ,¯ P p (cid:0) P v T \T v ( η = ∞ ) < α | T [0 , (cid:1) ≤ ¯ P p (cid:18) R T \ T v ( v , ∞ ) > − α √ α Lp − p c (cid:12)(cid:12)(cid:12)(cid:12) T [0 , (cid:19) ≤ ¯ P p (cid:18) Lp − p c + R T w ( w, ∞ ) > − α √ α Lp − p c (cid:12)(cid:12)(cid:12)(cid:12) T [0 , (cid:19) ≤ √ α p − p c L ¯ E p [ R T w ( w, ∞ )] , where we have used the independence of T w and T [0 , and the fact that α < /
4. Weestablished in (6.11) above that ¯ E p [ R T w ( w, ∞ )] ≤ C ( p − p c ) − . Inserting this and(6.13) into (6.12) to obtain¯ E p (cid:88) v ∈ G ( T ) { P v T \T v ( η = ∞ ) <α } ≤ a + C √ α, (6.14)which completes the proof. (cid:3) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 27
Proof of Lemma 3.3.
The main idea in this proof is to decompose the randomwalk on T [0 , into random walks on T [0 , and walks on the disjoint trees of T [1 , . Let( Y n ) n ≥ be a simple random walk on T [0 , started at v , and write H = σ ( Y , Y , . . . , Y η )for the history of this process until it hits the root (cid:37) . For i ≥
1, let h i be the i -thtime that ( Y n ) n ≥ visits a vertex in G [1] ∪ { (cid:37) } and let V i = Y h i . Furthermore, let H denote the number of visits of ( Y n ) n ≥ to G [1] ∪ { (cid:37) } until its first visit to the root, i.e., H = inf { i : V i = (cid:37) } . We can decompose the walk on T [0 , into the walk ( Y n ) n ≥ andthe walks on the branches of T [0 , \ T [0 , , and write P vp ( η < η , sib ( X η ) = 0 | T [0 , = T [0 , )= E p (cid:34) E p (cid:34) H − (cid:88) m =1 P V m T ( η < η p1 ) {| G [1] ( T Vm ) | =1 } m − (cid:89) i =1 P V i T ( η p1 < η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T [0 , = T [0 , (cid:35) , (6.15)where we write η p1 for the hitting time of a parent vertex of G [1] (i.e., the hitting time oflevel (cid:98) L ( p − p c ) − (cid:99) − V i , P V i T ( η < η p1 ) and {| G [1] ( T Vi ) | =1 } are independent of T [0 , . By (6.2) we have P V m T ( η < η p1 ) {| G [1] ( T Vm ) | =1 } = C T Vm ( V m , G [1] ( T V i ))1 + C T Vm ( V m , G [1] ( T V i )) {| G [1] ( T Vm ) | =1 } ≥ c ( p − p c ) {| G [1] ( T Vm ) | =1 } , (6.16)and similarly P V i T ( η p1 < η ) = 11 + C T Vm ( V m , G [1] ( T V i )) ≥ − c | G [1] ( T V i ) | ( p − p c ) . (6.17)Define for m < H and v ∈ G [1] H m ( v ) := |{ k < m : Y h k = v }| , (6.18)which counts the number of visits of ( Y n ) n ≥ to vertex v before the m -th visit to avertex in G [1] . Reordering the product in (6.15) according to the vertices and applyingthe bound in (6.16) gives the following lower bound, E p (cid:34) H − (cid:88) m =1 P V m T ( η < η p1 ) {| G [1] ( T Vm ) | =1 } m − (cid:89) i =1 P V i T ( η p1 < η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (cid:35) ≥ c ( p − p c ) E p H − (cid:88) m =1 {| G [1] ( T Vm ) | =1 } (cid:89) v ∈ G [1] P v T ( η p1 < η ) H m ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H . (6.19)Now note that H , H m ( v ) and V m are measurable with respect to H , and the probabilities P v T ( η p1 < η ) are independent of H and independent under ¯ P p ( · | T [0 , ) for different v , so E p (cid:34) H − (cid:88) m =1 P V m T ( η < η p1 ) {| G [1] ( T Vm ) | =1 } m − (cid:89) i =1 P V i T ( η p1 < η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (cid:35) ≥ c ( p − p c ) H − (cid:88) m =1 E p [ {| G [1] ( T Vm ) | =1 } P V m T ( η p1 < η ) H m ( V m ) | H ] × (cid:89) v ∈ G [1] : v (cid:54) = V m E p [ P v T ( η p1 < η ) H m ( v ) | H ] . (6.20)Applying now the bound (6.17), we see that for p sufficiently close to p c , (6.20) isbounded from below by c ( p − p c ) H − (cid:88) m =1 ¯ P p ( | G [1] | = 1)(1 − c ( p − p c )) H m ( V m ) (cid:89) v ∈ G [1] : v (cid:54) = V m (cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) H m ( v ) ≥ c ( p − p c ) ¯ P p ( | G [1] | = 1) H − (cid:88) m =1 (cid:89) v ∈ G [1] (cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) H m ( v ) = c ( p − p c ) ¯ P p ( | G [1] | = 1) H − (cid:88) m =1 (cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) m (6.21)= c ¯ E p [ | G [1] | ] − ¯ P p ( | G [1] | = 1) (cid:16)(cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) − (cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) H (cid:17) . By Lemma 2.3, ¯ P p ( | G [1] ( T ) | = 1) ≥ a and ¯ E p [ | G [1] ( T ) | ] ≤ a . So to show a uniformlower bound for (6.15), it remains to show that E p (cid:104) (cid:0) − c ¯ E [ | G [1] | ]( p − p c ) (cid:1) H (cid:12)(cid:12)(cid:12) T [0 , = T [0 , (cid:105) < − a (6.22)for an a > p . While the exact distribution of H depends on T [0 , , wemay give an easy lower bound. Recall that H − Y n ) n ≥ to G [1] until it hits the root, starting at some v ∈ G [1] . For any tree T with v ∈ G [1] ( T )we have P vT ( η < η +1 ) ≤ C T ( v, (cid:37) ) ≤ c ( p − p c ) , (6.23)where η +1 denotes the first hitting time of G [1] after time 0. This bound implies that H stochastically dominates a Geometric random variable H (cid:48) with success probability c ( p − p c ) and generating function E p [ θ H (cid:48) ] = c ( p − p c ) θ − θ + cθ ( p − p c ) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 29 for 0 ≤ θ ≤
1, which in turn implies that E p (cid:104) (cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) H (cid:12)(cid:12)(cid:12) T [0 , = T [0 , (cid:105) ≤ E p (cid:104)(cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) H (cid:48) (cid:105) = 1 − c ¯ E p [ | G [1] | ] c ¯ E p [ | G [1] | ] + c (cid:0) − c ¯ E p [ | G [1] | ]( p − p c ) (cid:1) . (6.24)The last term is bounded away from 0 uniformly in p , which shows (6.22) and completesthe proof. (cid:3) Moment bounds on regeneration distances: proof of Lemma 4.5
This proof is inspired by a similar moment estimate of [DGPZ02] and a regularityestimate for trees from [GK01]. By Lemmas 2.3, 3.1, and 4.4, E p [( | X τ | − | X τ | ) q ] = E vp [ | X τ | q | G [1] ( T ) = { v } , η = ∞ ] ≤ C E p [ | X τ | q ] , so it suffices to find a moment estimate for | X τ | . Recall that by Definition 4.1, the firstregeneration occurs at one of the potential regeneration times S k . More precisely, theregeneration time is set to equal the last S k that is finite, so that E p [ | X τ | q ] ≤ (cid:88) k ≥ E p [ | X S k | q { S k < ∞} ] ≤ (cid:88) k ≥ E p [ | X S k | q { S k < ∞} ] / P p ( S k < ∞ ) / . For S k to be finite, it has to occur at least k times that the walker sees a new part ofthe tree but then backtracks a generation, so by Lemma 3.1, P p ( S k < ∞ ) ≤ (cid:16) sup T : v ∈ G [1] ( T ) P vp (cid:0) η < ∞ | T [0 , = T [0 , (cid:1)(cid:17) k ≤ (cid:0) (cid:1) k . By Definition 4.1, we can also write | X S k |(cid:98) L/ ( p − p c ) (cid:99) = M k =: N k − + H k , M =: H (7.1)so that H k = M k − N k − . That is, H k counts the number of L -levels that are visited bythe walk after it surpasses the previous maximum generation N k − , until it arrives atthe next possible regeneration point, i.e., at a vertex v in an L -level with sib ( v ) = 0.Then H k can be stochastically dominated by 2 + ˜ H k , where ˜ H k is a Geometric randomvariable with success probabilityinf T : v ∈ G [1] ( T ) P vp ( η < η , sib ( X η ) = 0 | T [0 , = T [0 , ) ≥ a , where the lower bound holds by Lemma 3.3. Since Geometric random variables haveall moments bounded, it also holds for any q > E p [ H qk ] < ∞ . Write ˜ N k := N k − M k . Iterating the recursion (7.1), and noting that { S k < ∞} ⊂{ R k − < ∞} , we arrive at( p − p c ) q E p [ | X τ | q ] ≤ L q (cid:88) k ≥ E p (cid:32) H + k − (cid:88) i =1 ( ˜ N i + H i +1 ) (cid:33) q { R k − < ∞} / (cid:0) (cid:1) k/ ≤ L q (cid:32)(cid:88) k ≥ (2 k ) q − (cid:32) k (cid:88) i =1 E p [ H qi ] + k − (cid:88) i =1 E p [ ˜ N qi { R i < ∞} ] (cid:33)(cid:33) / (cid:0) (cid:1) k/ . Since the (2 q ) − th moment of H i is uniformly bounded, the proof will be completedonce we show that E p (cid:104) e s ˜ N i { R i < ∞} (cid:105) ≤ C < ∞ , (7.2)for some s >
0. Observe that ˜ N i counts how many new L -levels are reached by the walkerat X η Mi before backtracking to its L -ancestor. Since for different i these excursionshappen in disjoint parts of the tree, the ˜ N i are in fact i.i.d. under P p . It thus suffices tobound E p (cid:104) e s ˜ N { R < ∞} (cid:105) = (cid:88) n ≥ e sn P p ( ˜ N = n, R < ∞ ) . On the event { R < ∞} , a large value for ˜ N implies that the random walk backtracksa long distance towards the root. We will bound the probability by decomposing thistrajectory into level-sized chunks. Since the segment of the backbone tree between X η Mi and its L -ancestor is by definition of η M i isomorphic to a line graph of length (cid:98) L/ ( p − p c ) (cid:99) , the first step in this decomposition is to bound P p ( ˜ N = n, R < ∞ ) ≤ ¯ E p (cid:88) v ∈ G [ n ] ( T ) P v T ( X η n = v ) P v T ( η < ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | G [1] ( T ) | = 1 , where v is the unique vertex of G [1] ( T ). Note that Lemma 3.1 is not sufficient tobound P v T ( η < ∞ ), because the lemma gives an annealed bound, whereas we need aquenched bound. We instead use Lemmas 2.3 and 3.2 to show that the event B n ( α, β ) := (cid:40) n (cid:88) i =1 { P vi T \T vi ( η i − = ∞ ) ≥ α } ≥ βn for all v ∈ G [ n ] ( T ) (cid:41) , has a high probability provided α and β are small enough, where, for v = v n ∈ G [ n ] ( T ),we denote by v i its ancestor in G [ i ] ( T ) for 0 ≤ i < n . If we show this, then we are done,because, for T ∈ B n ( α, β ) and v ∈ G [ n ] ( T ), P v T ( η < ∞ ) ≤ n (cid:89) i =1 P v i T \T vi ( η i − < ∞ ) ≤ (1 − α ) βn , (7.3) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 31 so that P p ( ˜ N = n, R < ∞ ) ≤ (1 − α ) βn ¯ E (cid:20) (cid:88) v ∈ G [ n ] ( T ) P v T ( X η n = v ) (cid:12)(cid:12)(cid:12)(cid:12) | G [1] ( T ) | = 1 (cid:21) + ¯ P p (cid:0) B n ( α, β ) c | | G [1] ( T ) | = 1 (cid:1) = (1 − α ) βn + ¯ P p (cid:0) B n ( α, β ) c | | G [1] ( T ) | = 1 (cid:1) , (7.4)where for the equality we have used that conditionally on sib ( v ) = 0, the probabilitiesadd to 1 because G [ n ] ( T ) is a cutset on T separating the root from infinity. To showthat B n ( α, β ) c also has exponentially small probability, we introduce A n ( α, v ) := n (cid:88) i =1 { P vi T \T vi ( η i − = ∞ ) ≥ α } , and Z n ( α, θ ) := (cid:88) v ∈ G n ( T ) e − θA n ( α,v ) . We want to show that Z n ( α, θ ) decays exponentially for θ large enough. Note that byasking for the event { η i − = ∞} only on the cropped tree T \ T v i we have independenceof P v i T \T vi ( η i − = ∞ ) for different i under the environment law, which allows us tocalculate recursively¯ E p [ Z n ( α, θ ) | sib ( v ) = 0]= ¯ E p (cid:34) (cid:88) v ∈ G [ n − ( T ) e − θA n − ( α,v ) ¯ E p (cid:20) (cid:88) w ∈ G [1] ( T v ) { P w T \T w ( η i − = ∞ ) <α } + e − θ { P w T \T w ( η i − = ∞ ) ≥ α } (cid:12)(cid:12)(cid:12)(cid:12) | G [ n − ( T ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | G [1] ( T ) | = 1 (cid:35) = ¯ E p [ Z n − ( α, θ ) | | G [1] ( T ) | = 1] ζ ( α, θ ) = ζ ( α, θ ) n − , where ζ ( α, θ ) := ¯ E p (cid:88) v ∈ G [1] ( T ) { P v T \T v ( η = ∞ ) <α } + e − θ { P v T \T v ( η = ∞ ) ≥ α } . In the last step in the recursion we have used that ¯ E p [ Z ( α, θ ) | | G [1] ( T ) | = 1] = 1, since | G [1] ( T ) | = 1 implies that P v T \T v ( η = ∞ ) = 0.We proceed by bounding ζ ( α, θ ) as ζ ( α, θ ) ≤ ¯ E p (cid:88) v ∈ G ( T ) { P v T \T v ( η = ∞ ) <α } + e − θ ¯ E p [ | G [1] ( T ) | ] . (7.5) By Lemma 2.3 we have ¯ E p [ | G [1] ( T ) | ] ≤ a . With the bound of Lemma 3.2, this meansthat we can bound ζ ( α, θ ) ≤ a + h ( α ) + e − θ a =: γ ( α, θ ) , with γ ( α, θ ) < α is small enough and θ is large enough. Fix such a sufficient choiceof α and θ , then, by Markov’s inequality,¯ P p ( B n ( α, β ) c | | G [1] ( T ) | = 1) = ¯ P p (cid:18) min v ∈ G [ n ] ( T ) A n ( α, v ) < βn (cid:12)(cid:12)(cid:12)(cid:12) | G [1] ( T ) | = 1 (cid:19) = ¯ P p (cid:0) e − θ min v A n ( α,v ) > e − θβn (cid:12)(cid:12) | G [1] ( T ) | = 1 (cid:1) ≤ e nβθ ¯ E p (cid:2) e − θ min v A n ( α,v ) (cid:12)(cid:12) | G [1] ( T ) | = 1 (cid:3) ≤ e nβθ ¯ E p [ Z n ( α, θ ) | | G [1] ( T ) | = 1] ≤ e nβθ γ ( α, θ ) n − , and this bound decays exponentially in n if β is small enough. Inserting this boundinto (7.4) we see that (7.2) holds for some s > (cid:3) Moment bounds on the regeneration times: proof of Lemma 4.6
We start by establishing moment bounds for the time spent in the finite trees T trap v ,which as the name suggests, act as traps for the random walk. Let H v := inf { n ≥ X n = v } be the hitting time of vertex v and H + v = inf { n > X n = v } . When v = (cid:37) we will suppress the subscript. If π T is the invariant distribution for random walk on atree T , then the well-known formula E vT [ H + v ] = π T ( v ) − (8.1)holds. For the second moment we have the following identity, see [AF02, (2.21)], E vT [( H + v ) ] = π T ( v ) − (cid:32) (cid:88) w ∈ T π T ( w ) E wT [ H v ] + 1 (cid:33) . (8.2)Furthermore, we may bound E wT [ H v ] by the commute time E wT [ H v ] + E vT [ H w ]. TheCommute Time Identity of [CRR +
97] applied to simple random walk on a finite tree T gives E wT [ H v ] + E vT [ H w ] = 2( | T | − d T ( v, w ) , (8.3)with | T | the vertex cardinality of the tree and d T ( v, w ) the graph distance between v and w on T . This implies E vT [( H + v ) ] ≤ π T ( v ) − ( | T | − (cid:88) w ∈ T π T ( w ) d T ( v, w ) . (8.4)Moreover, for simple random walk on a finite connected tree T , π T ( v ) = deg T ( v ) (cid:80) w ∈ T deg T ( w ) = deg T ( v )2( | T | − . ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 33
So we arrive at E vT [( H + v ) ] ≤ T ( v ) − ( | T | − (cid:88) w ∈ T deg T ( w ) d T ( v, w ) . (8.5)Now, from here on, consider instead of T the tree T trap v located at v ∈ T Bb , i.e., atree rooted at v where v has degree U v (given by (2.2)) and where to the i th edge, i ∈ { , . . . , U v } , coming out of the root, there is an independent tree T ∗ i with generatingfunction f ∗ ( s ) as defined in (2.1) and with root (cid:37) i attached to the other end of the i thedge. We know that T trap v is almost surely finite. Write H trap v for H + v of random walkstarted at v , restricted to T trap v , with the convention that H trap v = 0 if T trap v = { v } , thatis, when U v = 0. Then, by (8.1), E p [ H trap v | U v ] = E p [ π T trap v ( v ) − | U v ] { U v (cid:54) =0 } = 2 E p (cid:34) U − v (cid:32)(cid:32) U v (cid:88) i =1 |T ∗ i | (cid:33) − (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U v (cid:35) { U v (cid:54) =0 } = 2 E p [ |T ∗ | − U − v ] { U v (cid:54) =0 } . (8.6)Applying Lemma 2.1 we obtain E p [ H trap v | U v ] = 2(1 − µ ∗ ) − { U v (cid:54) =0 } + O (1) = C ( p − p c ) − { U v (cid:54) =0 } + o (( p − p c ) − ) . (8.7)From (8.5) we get for the second moment E p [( H trap v ) | U v ] ≤ E p U − v (cid:32) U v (cid:88) i =1 |T ∗ i | (cid:33) U v (cid:88) i =1 (cid:88) w ∈T ∗ i deg T trap v ( w ) d T trap v ( w, v ) { U v (cid:54) =0 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U v ≤ ∆ (cid:88) i =1 E p |T ∗ i | (cid:88) w ∈T ∗ i (deg T ∗ i ( w ) + 1)( d T ∗ i ( w, (cid:37) i ) + 1) + 4 (cid:88) i (cid:54) = j E p |T ∗ i | (cid:88) w ∈T ∗ j (deg T ∗ j ( w ) + 1)( d T ∗ j ( w, (cid:37) j ) + 1) = 4∆ E p (cid:34) |T ∗ | (cid:88) w ∈T ∗ (deg T ∗ ( w ) + 1)( d T ∗ ( w, (cid:37) ) + 1) (cid:35) + 4∆(∆ − E p [ |T ∗ | ] E p (cid:34) (cid:88) w ∈T ∗ (deg T ∗ ( w ) + 1)( d T ∗ ( w, (cid:37) ) + 1) (cid:35) =: ( I ) + ( II ) , (8.8)where we have used U v ≤ ∆, the independence of the different GW-trees in a trap andin the second step we added one to the degree because deg T trap v ( (cid:37) i ) = deg T ∗ i ( (cid:37) i ) + 1. Now write Z ∗ n for the size of the n th generation of T ∗ and observe that |T ∗ | = (cid:80) w ∈T ∗ (cid:80) n ≥ Z ∗ n , and that (cid:80) w ∈T ∗ deg T ∗ ( w ) = (cid:80) n ≥ ( Z ∗ n + Z ∗ n +1 ) (each vertex has degree equalto the number of its offspring plus one, the latter accounting for its ancestor), so thatwe may bound ( I ) ≤ E p (cid:34)(cid:32)(cid:88) n ≥ Z ∗ n (cid:33) (cid:32)(cid:88) n ≥ (2 Z ∗ n + Z ∗ n +1 )( n + 1) (cid:33)(cid:35) ≤ C E p (cid:34)(cid:32)(cid:88) n ≥ Z ∗ n (cid:33) (cid:32)(cid:88) n ≥ nZ ∗ n (cid:33)(cid:35) , where in the last step we used the assumption that ∆ < ∞ . By conditioning on theearlier generations we can write the right-hand side as( I ) ≤ C (cid:88) ≤ m
0) = 1 − f (cid:48) p (0) /f (cid:48) p ( q p ) ≥ c. Combining this bound with (8.16), inserting that into (8.14), and writing η Bb j for thehitting time of G [ j ] for ( X Bb m ) m ≥ , we obtain E p [ τ − τ ] ≥ cp − p c E wp (cid:2) η Bb − η Bb − | G [1] = { w } , | G [2] | = 1 , η Bb < η Bb (cid:3) ≥ c (cid:48) ( p − p c ) , where, in the last step, we have used that ( X Bb m ) m ≥ restricted to T Bb [0 , , conditionally on {| G [1] | = | G [2] | = 1 } , is equivalent to a simple random walk on a line graph of length2 (cid:98) L/ ( p − p c ) (cid:99) . This proves the lower bound in Lemma 4.6.For the upper bounds, we first notice that by the same reasoning as in Lemma 4.5 itsuffices to show the bound for τ . Let us again first consider L v as in (8.12), for whichwe can bound E vp [ L v | T Bb ] ≤ E vp [ Y v | T Bb ] E vp [( H trap v,i ) | T Bb ] . (8.17)For the Geometric random variable Y v , the parameter satisfies 1 − U v / deg T ( v ) ≥ / ∆,which implies E vp [ Y v | T Bb ] ≤ ∆ , (8.18)while for the second expectation in (8.17) we may apply (8.11) to conclude E vp [ L v | T Bb ] ≤ C ( p − p c ) − . (8.19) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 37
We can use this upper bound to state a bound on the second moment of a genericrandom time τ that is measurable with respect to σ ( X Bb n , n ≥ E p [ τ ] = E p (cid:88) v ∈T Bb (cid:96) τ ( v ) (cid:88) i =1 L ( i ) v = E p (cid:88) v ∈T Bb (cid:96) τ ( v ) (cid:88) i =1 L ( i ) v + E p (cid:88) v (cid:54) = w ∈T Bb (cid:96) τ ( v ) (cid:88) i =1 L ( i ) v (cid:96) τ ( w ) (cid:88) i =1 L ( i ) w = E p (cid:34) (cid:88) v ∈T Bb (cid:96) τ ( v ) E v [ L v | T Bb ] + (cid:96) τ ( v )( (cid:96) τ ( v ) − E vp [ L v | T Bb ] (cid:35) + E p (cid:88) v (cid:54) = w ∈T Bb (cid:96) τ ( v ) (cid:96) τ ( w ) E vp [ L v | T Bb ] E wp [ L w | T Bb ] ≤ C ( p − p c ) − E p [ τ Bb ] + C ( p − p c ) − E p [( τ Bb ) ] , (8.20)where, for the third equality, we have used that traps at distinct vertices of the backbonetree are independent, and for the final inequality we have used (8.17). Applying thisupper bound to τ Bb , which denotes the number of steps of ( X Bb n ) n until τ , the upperbounds in Lemma 4.6 will therefore follow once we show E p [( τ Bb ) ] ≤ C ( p − p c ) − . (8.21)Since | X Bb n | stochastically dominates a simple random walk on N reflected at theorigin, η Bb m is stochastically dominated by the time the simple random walk hits thevertex m (cid:98) L/ ( p − p c ) (cid:99) . We have E p [( η Bb m ) k ] ≤ Cm k ( p − p c ) − k for any k ≥
1, so E p [( τ Bb ) q ] = ∞ (cid:88) m =1 E p [( η Bb m ) q { X τ ∈ G [ m ] } ] ≤ ∞ (cid:88) m =1 E p [( η Bb m ) q ] / P p ( X τ ∈ G [ m ] ) / ≤ C ( p − p c ) − q ∞ (cid:88) m =1 m q P p ( X τ ∈ G [ m ] ) / . (8.22)By the moment bound of Lemma 4.5, P p (( p − p c ) | X τ | ≥ m ) ≤ Cm − q − . This implies(8.21), which concludes the proof. (cid:3) The maximal displacement of a trap: proof of Lemma 5.1
The claimed moment bound follows from the following bound for the projectedprocesses, max ≤ δ ≤ ∆ − E p (cid:34) exp (cid:32) γ √ p − p c sup w ∈T trap (cid:37) ϕ ( w ) · e (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) deg T trap (cid:37) ( (cid:37) ) = δ (cid:35) ≤ C, (9.1) for constants γ, C , arbitrary unit vectors e with (cid:107) e (cid:107) = 1 and where, without loss ofgenerality, we consider a trap at the root, such that ϕ ( (cid:37) ) = 0.Recall from the decomposition in (2.1) that T trap (cid:37) consists of at most ∆ − (cid:37) , each tree having generating function f ∗ and meannumber µ ∗ p of offspring. Thus, it suffices to show (9.1) with T trap (cid:37) replaced by a singletree T ∗ with generating function f ∗ . Let W denote a Z d -valued random variable withdistribution D , and write K ( s ) = E p [ s W · e ] for the probability generating function of W · e . For x >
1, let θ ( x ) be the unique solution in (1 , ∞ ) to K ( θ ( x )) = x. (9.2)Neuman and Zheng [NZ17, Theorem 1.2] show thatlim sup n →∞ θ (( µ ∗ p ) − ) n P p (cid:18) sup w ∈T ∗ ϕ ( w ) · e ≥ n (cid:19) ≤ E p [e c || W || ] = (cid:80) x ∈ Z d e c (cid:107) x (cid:107) D ( x ) < ∞ for all c > θ (( µ ∗ p ) − ) − n ( p − p c ) − / ≤ e − cn , (9.4)for n sufficiently large, uniformly in p close enough to p c . We proceed by showing thatthis is the case.From Lemma 2.1 we obtain( µ ∗ p ) − = 1 + c ( p − p c )(1 + o (1)) . (9.5)Since E p [ W ] = (cid:80) x ∈ Z d xD ( x ) = 0 by the assumption in Theorem 1.1 we have E p [ W · e ] =0, which implies K (cid:48) (1) = 0, so that expansion around s = 1 gives K ( s ) ≤ c ( s − , (9.6)for s close enough to 1 (again using that E p [e c || W || ] < ∞ ). This in turn implies that for x close enough to 1, θ ( x ) ≥ c ( x − / . (9.7)Combining (9.5) and (9.7), as well as ( µ ∗ p ) − ≥ c ( p − p c ), we arrive at θ (( µ ∗ p ) − ) − n ( p − p c ) − / ≤ (cid:0) c ( p − p c ) / (cid:1) − n ( p − p c ) − / ≤ e − cn , (9.8)for p close enough to p c . This implies (9.4), which finished the proof. (cid:3) The trace of the walk on the backbone: proof of Lemma 5.2
We will show a general moment bound for the trace until the second regeneration,that is, a moment bound for the cardinality ofBBT ∪ BBT = { X , X , . . . , X τ } ∩ T Bb = { X Bb , X Bb , . . . , X Bb τ Bb } . (10.1)The bound on E p [ | BBT k | ] for any k ≥ B = B ( T ) the set of branch points of the backbone tree, that is, B ( T ) := { (cid:37) } ∪ { v ∈ T Bb : deg T Bb ( v ) ≥ } . ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 39
The bound for the trace on the backbone tree will follow from a bound on the numberof visited branch points of the backbone. Between those branchpoints, the random walkhas to travel across a section of the tree isomorphic to a line segment. For v ∈ T Bb , let anc B ( T ) ( v ) denote the first vertex on the shortest path between v and (cid:37) that is in B ( T ).Given v ∈ B ( T ), we call the path in T between v and anc B ( T ) ( v ) the backbone branchto v .Fix a small δ > S ( T ) be the set of all vertices v ∈ T Bb for which there exists a path connecting v to the root (cid:37) , such that for anyvertex w on this path, d T ( w, anc B ( T ) ( w )) ≤ (cid:98) δ/ ( p − p c ) (cid:99) , i.e., v is in S ( T ) if the path from v to anc B ( T ) ( v ) and all backbone branches along thepath to (cid:37) have length at most (cid:98) δ/ ( p − p c ) (cid:99) .The motivation for these definitions is as follows: Until the random walk exits S ( T ),the trace of the random walk on the backbone is a subset of S ( T ), and in order to exit S ( T ) the random walk has to cross a backbone branch of length at least (cid:98) δ/ ( p − p c ) (cid:99) .We can find a lower bound for the time it takes to traverse such a backbone branch,which implies an upper bound for the size of the trace on the backbone until that time.When the random walk exits S ( T ) and crosses a long backbone branch, it enters a new,independent subtree, and in this new tree we can iterate this upper bound.So set S := S ( T ) and let E := H Bb S c be the exit time of the set S for the randomwalk ( X Bb n ) n ≥ restricted to the backbone tree. Define recursively for k > S k := S k − ∪ S ( T X Ek − ) and E k := H Bb S ck . (10.2)Then E k stochastically dominates the sum ˜ E k = ˜ E (1)1 + · · · + ˜ E ( k )1 of k i.i.d. copies of˜ E = inf { n ≥ Y n = (cid:98) δ/ ( p − p c ) (cid:99)} , (10.3)where ( Y n ) n ≥ is a simple random walk on { , . . . , (cid:98) δ/ ( p − p c ) (cid:99)} starting in and reflectedat 0. Moreover, |S k | stochastically dominates |{ X , . . . , X E k } ∩ T Bb | . Since at eachtime E i the set S ( T X Ei ) is independent of the tree explored so far, we can bound E p [ |{ X , . . . , X E k } ∩ T Bb | q ] ≤ E p [ |S k | q ] ≤ E p (cid:34)(cid:32) k (cid:88) i =1 ˜ S ( i ) (cid:33) q (cid:35) ≤ k q E p [ |S | q ] , (10.4)where we wrote ˜ S ( i ) for i.i.d. copies of S .The next step is to show that, for any q ≥ E p [ |S | q ] ≤ C q ( p − p c ) − q . (10.5)Recall that ∆ is the maximal number of children in the original Galton-Watson tree.By definition of S , |S | ≤ ∆ · |S ∩ B ( T ) | · (cid:98) δ/ ( p − p c ) (cid:99) , (10.6)where we have used that ∆, the maximal number of offspring in the unpercolated tree,is finite. By construction, the set |S ∩ B ( T ) | is stochastically dominated by the total progeny of a Galton-Watson process, denoted by ˜ Z , with offspring distribution givenby the law of ˜ ξ := ∆ (cid:88) i =1 { (cid:96) i ≤(cid:98) δ/ ( p − p c ) (cid:99)} . (10.7)Here, (cid:96) i are independent random variables corresponding to the length of a backbonebranch, so that each (cid:96) i is an independent Geometric random variable, having mean c ( p − p c ) − (see Remark 2.2). Then ˜ ξ is a Binomial random variable with ∆ trials andprobability of success P p ( (cid:96) i ≤ (cid:98) δ/ ( p − p c ) (cid:99) ) ≤ − (1 − c ( p − p c )) (cid:98) δ/ ( p − p c ) (cid:99) ≤ − e − cδ , (10.8)where we have used Lemma 2.1. This means that we can choose δ > p , E p [ ˜ ξ q ] ≤ c < , (10.9)i.e., ˜ Z is a subcritical Galton-Watson process. Writing ˜ Z n for the size of the n -thgeneration of ˜ Z and using E p [ ˜ Z qn ] ≤ E p [ ˜ Z qn − ] E p [ ˜ ξ q ] ≤ E p [ ˜ ξ q ] n , we can bound themoments of the total progeny | ˜ Z| as E p [ | ˜ Z| q ] = ∞ (cid:88) N =0 E p (cid:34)(cid:32) N (cid:88) n =0 ˜ Z n (cid:33) q { ˜ Z N (cid:54) =0 , ˜ Z N +1 =0 } (cid:35) ≤ ∞ (cid:88) N =0 N (cid:88) n =0 N q − E p (cid:104) ˜ Z qn { ˜ Z N (cid:54) =0 } (cid:105) . (10.10)Applying the Cauchy-Schwarz inequality and using P p ( ˜ Z N (cid:54) = 0) ≤ E p [ ˜ Z N ] = E p [ ˜ ξ ] N , E p [ | ˜ Z| q ] ≤ ∞ (cid:88) N =0 N (cid:88) n =0 N q − E p [ ˜ Z qn ] / P p ( ˜ Z N (cid:54) = 0) / ≤ ∞ (cid:88) N =0 N q − (cid:32) ∞ (cid:88) n =0 E p [ ˜ ξ q ] n/ (cid:33) P p ( ˜ Z N (cid:54) = 0) / ≤ (1 − E p [ ˜ ξ q ] / ) − ∞ (cid:88) N =0 N q − E p [ ˜ ξ ] N/ . (10.11)Since E p [ ˜ ξ ] ≤ E p [ ˜ ξ q ] < p , (10.10) is bounded uniformly in p also. Thisimplies that by the Cauchy-Schwarz inequality, (10.4) and (10.5), E p [ |S ∩ B ( T ) | q ] ≤ E p [ | ˜ Z| q ] ≤ C q , (10.12)and, by (10.6), (10.12) shows that (10.5) holds. ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 41
So far, we have shown that (10.4) is bounded by C q k q ( p − p c ) − q . In order to turnthis into a bound for (10.1), we note that E p [ |{ X Bb , X Bb , . . . , X Bb τ Bb }| q/ ] ≤ ∞ (cid:88) k =1 E p [ |{ X Bb , X Bb , . . . , X Bb E k }| q/ { E k − ≤ τ Bb
In this section we prove Theorem 1.2 and Lemma 2.1. We start with Lemma 2.1,and also derive a result that will be crucially used in the proof of Theorem 1.2. In thefollowing, let m p,k := f ( k ) p (1) = E p [ ξ p ( ξ p − · · · ( ξ p − k + 1)]denote the k -th factorial moment of ξ p . We abbreviate m k = m ,k , noting that m p,k = p k m k . Recall that q p is the extinction probability, i.e., the unique fixed point of f p in (0 , Proof of Lemma 2.1.
We start by deriving the generating function f p , for whichwe use that, conditioned on { ξ = n } , the number of offspring ξ p in the pruned tree isdistributed as a Binomial with n trials and success probability p . Therefore, f p ( s ) = E p [ ξ sp ] = ∞ (cid:88) k =0 E p [ ξ sp | ξ = n ] p k = ∞ (cid:88) k =0 ( ps + (1 − p )) k p k = f ( ps + (1 − p )) , (11.1) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 43 where f is the generating function of the offspring distribution ( p k ) k ≥ . Thus, we obtainthat f (cid:48) p (0) = pf (cid:48) (1 − p ) ≥ c > , (11.2)uniformly for p ≥ p c , as required. Furthermore, setting s = 1 yields f (cid:48) p (1) = pf (cid:48) (1) = pm = p/p c . This proves the two equalities in (2.3).We proceed with the three asymptotic relations in (2.4), where we start with thefirst.Since q p is the fixed point of f p , we have1 − q p = 1 − f p ( q p ) = 1 − f ( pq p + 1 − p ) = 1 − f (1 − p (1 − q p )) . (11.3)Taking the Taylor expansion of the right-hand side in 1 − q p and using that q p → p (cid:38) p c yields1 − q p = pf (cid:48) (1)(1 − q p ) − p f (cid:48)(cid:48) (1)(1 − q p ) + p f (cid:48)(cid:48)(cid:48) (1)(1 − q p ) + o ((1 − q p ) ) . (11.4)Here we have used the assumption that E p [ ξ ] < ∞ . Using that f ( k ) (1) = m k anddividing through by 1 − q p > p m (1 − q p ) = ( pm −
1) + p m (1 − q p ) + o ((1 − q p ) ) . (11.5)We will use these asymptotics in the proof of Theorem 1.2 below. Replacing the lasttwo terms by O ((1 − q p ) ), the first asymptotics in (2.4) follows, as1 − q p = 2 m p ( pm −
1) + O ((1 − q p ) ) = 2 m m ( p − p c ) + O ((1 − q p ) ) , (11.6)with c = 2 m /m , since p = p c + O ( p − p c ) = 1 /m + O ( p − p c ).To prove the second asymptotics in (2.4), we note that f (cid:48) p ( q p ) = pf (cid:48) ( q p ) = pf (cid:48) (1 − p (1 − q p )) = pf (cid:48) (1) − p f (cid:48)(cid:48) (1)(1 − q p ) + O ((1 − q p ) ) , so that, by (11.6), f (cid:48) p ( q p ) = pm − (1 − q p ) p m + O ((1 − q p ) )= 1 + m ( p − p c ) − m ( p − p c ) + O (( p − p c ) )= 1 − ( p − p c ) m + O (( p − p c ) ) . (11.7)The third asymptotics in (2.4) similarly follows, sinceˆ f (cid:48)(cid:48) p (0) = (1 − q p ) f (cid:48)(cid:48) p ( q p ) = (1 − q p ) p f (cid:48)(cid:48) (1 − p (1 − q p ))= (1 − q p ) p c f (cid:48)(cid:48) (1)(1 + o (1)) = 2 m ( p − p c )(1 + o (1)) . (cid:3) We proceed with the proof of Theorem 1.2, for which we rely on an explicit formulafor the effective speed due to Lyons, Pemantle and Peres [LPP95, Page 601], v ( p ) = ∞ (cid:88) k =0 k − k + 1 p k ( p ) 1 − q k +1 p − q p , with p k ( p ) = P p ( ξ p = k ). Using Lemma 2.1, we expand this expression for p close to p c . Proof of Theorem 1.2.
We again expand in terms of 1 − q p . We rewrite(1 + q p ) v ( p ) = ∞ (cid:88) k =0 k − k + 1 p k ( p ) k (cid:88) i =0 q ip = ∞ (cid:88) k =0 k − k + 1 p k ( p ) k (cid:88) i =0 (1 − (1 − q p )) i . (11.8)Substituting the expansion(1 − (1 − q p )) i = 1 − i (1 − q p ) + i ( i − − q p ) + o (cid:0) i (1 − q p ) ) . (11.9)gives three terms and an error term. Using p c = 1 /m = 1 / E p [ ξ ], the first term equals ∞ (cid:88) k =0 k − k + 1 p k ( p )( k + 1) = E p [ ξ p −
1] = p E p [ ξ ] − m ( p − p c ) . (11.10)Using (cid:80) ki =0 i = k ( k + 1), the second equals − (1 − q p ) ∞ (cid:88) k =0 k ( k − p k ( p ) = − (1 − q p ) m p, = − (1 − q p ) p m . (11.11)Using (cid:80) ki =0 i ( i −
1) = ( k + 1) k ( k − (1 − q p ) ∞ (cid:88) k =0 k − k + 1 p k ( p ) k (cid:88) i =2 i ( i −
1) (11.12)= (1 − q p ) ∞ (cid:88) k =0 k − k + 1 p k ( p ) ( k + 1) k ( k − (1 − q p ) (cid:16) ∞ (cid:88) k =0 k ( k − k − p k ( p ) + ∞ (cid:88) k =0 k ( k − p k ( p ) (cid:17) = (1 − q p ) (cid:0) m p, + m p, (cid:1) . Using E p [ ξ ] < ∞ and | ( k − / ( k + 1) | ≤
1, the error term can be estimated as o ((1 − q p ) ) ∞ (cid:88) k =0 p k ( p ) k (cid:88) i =0 i = o ((1 − q p ) ) . (11.13)We conclude that(1 + q p ) v ( p ) = m ( p − p c ) − (1 − q p ) p m (11.14)+ (1 − q p ) (cid:0) m p, + m p, (cid:1) + o ((1 − q p ) ) . Substituting (11.5) for the second term, we obtain m ( p − p c ) − (1 − q p ) p m = − p m (1 − q p ) + o (( p − p c ) ) , (11.15)which cancels up to leading order with the term (1 − q p ) m p, . (We have no intuitiveexplanation for why the third moment drops out.) ANDOM WALK ON BARELY SUPERCRITICAL BRANCHING RANDOM WALK 45
By (11.6), we arrive at(1 + q p ) v ( p ) = (1 − q p ) m p, + o (cid:0) ( p − p c ) (cid:1) (11.16)= ( p − p c ) m m + o (cid:0) ( p − p c ) (cid:1) . Using that 1 + q p = 2 + O ( p − p c ) completes the proof. (cid:3) Acknowledgements.
The work of JN was supported by the Deutsche Forschungs-gemeinschaft (DFG) through grant NA 1372/1. RvdH was supported by NWO throughVICI-grant 639.033.806. The work of RvdH and TH is also supported by the NetherlandsOrganisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003.
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