Random Walks in a Sparse Random Environment
Anastasios Matzavinos, Alexander Roitershtein, Youngsoo Seol
aa r X i v : . [ m a t h . P R ] N ov Random walks in a sparse random environment
Anastasios Matzavinos ∗ Alexander Roitershtein † Youngsoo Seol ‡ September 5, 2016; Revised: November 17, 2016
Abstract
We introduce random walks in a sparse random environment on Z and investigatebasic asymptotic properties of this model, such as recurrence-transience, asymptoticspeed, and limit theorems in both the transient and recurrent regimes. The new modelcombines features of several existing models of random motion in random media andadmits a transparent physical interpretation. More specifically, a random walk in asparse random environment can be characterized as a “locally strong” perturbation ofa simple random walk by a random potential induced by “rare impurities,” which arerandomly distributed over the integer lattice. Interestingly, in the critical (recurrent)regime, our model generalizes Sinai’s scaling of (log n ) for the location of the randomwalk after n steps to (log n ) α , where α > MSC2010: primary 60K37; secondary 60F05.
Keywords : RWRE, sparse environment, limit theorems, Sinai’s walk.
We start with a general description of one-dimensional random walks in a random environ-ment. Let Ω = (0 , Z and let F be the Borel σ − algebra of subsets of the product space Ω . A random environment is a random element ω = ( ω n ) n ∈ Z of the measurable space (Ω , F ) . The environment determines the transition kernel of the underlying random walk. Namely,a random walk in a random environment ω = ( ω n ) n ∈ Z ∈ Ω is a Markov chain ( X n ) n ≥ on Z ,the transition kernel of which is given by P ω ( X n +1 = j | X n = i ) = ω i if j = i + 11 − ω i if j = i −
10 otherwise . ∗ Division of Applied Mathematics, Brown University, Providence, RI 02912, USA;e-mail: [email protected] † Dept. of Mathematics, Iowa State University, Ames, IA 50011, USA; e-mail: [email protected] ‡ Dept. of Mathematics, University of South Florida, Tampa, FL 33620, USA; e-mail: [email protected] X = x, x ∈ Z , is denoted by P x,ω and is referred to as the quenched law of the random walk.We denote the probability distribution of ω in (Ω , F ) by P, and we let E P denote the cor-responding expectation operator. That is, for a measurable function f ( ω ) of the environment ω , we have E P ( f ) = R Ω f ( ω ) dP ( ω ) . Let G be the cylinder σ -algebra on Z n . A random walkin a random environment (RWRE) associated with P is a process ( X, ω ) on the measurablespace (Ω × Z N , F ⊗ G ) equipped with the annealed probability law P = P ⊗ P ω , which isdefined by P x ( F × G ) = Z F P x,ω ( G ) P ( dω ) , F ∈ F , G ∈ G . The expectations under the laws P x,ω and P x are denoted by E x,ω and E x , respectively. Wewill usually omit the index 0 when x = 0 , which is to say that we will write P ω , E ω , P , and E for P ,ω , E ,ω , P , and E , respectively. Notice that, since the process “learns” theenvironment according to the Bayes rule as time progresses, X = ( X n ) n ≥ is not, in general,a Markov chain under the annealed measure P . We now describe in detail the specific model of random environment that we consider inthis paper. Let ( λ k , d k ) k ∈ Z be a stationary and ergodic sequence of pairs, such that λ k ∈ (0 , d k ∈ N . Throughout the paper we denote by P the joint law of the sequence of pairs( λ k , d k ) k ∈ Z . For n ∈ Z let a n = P nk =1 d k if n >
00 if n = 0 − P k = − n d k if n < . The random variables a n serve as locations of random impurities in the, otherwise homoge-neous, random medium. More precisely, the sparse random environment ω associated with P is defined by ω n = (cid:26) λ k if n = a k for some k ∈ Z , / . (1)For future reference, we also define ρ n = 1 − ω n ω n and ξ n = 1 − λ n λ n , n ∈ Z . (2)We refer to the random walk in the environment defined by (1) as a random walk in asparse random environment (RWSRE). The primary focus of this paper is to illuminate thedependence of the basic properties of RWSREs on the distribution of the sequence ( d n ) n ∈ Z , and compare the dynamics of RWSREs to the dynamics of the classical RWRE, whichcorresponds to the special case d = 1 a. s.In the classical RWRE model, ω is a stationary and ergodic sequence under P [37]. It isknown (see [16, 28, 32, 37] and, for instance, [1, 3, 7] and [9, 10, 21] for some recent advancesin the recurrent and transient cases, respectively) that asymptotic results for one-dimensionalRWREs can usually be stated in terms of certain averages of functions of ρ and explained2y means of typical “landscape features” (such as traps and valleys ) of the random potential ( R n ) n ∈ Z , which is associated with the random environment as follows: R n = n P k =1 log ρ k if n > , − | n − | P k =0 log ρ − k if n < . (3)We remark that interpreting a RWRE as a random walk in the random potential (3) servesto provide heuristic explanations to most results about RWREs, including those discussedin this paper.In our model, the sparse environment ω n is defined as a simple functional of the markedpoint process ( a n , λ n ) n ∈ Z , and it is in general non-stationary. However, it is well known that if E p ( d ) < ∞ , the underlying probability space can be enlarged to include a random variable M , such that the random shift ( a n + M , λ n + M ) n ∈ Z of the sequence ( a n , λ n ) n ∈ Z is stationaryand ergodic. Therefore, one should expect that if E P ( d ) < ∞ , basic zero-one laws, suchas recurrence-transience, existence of asymptotic speed, and ballisticity, are similar to thecorresponding features of the RWRE associated with the random environment ω = ( λ n ) n ∈ Z . However, an analogous claim about the similarity of limit theorems for random walksin the environments λ and ω is less obvious. Indeed, the dependence structure associatedwith the environment plays a crucial role in such theorems, and it is clearly not preservedunder the transformation λ ω. We remark, for instance, that the transformation of ani.i.d. environment yields a stationary and ergodic transformation of a Markov chain. SeeSections 2 and 3.3 for a more detailed discussion, and Section 3.5 for the case E ( d ) = ∞ .In the continuous setting, a model which is closely related to the RWSREs discussed inthis paper is the multi-skewed Brownian motion introduced in [24]. A direct discrete-timeanalogue of the multi-skewed Brownian motion is a multi-skewed random walk , which canbe introduced as a quenched variant of our model when ( λ n ) n ∈ Z is a certain deterministicsequence of constants. In accordance to the physical motivation of the model in [24], the au-thor refers to the marked sites (i.e., the elements of A in the author’s notation) as interfaces,while the long stretches of “regular” sites between interfaces are referred to by the authoras layers.We remark that certain random environments that consist of alternating stretches ofsites of two different types, and induce sub-linear growth rates on the underlying randomprocesses, have been considered in [23, 31] and, in a slightly different context, in [4]. Theoverlap between results and proof methods in this work and in [23, 31] is minimal, and it isdiscussed in more detail in Section 3.Somewhat related to our work is the study of [20, 19], where it is shown that an X n ∼ (log n ) α asymptotic behavior of the random walk can occur under a perturbation of ani.i.d. recurrent environment ( ω n ) n ∈ Z of the form ω new n = ω n + f n , where E P (cid:0) log − ω ω (cid:1) = 0 and f n converges to zero in probability as | n | → ∞ . A heuristic explanation of this phenomenoncan be provided by considering that both recurrent and transient random walks “spreadout,” and hence, for a large class of perturbations f n , a typical landscape of the environmentas viewed from the particle at time n can be identified. Moreover, it can be shown thatthis typical landscape is a dominant factor in determining the asymptotic behavior of therandom walk as n goes to infinity. 3he rest of the paper is organized as follows. In Section 2, we introduce a notion of adual stationary environment along with the Palm dualities that are used in the proof of ourresults. In Section 3 we state and discuss our main results for the asymptotic behavior ofRWSREs. These include recurrence and transience criteria, asymptotic speed, stable laws,and a Sinai-type result for RWSREs. Finally, in Section 4 we present the proofs of ourresults. The sparse environment introduced in Section 1 is in general a non-stationary sequence. Theaim of this section is to introduce a dual stationary environment and relate the propertiesof the RWSRE to the corresponding properties of the RWRE in the dual environment. If E p ( d ) < ∞ , then the underlying probability space can be enlarged to include a non-negativerandom variable M, such that the random shift ( ω n − M ) n ∈ Z of the environment ( ω n ) n ∈ Z isstationary and ergodic. Furthermore, the distribution of the sparse environment turns outto be the distribution of its stationary version conditioned on the event 0 ∈ A . In contrast to the usual RWRE, ω = ( ω n ) n ∈ Z is in general a non-stationary sequence inthe RWSRE model. In fact, ω is cycle-stationary under P , namely θ d n ω = D ω under P for all n ∈ Z , where X = D Y means that the distributions of the random variables X and Y coincide, andthe shift θ k is a measurable mapping of (Ω , F ) into itself which is defined for any (possiblyrandom) k ∈ Z by ( θ k ω ) n = ω n + k , n ∈ Z . If E P ( d ) < ∞ one can define a “stationary dual” e ω of the environment ω as follows [35, 36].Without loss of generality, we assume that the underlying probability space supports arandom variable U , which is independent of ω and is distributed uniformly on the interval[0 , ⊂ R . For x ∈ R , let ⌊ x ⌋ denote the integer part of x, that is ⌊ x ⌋ = sup { n ∈ Z : n ≤ x } . We now define (cid:0)e a n , e ω n ) n ∈ Z by setting e a n = a n + ⌊ U d ⌋ and e ω n = (cid:26) λ k if n = e a k for some k ∈ Z , / . Let A = ( a n ) n ∈ Z be the set of marked sites of the integer lattice and let e A = ( e a n ) n ∈ Z denoteits randomly shifted version introduced above. Furthermore, let e n = { n ∈A} and e e n = { n ∈ e A} , n ∈ Z , and let Υ := ( e n , ω n ) n ∈ Z and e Υ := (cid:0)e e n , e ω n (cid:1) n ∈ Z . Notice that this construction implies theidentity (cid:0)e e n , e ω n (cid:1) n ∈ Z = (cid:0) θ − M e n , θ − M ω n (cid:1) n ∈ Z , where M := ⌊ U d ⌋ . For sparse environments induced by a renewal sequence a n , the dual environment canbe defined equivalently in a rather explicit manner as a functional of an auxiliary Markov4hain. We will exploit this alternative construction in Section 4.3. The uniqueness of thedual environment (which implies, in particular, that the alternative construction yields thesame dual) follows from the reverse “stationary to cycle-stationary” Palm duality described,for instance, in [36, Theorem 1].The following theorem is an adaptation to our setting of the classical Palm dualities [35,Chapter 8] between the distribution of Υ under P and the distribution of e Υ under a measure Q equivalent to P . Theorem 2.1 (see Theorem 2 in [36]) . Assume that ( λ n , d n ) n ∈ N is a stationary ergodicsequence under P and E P ( d ) < ∞ . Define a new probability measure Q on the Borel subsetsof the product set (cid:0) { , } × (0 , (cid:1) Z by setting dQdP ( υ ) = d ( υ ) E P ( d ) , υ ∈ (cid:0) { , } × (0 , (cid:1) Z . Then:(a) ( e e n , e ω n ) n ∈ Z is a stationary and ergodic sequence under Q. (b) P ( A ∈ · ) = Q (cid:0) e A ∈ · | ∈ e A (cid:1) . We remark that although the claim that the sequence e Υ n = ( e e n , e ω n ) is ergodic is notexplicitly made in [36], it can be deduced, for instance, from the result of Exercise 1 in[22, p. 56]. The following is a straightforward corollary to Theorem 2.1. For the sake ofcompleteness, the proof is given in the Appendix. Corollary 2.2.
Under the conditions of Theorem 2.1, we have:(a) E P ( d ) = E P ( d ) · E Q ( d ) . (b) E Q ( d ) = 2 E Q ( e a ) + 1 . We remark that the identity E Q ( d ) = 2 E Q ( a ) + 1 can be thought of as a variation ofthe “waiting time paradox” of the classical renewal theory [8]. In this section, we state the basic limit theorems that describe the asymptotic behavior ofthe random walk X n , while the proofs are provided in Section 4. We first state recurrenceand transience criteria for RWSRE. Let σ = 0 and σ n = inf { k ∈ N : k > σ n − and X k ∈ A} . Thus ( σ n ) n ∈ N consists of the times of successive visits of X n to the random point set A . Define a nearest-neighbor random walk ( X n ) n ≥ on Z by setting X n = k if and only if X σ n = a k . (4)5aking into account the solution of the gambler’s ruin problem for the simple symmetricrandom walk, we note that X n is a RWRE with quenched transition probabilities given by P ω ( X n +1 = j | X n = i ) = ξ i · d i if j = i + 1(1 − ξ i ) · d i − if j = i − ξ i · d i − d i + (1 − ξ i ) · d i − − d i − if j = i . (5)Moreover, lim sup n →∞ X n = lim sup n →∞ X n and lim inf n →∞ X n = lim inf n →∞ X n , P − a. s.Thus, under very mild conditions, recurrence and transience criteria for the RWSRE X n canbe derived directly from the corresponding criteria for the RWRE X n (see, for instance, [37,Theorem 2.1.2] for the latter). More precisely, we have: Theorem 3.1.
Suppose that the following three conditions are satisfied:1. The sequence of pairs ( d n , λ n ) n ∈ Z is stationary and ergodic2. E P (log ξ ) exists (possibly infinite)3. E P (cid:0) log d ) < ∞ . Then:(a) E P (log ξ ) < implies lim n →∞ X n = + ∞ , P − a. s. (b) E P (log ξ ) > , implies lim n →∞ X n = −∞ , P − a. s. (c) E P (log ξ ) = 0 implies lim inf n →∞ X n = −∞ and lim sup n →∞ X n = + ∞ , P − a. s.Theorem 3.1 implies that as long as E P (log d ) is finite, the sparse environment ω in-duces the same recurrence-transience behavior as the underlying random environment λ. The following theorem shows that the opposite phenomenon occurs when E P (log d ) = + ∞ .Namely, the properties of λ are essentially irrelevant to the basic asymptotic behavior of X n . Theorem 3.2.
Suppose that the following conditions hold:1. The sequence of pairs ( d n , λ n ) n ∈ Z is stationary and ergodic2. The random variables d n are i.i.d.3. E P (cid:0) | log ξ | (cid:1) < + ∞ while E P (log d ) = + ∞ . Then, lim inf n →∞ X n = −∞ and lim sup n →∞ X n = + ∞ , P − a. s.We remark that ( d n ) n ∈ N is not necessarily independent of ( λ n ) n ∈ Z . The proof of Theo-rem 3.2 is given in Section 4.1. 6 .2 Transient RWSRE: asymptotic speed We now turn our attention to the law of large numbers for X n . Whenever it exists, lim n →∞ X n /n is referred to as the asymptotic speed of the random walk. Let T = 0 and for n ∈ N ,T n = inf { k ≥ X k = n } and τ n = T a n − T a n − . (6)Let S = 1 + 2 ∞ X i =0 i Y j =0 ξ j and F = 1 + 2 ∞ X i =1 i − Y j =0 ξ − − j . (7)We have the following: Theorem 3.3.
Let the conditions of Theorem 3.1 hold. Suppose in addition that ( d n ) n ∈ Z isindependent of ( λ n ) n ∈ Z under P. Then the asymptotic speed of the RWSRE exists P − a. s. Moreover, P (cid:0) lim n →∞ X n /n = v P (cid:1) = P (cid:0) lim n →∞ T n /n = 1 / v P (cid:1) = 1 , where v P ∈ ( − , is a constant whose reciprocal v − P is equal to P = { lim n →∞ X n =+ ∞} h VAR P ( d ) E P ( d ) + E P ( S ) · E P ( d ) i (8) − { lim n →∞ X n = −∞} · h VAR P ( d ) E P ( d ) + E P ( F ) · E P ( d ) i , P − a. s.Notice that if λ i (and hence ξ i ) are i.i.d., then (8) reduces to1v P = { lim n →∞ X n =+ ∞} · h VAR P ( d ) E P ( d ) + E P ( d ) · E P ( ξ )1 − E P ( ξ ) i − { lim n →∞ X n = −∞} · h VAR P ( d ) E P ( d ) + E P ( d ) · E P ( ξ )1 − E P ( ξ ) i , P − a. s.In order to compare (8) with the corresponding result for the regular RWRE, note that iflim n →∞ X n = + ∞ , P − as, then (8) reduces to1v P = VAR P ( d ) E P ( d ) + E P ( d ) · E P ( S ) , P − a. s.Recall the dual environment e ω defined in Section 2, and let e ρ n = 1 − e ω n e ω n , n ∈ Z , and e S = ∞ X i =1 (1 + e ρ − i ) i − Y j =0 e ρ − j + 1 + e ρ . (9)It is well known that the asymptotic speed of the usual RWRE is given by 1 (cid:14) E P (cid:0) e S (cid:1) (see,for instance, [37, Theorem 2.1.9]). The proof of the following proposition is straightforward,and it is provided in the Appendix. 7 roposition 3.4. Let the conditions of Theorem 3.3 hold. Suppose in addition that1. lim n →∞ X n = + ∞ , P − a. s. E P ( d ) < ∞ . Then, v P = 1 (cid:14) E Q (cid:0) e S (cid:1) . We remark that a proposition similar to Proposition 3.4 can be obtained when the randomwalk is transient to the left (i.e., when lim n →∞ X n = −∞ , P − a. s.) by replacing e ρ − k with ρ − k in the formula (9) for e S. Theorem 3.3 immediately yields the following version of Theorem 1.3 and Corollary 1.4in [23]. For any constants µ > ν ≥
0, we denote by P ◦ µ,ν the set of distributions( λ n , d n ) n ∈ Z for which the conditions of Theorem 3.3 hold and E P ( d ) = µ and 1 (cid:14) E P ( S ) = ν. We then have:
Corollary 3.5. max P ∈P ◦ µ,ν v P = ν/µ. Furthermore, the maximum is attained at P ∈ P ◦ µ,ν ifand only if VAR P ( d ) = 0 . Combining this result with [34, Theorem 4.1], we obtain the following corollary. For anyconstants µ > b <
0, we denote by P ∗ µ,ν the set of distributions ( λ n , d n ) n ∈ Z for whichthe conditions of Theorem 3.3 hold and E P ( d ) = µ and E P (log ξ ) = b. Corollary 3.6.
We have: max P ∈P ∗ µ,ν v P = 1 µ · − e b e b . Furthermore, the maximum is attained at P ∈ P ◦ µ,ν if and only if VAR P ( d ) = VAR P ( λ ) = 0 , in which case λ = e b , P − a. s.The slowdown of a one-dimensional random walk in a random environment, as comparedto a simple random walk, is a well-known general phenomenon [11, 32, 37] that can beexplained heuristically by fluctuations in the associated random potential. For example,a random walk transient to the right will quickly pass stretches of the environment that“push” it forward, but will be “trapped” for a long time in atypical stretches that “push” itbackward. The situation is different in higher dimensions. See, for instance, [26]. The aim of this section is to derive non-Gaussian limit laws for transient random walks in asparse random environment. The existence of the stationary dual environment suggests thatthe limit theorems can be first obtained for the random walk in the dual environment andthen translated into the corresponding results for the RWSRE. In what follows, we adopt8his approach even though it has the shortcoming of restricting our derivation to a class ofi.i.d. environments for which stable laws in the dual setting are known. It appears plausiblethat alternative methodologies, which would be considerably more technically involved, suchas a direct generalization of the “branching process” approach of [16, 18], or an adaptationof the “random potential” method developed in [10], would allow to extend the resultspresented in this chapter to a larger class of i.i.d. environments (and also perhaps to someMarkov-dependent environments).We will adopt here the following set of assumptions:
Assumption 3.7. (A1) ( λ n ) n ∈ Z is an i.i.d. sequence(A2) ( d n ) n ∈ Z is an i.i.d. sequence independent of ( λ n ) n ∈ Z (A3) P ( ǫ < λ < − ǫ ) = 1 for some ǫ ∈ (0 , / . (A4) For some κ > , E P ( ξ κ ) = 1 (10) (A5) There exists a constant M > such that P ( d < M ) = 1 . (A6) The distribution of log ξ is non-arithmetic, that is P (log ξ ∈ α Z ) < for all α ∈ R . Notice that (A4) implies by Jensen’s inequality that E P (log ξ ) ≤ . In view of (A6),the inequality is strict and hence the random walk is transient to the right. We remarkthat although condition (A5) appears to be required for our proof, it is likely that it can berelaxed or even omitted.For any κ ∈ (0 ,
2] and b >
0, we denote by L κ,b the stable law of index κ with thecharacteristic function log b L κ,b ( t ) = − b | t | κ (cid:16) i t | t | f κ ( t ) (cid:17) , (11)where f κ ( t ) = − tan π κ if κ = 1 and f ( t ) = 2 /π log t. With a slight abuse of notation weuse the same symbol for the distribution function of this law. If κ < , L κ,b is supported onthe positive reals, and if κ ∈ (1 , , L κ,b has zero mean [27, Chapter 1]. For κ = 2 , the law L ,b is a normal distribution with zero mean and variance equal to 2 b. We have:
Theorem 3.8.
Suppose that Assumption 3.7 is satisfied. Then the following hold for some b > (i) If κ ∈ (0 , , then lim n →∞ P ( n − κ X n ≤ z ) = 1 − L κ,b ( z − /κ ) , (ii) If κ = 1 , then lim n →∞ P (cid:0) n − (log n ) ( X n − δ ( n )) ≤ z (cid:1) = 1 − L ,b ( − z ) , for suitable A > and δ ( n ) ∼ ( A log n ) − n, (iii) If κ ∈ (1 , , then lim n →∞ P (cid:0) n − /κ ( X n − n v P ) ≤ z (cid:1) = 1 − L κ,b ( − z ) , (iv) If κ = 2 , then lim n →∞ P (cid:0) ( n log n ) − / ( X n − n v P ) ≤ z (cid:1) = L ,b ( z ) . κ > , then the standard CLTholds (it follows, e.g., from [37, Theorem 2.2.1]).For the hitting times T n , we have: Proposition 3.9.
Let the conditions of Theorem 3.8 hold. Then the following hold for some ˜ b > (i) If κ ∈ (0 , , then lim n →∞ P (cid:0) n − /κ T n ≤ t (cid:1) = L κ, ˜ b ( t ) , (ii) If κ = 1 , then lim n →∞ P (cid:0) n − ( T n − nD ( n )) ≤ t (cid:1) = L , ˜ b ( t ) , for suitable c > and D ( n ) ∼ c log n, (iii) If κ ∈ (1 , , then lim n →∞ P (cid:0) n − /κ (cid:0) T n − n v − P (cid:1) ≤ t (cid:1) = L κ, ˜ b ( t ) , (iv) If κ = 2 , then lim n →∞ P (cid:0) ( n log n ) − / ( T n − n v − P ) ≤ t (cid:1) = L , ˜ b ( t ) . It can be shown that the value of the parameter b in the statement of Theorem 3.8 is solelydetermined by the distribution of λ ; in particular, it is independent of the distribution of d ,provided that d satisfies the conditions of the theorem. This result may appear surprisingat first, especially in view of a large deviation interpretation of κ given in [37, Section 2.4](it is not hard to see that the rate functions of the random potentials associated with thesequences ξ n and ρ n are actually different). However, it can be explained in terms of theassociated branching process and the corresponding interpretation for κ . Furthermore, acareful inspection of the proof given in Section 4.3 shows that both parameters b and b of the limiting distributions are decreasing functions of E P ( d ) and increasing functions of V AR ( d n ) . This can be explained by the fact that b , in some rigorous sense, plays the role ofthe variance for the stable laws L κ,b ; see, for instance, the form of the characteristic functionin (11) and compare it to the characteristic function of a normal distribution. The goal of this section is to obtain a generalization of Sinai’s limit theorem for a classof recurrent RWSREs. The main result is stated in Theorem 3.11. A suitable normalizedrandom potential for the RWSRE is introduced in Lemma 4.5. The notion of a valley of therandom potential, which is essential for understanding the behavior of Sinai’s model [28, 37],is directly carried over to our setup. The proof of the main result is presented in Section 4.4.Sinai [28] studied a recurrent RWRE X n and showed that σ (log n ) X n ⇒ b ∞ , where b ∞ is a random variable which can be described as the “location of the deepest valley”of a Brownian motion. The proof of Sinai [28] uses a construction that implements the ideathat a properly scaled recurrent RWRE can be thought of as the motion of a particle in asuitably normalized random potential W n . The normalized potential converges to a Brownianmotion, and Sinai’s result shows a remarkable slowing down of the diffusive time scale. Thedensity function of the limit distribution b ∞ was characterized independently by Kesten [15]10nd Golosov [12, 13], who obtained that P ( b ∞ ∈ dx ) = 2 π ∞ X k =0 ( − k k + 1 exp n − (2 k + 1) π | x | o dx In this paper, we derive a limit theorem for a recurrent random walk in a sparse randomenvironment under the following assumption: Let α ∈ (0 ,
1) and assume that d is in thedomain of attraction of a stable law with index α. Namely, P ( d > t ) = t − α h ( t ) , t ≥ , (12)where h ( t ) is slowly varying at infinity, that is h ( λt ) ∼ h ( t ) as t goes to + ∞ for all λ > . In particular, we define S n = n − P n − k =1 log ξ k and assume the following: Assumption 3.10. E P (log ξ ) = 0 (recurrence)2. σ P := E P (log ξ ) ∈ (0 , ∞ ) √ n P [ nt ] k =1 log ρ k ⇒ B ( t ) P ( d > t ) = t − α h ( t ) , where α ∈ (0 , and h ( t ) is slowly varying. Recall that a function f : R + → R is said to be regularly varying of index α ∈ R if f ( t ) = t α h ( t ) for a slowly varying h : R + → R . We denote the set of all regularly varyingfunctions of index α by R α . We have the following:
Theorem 3.11.
Let Assumption 3.10 hold and fix any δ > . Then, there is a function u ∈ R /α such that that the following holds: For any ε > and δ ∈ (0 , , there is an integer n such that for all n > n there exist a set of environments C n ⊂ Ω and a random variable b n = b n ( ω ) such that P ( C n ) ≥ − δ and lim n →∞ P ω (cid:16)(cid:12)(cid:12)(cid:12) X n u (log n ) − b n (cid:12)(cid:12)(cid:12) > ε (cid:17) = 0 uniformly in ω ∈ C n . Moreover, as n → ∞ the probability distribution for b n convergesweakly to a non-degenerate limiting distribution b ∞ . We remark that Sineva [29, 31] obtained similar limit laws for different variations ofSinai’s model. In all these results, the limiting distribution of a properly scaled random walk X n admits a representation as the deepest valley of an auxiliary process, which in turn isobtained as the weak limit of a suitably defined random potential. For a definition of thenotion of a valley in this context, we refer the reader to [28] or [37]. In this section we study the “environment viewed from the particle” process ( θ X n ω ) n ≥ fora transient RWSRE. It is not hard to see that the pair ( θ X n ω, X n ) forms a Markov chain,which allows to consider X n as a functional (projection into the second coordinate) of a11arkov process. Even though the state space of this Markov chain is considerably large,the representation is useful due to the fact that the underlying Markov chain turns out tobe stationary and ergodic in the transient regime. The concept of the environment viewedfrom the particle was introduced by S. Kozlov in a broader context in [17] (see also [32] and[5, 6, 33]). In Section 3.2, we proved the existence of the asymptotic speed v P := lim n →∞ X n for RWSREs associated with a stationary and ergodic environment ( d n , λ n ) n ∈ Z by using adirect approach. In fact, using the techniques described in [37, Section 2.2] and the existenceof the dual environment, one can prove the following result. Similarly to (2), let ξ n = 1 − λ n λ n , n ∈ Z . (13)We have: Theorem 3.12.
Consider a random walk X n in a stationary and ergodic sparse environment ( λ n , d n ) n ∈ Z . Assume that E P (log ξ ) is well defined (possibly infinite) and E P ( d ) < ∞ . Then(a) v P > if and only if there exists a stationary distribution P ◦ equivalent to P for theMarkov chain ω n = θ X n ω, n ≥ . If such a distribution P ◦ exists it is unique and isgiven by the following formula: P ◦ ( B ) = v P E Q h E ω (cid:16) T − X n =0 { ω n ∈ B } (cid:17)i , B ∈ F , (14) where ω n := θ ξ n ω and Q is the distribution of the dual environment.(b) ( ω n ) n ≥ is an ergodic process under P ◦ := P ◦ ⊗ P ω . (c) dP ◦ dP = dQdP × Λ( ω ) = d · Λ( ω ) E P ( d ) , where Λ( ω ) := 1 ω h ∞ X i =1 i Y j =1 ρ j i = 1 ω h d + ∞ X i =1 d i +1 × i Y j =1 ξ j i . (15) (d) v P = 1 /E Q (Λ) = E P ( d ) E P ( d Λ) . With one exception, the proof of Theorem 3.12 follows along the lines of the correspondingresults in [37, Section 2.1] (namely, Lemmas 2.1.18, 2.1.20, 2.1.25, and Corollary 2.1.25therein). The only exception is the proof that the existence of the environment viewedfrom the position of the particle actually implies v P > . The latter can be obtained by astraightforward modification of the proofs of [5, Theorem 3.5 (ii)] or [25, Theorem 2.3] forinstance. The proof of Theorem 3.12 is therefore omitted.
Remark 3.13.
The asymptotic speed for the simple nearest-neighborhood random walk on Z with probability of jumps forward p and jumps backward q = 1 − p is ( p − q ) = 2 p − . Although v P is not equal to E Q (2 ω − , quite remarkably it turns out to be equivalent to E P ◦ (2 ω − (compare, for instance, with formula (2.1.29) in [37]). Proofs
Recall the definitions of ρ n and ξ n in (2). Furthermore, for a non-zero integer n let η n bethe number of marked sites within the closed interval I n = sign( n ) · [1 , n ] . More precisely, let η = 0 and η n = χ ( I n ∩ A ) = P nk =1 { k ∈A} if n > P nk =1 {− k ∈A} if n < . (16)Notice that by the ergodic theorem,lim | n |→∞ η n | n | = Q (0 ∈ A ) = 1 E P ( d ) , P − a. s. and Q − a. s. (17)Denote S ( ω ) = ∞ X k =1 ρ ρ · · · ρ k and F ( ω ) = ∞ X k =0 ρ − ρ − − · · · ρ − − k . To prove Theorem 3.2 it suffices (see, for instance, the proof of [37, Theorem 2.1.2]) to showthat the conditions of the theorem imply P (cid:0) S ( ω ) = F ( ω ) = + ∞ (cid:1) = 1 (18) Remark 4.1.
The functions S ( ω ) and F ( ω ) appear in the solution of the gambler’s ruinproblem for an infinite box. Therefore, they are related to the basic recurrence-transienceproperties of the random walk (see, e.g., [37, Theorem 2.1.2]). In particular, (18) impliesrecurrence. Toward this end, note that η a n = η a n +1 = . . . = η a n +1 − = n for n ≥ , and hence S ( ω ) = ∞ X k =1 ξ ξ · · · ξ η k = ( a −
1) + ∞ X n =1 ξ ξ · · · ξ n · d n +1 , (19)where, to claim the first identity, we used the standard convention that ξ ξ · · · ξ η k = 1 if η k = 0 . Similarly, η a n = η a n +1 = . . . = η a n +1 − = n + 1 for n < , and hence F ( ω ) = ∞ X k =1 ξ − ξ − − ξ − − · · · ξ − − η k = ∞ X n =0 ξ − ξ − − · · · ξ − − n · d − n . (20)Furthermore, the condition E P (log d ) = + ∞ implies that P ∞ n =1 P (log d > M · n ) = ∞ for any M > . Thus, since d n are i.i.d., it follows from the second Borel-Cantelli lemmathat P (log d n > M · n i. o.) = 1 for any M > . Hence, the ergodic theorem along with thecondition E P (cid:0) | log ξ | (cid:1) < + ∞ imply that, P − a. s. , lim sup n →∞ n log (cid:16) d n +1 · n Y k =1 ξ k (cid:17) = lim sup n →∞ n (cid:16) n X k =1 log ξ k + log d n +1 (cid:17) = + ∞ , which yields P (cid:0) S ( ω ) = + ∞ (cid:1) = 1 . A similar argument shows that, under the conditions ofthe theorem, P (cid:0) F ( ω ) = + ∞ (cid:1) = 1 and hence (18) holds, as desired.13 .2 Proof of Theorem 3.3 The proof is an adaption of the corresponding arguments for the regular RWRE. See, forinstance, [37, Section 2.2].Recall the definitions of T n and τ n in (16). We have: Lemma 4.2.
Assume that the conditions of Theorem 3.1 hold and suppose, in addition, that P (lim sup n →∞ X n = 1) . Then ( τ n ) n ∈ N is a stationary and ergodic sequence under the law P . Proof of Lemma 4.2.
Enlarge, if needed, the underlying probability space to include a se-quence of i.i.d. random variables γ = ( γ x,n ) x ∈ Z ,n ∈ N such that1. γ is independent of ω under the law P, and2. each random variable γ x,n is distributed uniformly on the interval [0 , . Let N denote the set of non-negative integers, that is N = N ∪ { } . For x ∈ Z and n ∈ N , let l x ( n ) = P nt =0 { X t = x } be the number of visits of the random walk to the site x by the time n. For n ∈ N , denote l n = l X n ( n ) , γ n = γ X n ,l n , and ω n = ω X n . Without loss of generality,we can assume that X n is defined recursively as follows: X n +1 = X n + { γ n <ω n } − { γ n ≥ ω n } . For n ∈ N , let C n = ( ξ x , d j , Γ x ) j,x ≤ n , where Γ x = ( γ k,i ) k ≤ x,i ∈ N . The sequence ( τ n ) n ∈ N definedby (6) is stationary under P because ( λ n , d n ) n ∈ Z is stationary and P ( T n < ∞ ) = 1 for all n > τ n ) n ∈ N becomes a deterministic function of ( C k ) k ≤ n . This completes the proof of the lemmasince the sequence ( C n ) n ∈ N is stationary and ergodic under P in the enlarged probabilityspace.Under the conditions of Lemma 4.2, the ergodic theorem yields T a n n = 1 n n X i =1 τ a i → E ( τ a ) as n → ∞ , P − a. s.We have Lemma 4.3.
Assume that the conditions of Theorem 3.1 hold and suppose, in addition, that λ and A are independent under P. Then:(a) E ( T a ) = VAR P ( d ) + E P ( S ) · [ E P ( d )] , (b) E ( T a − ) = VAR P ( d ) + E P ( F ) · [ E P ( d )] . Proof of Lemma 4.3.
We will only prove the result in (a), the proof of (b) being similar. Toevaluate T a , we will use a decomposition of the paths of the random walk according to itsfirst step: T a = 1 + { X =1 } [ { e T < e T a } ( e T + T ′ a ) + { e T > e T a } e T a ] (21)+ { X = − } [ { b T < b T a − } ( b T + T ′′ a ) + { b T > b T a − } ( b T a − + T ′ + T ′′′ a )] , e T = inf { n > T : X n = 0 } , e T + T ′ a = inf { n > e T : X n = a } , e T a = inf { n > T : X n = a } , b T = inf { n > T − : X n = 0 } , b T + T ′′ a = inf { n > b T : X n = a } , b T a − = inf { n > T − : X n = a − } , b T a − + T ′ = inf { n > b T a − : X n = 0 } , b T a − + T ′ + T ′′′ a = inf { n > T a − + T ′ : X n = a } . Taking quenched expectations E ω ( · ) in both sides of (6) yields E ω ( T a ) = 1 + λ [ E ( T ∧ T a ) + P ( T < T a ) E ω ( T a )]+ (1 − λ )[ E − ( T ∧ T a − ) + E ω ( T a ) + P − ( T a − < T ) E ω,a − ( T )] . Using the solution of the gambler’s ruin problem under P , we obtain λ a E ω ( T a ) = 1 + λ ( a −
1) + (1 − λ )( | a − | −
1) + 1 − λ | a − | E ω,a − ( T )= λ a + (1 − λ ) | a − | + 1 − λ | a − | E ω,a − ( T ) . Thus, 1 a E ω ( T a ) = a + ξ | a − | + ξ · | a − | E ω,a − ( T ) . Iterating yields 1 a E ω ( T a ) = a + 2 ∞ X k =0 ξ ξ − · · · ξ − k · d − k . Taking expectations with respect to P , and using a truncation argument similar to thatgiven in the proof of [37, Lemma 2.1.12] in order to verify when E ( T a ) < ∞ , we obtain E ( T a ) = VAR P ( d ) + (cid:2) E P ( d ) (cid:3) · E P ( S ) , as desired. This completes the proof of (a) of the lemma. Part (b) can be derived along thesame lines, and hence its proof is omitted.In view of Lemma 4.3, we are now in a position to finish the proof of Theorem 3.3.Variations of Lemma 4.4 below have appeared in a number of references in the field ofrandom walks in random environments. Nonetheless, we provide a proof for the reader’sconvenience. Lemma 4.4.
Assume that the conditions of Theorem 3.1 hold and suppose, in addition, that lim n →∞ T an n = α, P − a. s. , for some constant α ≤ ∞ . Then, lim n →∞ T n n = αE P ( d ) and lim n →∞ X n n = E P ( d ) α , P − a. s.15 roof of Lemma 4.4. First, observe that (16) implies a η n ≤ n < a η n +1 , P − a. s.Thus, in view of (17),lim n →∞ T n n = lim n →∞ T a ηn +1 n = lim n →∞ T a ηn η n · η n n = αE P ( d ) , P − a. s.Let now ζ ( n ) ∈ Z be the unique nonnegative random number such that T a ζ ( n ) ≤ n < T a ζ ( n )+1 . (22)Since X n is transient to the right, P (cid:0) lim n →∞ ζ ( n ) = ∞ (cid:1) = 1 . Furthermore, (22) implies that X n < a ζ ( n )+1 and X n ≥ a ζ ( n ) − ( n − T a ζ ( n ) ) . Thus, a ζ ( n ) n − (cid:16) − T a ζ ( n ) n (cid:17) ≤ X n n < a ζ ( n )+1 n . But (22) along with the existence of lim n →∞ nT n yieldlim n →∞ a ζ ( n ) n = lim n →∞ a ζ ( n ) T a ζ ( n ) = lim n →∞ nT n = E P ( d ) α , P − a. s.Hence, E P ( d ) α ≤ lim inf n →∞ X n n ≤ lim sup n →∞ X n n ≤ E P ( d ) α , which implies the result in Lemma 4.4.The proof of Theorem 3.3 is complete. The proof uses the dual Markovian environment and the reduction to stable limit laws forrandom walks in a Markovian environment obtained in [18]. Recall the definition of T n in(6), and observe that the distribution of the trajectory ( X n ) n ∈ N under the law P coincideswith the distribution of ( X T a + n − a ) n ∈ N under Q. Moreover, (cid:12)(cid:12) X n − ( X T a + n − a ) (cid:12)(cid:12) ≤ a + (cid:12)(cid:12) X n − X n + T a (cid:12)(cid:12) ≤ a + T a . Since the random walk is transient to the right, Q ( T a < ∞ ) = P ( T a < ∞ ) = 1 . Therefore, c n · (cid:12)(cid:12) X n − ( X T a + n − a ) (cid:12)(cid:12) converges to zero in distribution (under the law Q ) for any sequenceof scaling factors ( c n ) n ∈ N such that lim n →∞ c n = ∞ . Thus it suffices to prove the stable limitlaws under Q. Y n = n − a η n = n − sup { k ∈ Z : k ≤ n and k ∈ A} , n ∈ Z . Notice that Y a k = 0 , k ∈ N , and Y n +1 − Y n = 1 if a η n ≤ n < a η n +1 . Let Z + denote the set of non-negative integers, that is Z + = N ∪ { } . If Assumption 3.7holds, then the sequence Y = ( Y n ) n ∈ Z is a positive-recurrent Markov chain on Z + under thelaw Q. Furthermore, the transition kernel H ( x, y ) = Q ( Y n +1 = y | Y n = x ) is given by H ( x, y ) = P ( d >x +1) P ( d >x ) if y = x + 1 , x ∈ Z + P ( d = x +1) P ( d >x ) if y = 0 , x ∈ Z + Y is non-homogeneous under the law P. It follows from The-orem 2.1 that Y is a stationary Markov chain on Z + under the law Q , and the initialdistribution of Y is the (unique) invariant distribution of H. That is, using the notation ofTheorem 2.1, Q ( Y = x ) = P (cid:0) ⌊ U d ⌋ = x (cid:1) = P ( d > x ) E P ( d ) , x ∈ Z + . It then follows that under Q, the sequence ( Y, ω ) constitutes a two-component Markov chainwith transition kernel depending only on the current value of the first component (but notof the second). More precisely, with probability one, Q ( Y n +1 = y, ω n +1 ∈ A | Y n = x, ω n = u ) = H ( x ; y, A ) , where the stochastic kernel H on Z + × ( Z + × Ω) is given by H ( x ; y, A ) = H ( x, y ) · (cid:0) { ∈ A } · { y =0 } + P ( λ ∈ A ) · { y> } (cid:1) , A ∈ B (cid:0) [0 , (cid:1) . Clearly, the reverse chain ( Y n ) n ∈ Z is an irreducible Markov chain in the finite state space { , , . . . , M − } . In view of the main results of [18], in order to establish that the claimsof Theorem 3.8 and Proposition 3.9 hold for the RWSRE X n and the hitting times T n , itsuffices to verify the following set of conditions for the stationary Markov chain Y n and theassociated Markov hidden model ( Y − n , λ − n , d − n ) n ∈ Z :(B1) lim sup n →∞ n log E Q (cid:16)Q n − i =0 ρ β − i (cid:17) ≥ n →∞ n log E Q (cid:16)Q n − i =0 ρ β − i (cid:17) < β > β > . (B2) The process q n = log ρ − n is non-arithmetic relative to ( x n ) in the following sense: theredo not exist a constant α > γ : S → [0 , α ) such that Q (cid:0) q ∈ γ ( x − ) − γ ( x ) + α · Z (cid:1) = 1 .
17o this end, observe that condition (B1) holds (compare with [18]) because the followingholds true: By virtue of Theorem 2.1lim sup n →∞ n log E Q n − Y i =0 ρ β − i ! = lim sup n →∞ n log E Q n Y i =1 ρ βi ! = lim sup n →∞ n log E P n Y i =1 ρ βi ! , the function Λ( β ) := lim sup n →∞ n log E (cid:16)Q n − i =0 ρ β − i (cid:17) is convex, Λ(0) = 0 , Λ ′ (0) = E p (log ρ ) < , the distribution of ρ is non-degenerate, and E P (cid:16) n Y i =1 ρ κi (cid:17) = E P (cid:16) η n Y i =0 ξ κi (cid:17) = E P (cid:16) E P (cid:16) η n Y i =0 ξ κi | η n (cid:17)(cid:17) = E P (cid:16)(cid:0) E P ( ξ κ ) (cid:1) η n (cid:17) = E P (1 η n ) = 1 . Furthermore, (A6) of Assumption 3.7 along with the fact that ξ n are i.i.d. trivially implies(B2). Notice that the measure Q is absolutely continuous with respect to P, and therefore P (cid:0) q ∈ γ ( x − ) − γ ( x ) + α · Z (cid:1) = 0 guarantees Q (cid:0) q ∈ γ ( x − ) − γ ( x ) + α · Z (cid:1) = 0 . The proofis complete.
Let D ( R ) denote the set of real-valued c`adl`ag functions on [0 ,
1] equipped with the Skorokhod J -topology. We use the notation ⇒ to denote the weak convergence in D ( R ) . We have: U n ( t ) := 1 r n ⌊ nt ⌋ X k =1 d k converges weakly to G α ( t ) , where G α ( t ) is a totally asymmetric stable process with E (cid:0) e iθG α (1) (cid:1) = exp n −| θ | α (cid:16) − i sign( θ ) tan (cid:16) πα (cid:17)(cid:17)o , θ ∈ R . The key ingredient of the proof is the following (functional) limit theorem for a suitablydefined random potential.
Lemma 4.5. (a) Assume that condition (12) holds with α ∈ (0 , . Then, as n → ∞ , n ⌊ u (log n ) t ⌋ X k =1 log ρ k ⇒ V α , for some sequence u ( n ) ∈ R α . b) If E ( d ) < ∞ , then √ n ⌊ nt ⌋ X k =1 log ρ k ⇒ µσ P W, where W is a standard Brownian motion and V α = σ P W ( G − α ) . Proof of Lemma 3. (a)
Let U n ( t ) := 1 r n ⌊ nt ⌋ X k =1 d k = 1 r n a ⌊ nt ⌋ and R n ( t ) := 1 √ n ⌊ nt ⌋ X k =1 log ξ k . It follows from the assumptions of the lemma that, as n → ∞ ,U n ( t ) ⇒ G α and R n ( t ) ⇒ σ P W. for some sequence r n ∈ R α +1 . Let U − n = n − · η ( ⌊ tr n ⌋ ) and G − α be the inverses in D ( R + , R ) of U n and G α , respectively. Then the convergence of U n and R n , along with their independenceof each other, imply (see, for instance, the derivation of the formula (2.29) in [14]) that in D ( R + , R ) , (cid:16) R n ( t ) , n η ( ⌊ tr n ⌋ ) (cid:17) ⇒ (cid:0) σ P W, G − α (cid:1) , as n → ∞ . Since the paths of the Brownian motion are continuous, it follows from a random changelemma in [2, p. 151] that R n (cid:0) n η ( ⌊ tr n ⌋ ) (cid:1) ⇒ σ P W ( G − α ( t )) in D ( R + , R ) . That is,1 √ n η ([ r n t ]) X k =1 log ξ k ⇒ σ P W ( G − α ) , as n → ∞ , and, passing to the subsequence n k = log k, k ∈ N , we obtain1log k η ([ r log2 k t ]) X i =1 log ξ i = 1log k [ r log2 k t ] X i =1 log ρ i ⇒ σ P W ( G − α ) , as k → ∞ . To conclude the proof of part (a), notice that r n ∈ R α implies that r log n = u (log n ) forsome sequence u ( n ) ∈ R α , as desired. (b) We now turn to the proof of part (b) of the theorem. Since E p ( d ) < ∞ , the renewaltheorem implies η n n → C = 1 E p ( d ) . Therefore (see, for instance, Theorem 14.4 in [2, p. 152]),1 √ n [ nt ] X k =1 log ρ k = 1 √ n η ([ nt ]) X k =1 log ξ k = p η ([ nt ]) √ n p η ([ nt ]) η ([ nt ]) X k =1 log ξ k ⇒ µσ P B ( · ) , where the convergence is the weak convergence in the Skorohod space D ( R + ) . b R n ( t ) = sign( t ) · n [ u (log n ) t ] X k =1 log ρ k = sign( t ) · n η [ u (log n ) t ] X k =1 log ξ k . (23)By Lemma 3, { b R n ( t ) : t ≥ } converges weakly in D ( R + , R ) to the process V α . One cannow proceed as in [29] in order to establish Theorem 3.11. In fact, the proof of the mainresult in [29] is an adaptation of the original argument of Sinai [28] to a situation wherethe random potential b R n in the form given by (23) converges weakly to a non-degenerateprocess in D ( R + , R ) . We remark that a somewhat shorter derivation of Theorem 3.11 canbe obtained by an adaptation of a version of Sinai’s argument given in [37, Section 2.5].The version of the proof in this paper follows the approach of [12] and is due to Dembo,Guionnet, and Zeitouni. (a)
By part (b) of Theorem 2.1, E Q ( d ) = Z d · d E P ( d ) dP = E P ( d ) E P ( d ) , which is equivalent to the the first claim in the corollary. (b) By definition, e a = ⌊ U d ⌋ , where U is independent of d under Q. We therefore have, E Q ( e a ) = E Q (cid:0) E Q ( e a | d ) (cid:1) = E Q (cid:16) d d − X k =0 j (cid:17) = 12 (cid:0) E Q ( d ) − (cid:1) , which verifies the second claim in the corollary. Recall S from (7) and e S from (9). First, notice that e S = ∞ X i =1 i − Y j =0 e ρ − j + ∞ X i =1 i Y j =0 e ρ − j + 1 + e ρ = 1 + 2 ∞ X i =0 i Y j =0 e ρ − j . Next, observe that the following version of the identity (19) can be stated in terms of the“tilde environment” ( e a n , λ n ) n ∈ Z : ∞ X n =0 e ρ e ρ · · · e ρ n = e a + ∞ X n =0 ξ ξ · · · ξ n · d n +1 . (24)20herefore, using the stationarity of the environment under Q along with the identity (24)and part (b) of Corollary 2.2, E Q (cid:0) e S (cid:1) = 1 + 2 E Q (cid:16) ∞ X i =0 i Y j =0 e ρ i (cid:17) = E Q (2 e a + 1) + E P ( d ) · (cid:2) E Q ( S ) − (cid:3) = E Q ( d ) + E P ( d ) · (cid:2) E Q ( S ) − (cid:3) = E P ( d ) E P ( d ) + E P ( d ) · (cid:2) E Q ( S ) − (cid:3) = VAR P ( d ) E P ( d ) + E P ( d ) · E P ( S ) , which together with Theorem 3.1 imply that the asymptotic speed of the RWSRE on theevent { lim n →∞ X n = + ∞} is 1 (cid:14) E Q (cid:0) e S (cid:1) , P − a. s. , under the conditions of Theorem 3.3. Acknowledgements
The research of Anastasios Matzavinos is supported in part by the National Science Founda-tion under Grants NSF CDS&e-mss 1521266 and NSF CAREER 1552903. The research ofAlexander Roitershtein is supported in part by the Simons Foundation under CollaborationGrant
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