Quenched tail estimate for the random walk in random scenery and in random layered conductance II

This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102–128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.


Introduction and main results
This paper is a continuation of our earlier work [16]. In that paper, we studied upper deviation estimates for the random walk in random scenery in the case where the random scenery is non-negative and has a Pareto distribution. The results were applied to establish tail estimates for a random conductance model with a layered structure.
There are two situations left incomplete in [16] for the random walk in random scenery. The first is the regime where the upper deviation probability exhibits a power law decay. Such a regime was proved to exist in d ≥ 3 but the precise decay rate has not been obtained. The second is the lower deviation estimate. In this paper, we give the precise estimates in the first regime (Theorem 1.1), that is needed for the asymptotic of the moderate deviation and the Green function for the random conductance model explained below, and a partial result for the second problem of lower deviation (Proposition 1.2).
As in the previous work, these results have applications to a random conductance model with a layered structure (see Section 1.2 for the precise formulation). In fact, one of our main motivation is to show that in contrast with the standard situation, where the random conductance are independent and identically distributed (cf. [4]), the upper tail of distribution of the conductance in a layered structure yields an anomalous behavior of the heat kernel and the Green function. The first application is to the ondiagonal behavior of the heat kernel. We determine the so-called spectral dimension of the integer lattice weighted by the random conductance, and prove a sharp criterion for the recurrence and transience (Theorem 1.3). The spectral dimension exhibits a non-standard behavior as soon as the first moment of the conductance is infinite. The second application is to a moderate deviation estimate. Similarly to the random walk in random scenery, we found a power law decay regime in [16], and in this paper the sharp asymptotics is determined (Theorem 1.6). To the best of our knowledge, this slow decay of moderate deviation probability is a new phenomenon in the context of random walk in random environment. The last application is to derive estimates for the Green function (Theorem 1.7). In contrast with the spectral dimension, the decay of the Green function exhibits a non-standard behavior even for finite mean conductance when d ≥ 5.

Results for random walk in random scenery
Let ({z(x)} x∈Z d , P) be a family of non-negative, independent and identically distributed random variables whose law satisfies P(z(x) > r) = r −α+o(1) as r → ∞ (1.1) for some α > 0. The random walk in random scenery is the additive functional of the continuous time simple random walk ((S t ) t≥0 , (P x ) x∈Z d ) on Z d defined as follows: A(t) = t 0 z(S u ) du. (1.2) For the background and related works, see the introduction of [22,16]. Among many results, the asymptotic behaviors of A(t) in our setting are studied in [23,12], which say that A(t) scales like t s(d,α)+o(1) with s(d, α) = α+1 2α ∨ 1, d = 1, under the product measure P⊗P 0 . In fact, finer distributional limit results are established in [10,11,23,12].
In this paper, we study quenched tail estimates for A(t), that is, conditionally on z.
The following upper tail estimates are proved in the previous work [16]: Theorem A (Theorem 1 in [16]). Let ρ > α+1 2α ∨ 1 for d = 1 and ρ > d 2α ∨ 1 for d ≥ 2. Then there exists p(α, ρ) > 0 such that P-almost surely, P 0 (A(t) ≥ t ρ ) = exp −t p(α,ρ)+o (1) (1.4) as t → ∞. Furthermore, when d = 1 and ρ < α+1 2α ∨ 1 or d ≥ 2 and ρ < d 2α ∨ 1, P-almost surely the above probability is bounded from below by a negative power of t. In this paper, we provide sharp estimates in the power law decay regime of Theorem A and an estimate for a lower deviation probability.
Let us start with the upper deviation in the case d ≥ 3. The reason why we have a power law decay in the regime d ≥ 3 and ρ < d 2α ∨ 1 can be explained as follows: The random walk can reach any point inside [−t 1/2 , t 1/2 ] d with probability ct −d/2 . Since the highest value of z-field in this box is t d 2α +o (1) , the deviation up to this value can be achieved with probability at least ct −d/2 . The first result in this paper provides a sharp estimate in this regime. Theorem 1.1. Let d ≥ 3 and α < d/2. Then for any ρ ∈ ( 1 α ∨ 1, d 2α ), (1) as t → ∞ (1.6) and P 0 (A(t) ≥ t ρ , S t = 0) = t −αρ+1− d 2 +o(1) as t → ∞. (1.7) If α > 1 in addition, then the same bounds hold with t ρ replaced by ct for any c > E[z(0)].
In the proof, we will see that these asymptotics are coming from a simple specific strategy: the random walk visits the intermediate level set {x ∈ Z d : z(x) ≥ t ρ } and spends a unit time there. The right-hand side of (1.6) turns out to be the probability for the random walk to visit that level set before time t.
In the remaining case where d = 1, 2 and the conditions in the last part of Theorem A are satisfied, we expect from (1.3) that the left-hand side of (1.4) tends to one as t → ∞. This is confirmed by the following lower tail estimate. It shows that, in contrast with the upper deviation, any small lower deviation causes a stretched exponential decay in all dimensions.
It would be an interesting problem to determine the precise decay rate as a function of .

Results for random walk in layered conductance
Let us define the conductance field on Z 1+d by This is the variable speed random walk in the random conductance field ω. One of the primary interests in this type of model is the central limit behavior. When E[z(0)] < ∞, weak convergence results, such as the quenched functional central limit theorem, are relatively easy to establish since the environment is balanced and reversible for the environment viewed from the particle process. A similar model in the discrete time is called "toy model" in [8,Section 2.3]. In this paper, we will focus on the local central limit theorem type results. Our result in fact covers the case E[z(0)] = ∞, where the heat kernel exhibits anomalous behaviors. This model is related to the random walk in random scenery through the following representation. Let ((S 1 t , S 2 t ) t≥0 , (P x ) x∈Z 1+d ) be the continuous time simple random walk on Z 1+d which jumps to each of the neighboring sites at rate one, where S 1 is the first one-dimensional component and S 2 is the remaining d-dimensional component. Then the process ((X t ) t≥0 , (P ω x ) x∈Z 1+d ) has the representation (1.11) where the clock process is defined by A 2 (t) = t 0 z(S 2 u ) du. In particular, it follows that X t scales like t s(d,α) 2 in e 1 -direction and t 1 2 in the other directions. If we define a distance between two points x = (x 1 , x 2 ) and y = (y 1 , with | · | denoting the Euclidean norm on Z d , then X t obeys the diffusive t 1 2 scaling in this distance. We will see that this distance plays a similar role as the so-called intrinsic distance; see [2] for a recent study in the random conductance setting. The representation (1.11) plays a key role in the proofs of following results.
Our first result concerns the behavior of the on-diagonal heat kernel. Roughly speaking, its behavior is similar to the case ω ≡ 1 when α ≥ 1, and different when α < 1. (1.14) as t → ∞. In particular, when d = 1, (X t ) t≥0 is transient if α < 1 and recurrent if E[z(0)] < ∞, and when d ≥ 2, it is always transient.

Remark 1.4.
(i) By irreducibility of the random walk, the return probability P ω x (X t = x) has the same asymptotics as in (1.13) for all x ∈ Z d , P-almost surely. This implies that when α < 1, the weighted graph (Z 1+d , ω) has the spectral dimension which is strictly greater than 1 + d.
(ii) The on-diagonal behavior of the heat kernel is often related to the volume (cardinality) of an intrinsic distance ball with radius t 1 2 . In our setting, the above asymptotics are the same as the volume of the ball with respect tod in (1.12) with radius t 1 2 .
The right-hand side of (1.14) coincides with the asymptotics of P E[ω] 0 (X t = 0), the return probability for the random walk in the averaged conductance. It is natural to ask when this extends to the local central limit theorem. The following result answers affirmatively when α > d 2 ∨ 1, and negatively when α < d 2 ∨ 1. Proposition 1.5. For any R > 0, the following hold P-almost surely: Our next result is about the decay rate of P ω 0 (X t = t δ e 1 ). When δ > s(d,α) 2 ∨ 1 2 , this is a large or moderate deviation probability, and in [16,Theorem 4], it is proved to decay stretched exponentially except when d ≥ 3, α < d 2 and δ < d 4α . In the last regime, unlike in the independent and identically distributed conductance case (cf. [4]), it is shown to exhibit a power law decay in [16,Theorem 4]. The following theorem determines the exponent.
While this power law decay is new for nearest neighbor walks, it is standard for random walks with unbounded jumps. In our setting, the unbounded conductance along lines play a similar role to a long distance jump. More precisely, the above asymptotics is the same as the probability of the strategy explained in Figure 1. Figure 1: A strategy of the random walk that gives the lower bound for P ω 0 (X t = t δ e 1 ).
The thick polyline represents the range of the random walk. The second coordinate walk X 2 visits y ∈ Z d with z(y) ≥ t 2δ , spends a unit time there, and then come back to the origin. While the second coordinate is y, the first coordinate walk X 1 moves in a speed faster than t 2δ and hence can reach t δ = √ t 2δ in the unit time.
Based on the above results on heat kernel asymptotics, we can derive estimates on the Green function defined by (1.20) Note that our random walk is transient and g ω is well-defined if α < 1 or d ≥ 2. When d ≥ 3, we have the standard exponent for α ≥ d 4 ∨ 1, and non-standard exponents for α < d 4 ∨ 1 depending on d and α. When d = 2, the decay exponent is always −1, which is the same as for the three dimensional simple random walk, regardless the value of α.
The non-standard behaviors are caused by the power law decay in Theorem 1.6.
(ii) In the case d ≥ 5 and α < d 4 , the Green function in the direction e 1 decays slower thand(0, ne 1 ) 2−ds+o(1) as n → ∞. In particular for α ∈ (1, d 4 ), the exponent is strictly smaller than 1 − d although the quenched functional central limit theorem holds. This looks strange but there is no contradiction since this slow decay is specific to the e 1 -direction. We elaborate more on this singularity in direction in Remark 9.1.
Let us compare Theorems 1.3 and 1.7 with recent results for more general ergodic random conductance models. The quenched local central limit theorem is proved in [1, Theorem 1.11, Remarks 1.5 and 1.12] under a moment condition which in our case reads as (1.23) The as t → ∞. We dropped the superscript in the last formula since it involves only the second random walk S 2 . If we formally substitute (1.3), that is, into (1.24), we obtain Theorem 1.3. Thus our task is to justify this substitution by controlling the upper and lower deviations of A(t) away from the above scaling for P-almost every z. In view of (1.24), the lower and upper bound for P ω 0 (X t = 0) are related to the upper and lower tail estimate for the random walk in random scenery, respectively. Before closing this introduction, we mention two variants of our model. The first is a multidimensional layer model which can be covered by our method. (1.26) All the above results have extensions to this setting and we list them below, though we give proofs only in the simplest Z 1+d case for brevity. In the following list, we assume d 1 ≥ 2 and state the results with o(1) in the exponents which is not always necessary.
On-diagonal estimate: For any d 1 ≥ 2, P-almost surely, (1.28) In particular, the random walk is always transient in this case.
Off-diagonal estimate: (1.30) Green function estimate: For any x 1 ∈ R d1 \ {0}, P-almost surely, This process jumps to a neighboring site with probability proportional to ω(x, ·) and with rate one, hence called the constant speed random walk. This can be realized as a time change of the variable speed random (1.33) In the last expression, A 2 (B −1 (t)) is comparable to t, up to multiplicative constant, and hence the first coordinate behaves more or less like a simple random walk on Z d1 for large time. The second coordinate is a well-studied process called the Bouchaud trap model. For quenched results, we refer the reader to [6] for functional limit theorems and [14] for a two-sided heat kernel estimate. Though the long time behaviors in Theorems 1.3 and 1.6 become different by the time change, the decay of Greens function in Theorem 1.7 is unchanged sinceg ω (x, y) = g ω (x, y)ω(y). This allows us to formally compute the spectral dimension for the constant speed random walk on Z d1+d2 as The detail is given in Remark 9.1 (iii).

Notation convention
In the proofs, we will use c and c to denote positive constants depending only on d and α, whose values may change from line to line. We write

Overview of the paper
Since we have various results depending on the parameters, the proofs split into many cases. We summarize the organization of the rest of the paper and also explain basic ideas of the proofs.
In Section 3, we recall standard estimates on the transition probability for the simple random walk.
In Section 4, we prove Theorem 1.1, that is, the power law decay of the random walk in random scenery. We first prove that for the random walk to achieve A(t) ≥ t ρ , it is almost necessary and sufficient to visit the relevant level set {x ∈ Z d : z(x) ≥ t ρ }.
More precisely, we prove that it is too difficult to get a contribution from lower level sets of z. Here we need to invoke some estimates from [16]. The upper bound on the hitting probability to the relevant level set is shown by a rather simple argument using the asymptotics of the random walk range. The lower bound is obtained by using the so-called second moment method. In Section 5, we prove the on-diagonal lower bounds in Theorem 1.3. Essentially, we simply substitute the following results on the random walk in random scenery into (1.24): the law of large numbers for A(t) when E[z(0)] < ∞, Theorem A when d = 1, 2 and E[z(0)] = ∞, and Theorem 1.1 when d ≥ 3 and E[z(0)] = ∞.
In Section 6, we prove the on-diagonal upper bounds in Theorem 1.3. In fact they follow immediately from Proposition 1.2 and most of the section is devoted to the proof of it. When α ≥ 1, we use certain truncations to reduce the problem to upper and lower deviations for the random walk in bounded scenery, which are rather well-understood.
For the case α < 1, it essentially amounts to proving that it is difficult to The argument for d = 1 is rather bare-handed and based on the path decomposition according to the successive moves over the points in the above level set. For the case d ≥ 2, we use the so-called method of enlargement of obstacles developed by Sznitman [27].
Sections 7, 8 and 9 are devoted to the proofs of Proposition 1.5, Theorems 1.6 and 1.7, respectively. The proofs are mostly straightforward applications of the results in the earlier sections.

A bound on the continuous time random walk
We frequently use the following estimate on the transition probability of the continuous time simple random walk p t (x, y) = P x (S t = y). This can be found in [15, for |x| > t.

Power law decay rate for random walk in random scenery
Proof of Theorem 1.1. We first prove the upper bounds. Let us define the level set for We often write H λ instead of H λ (t) for simplicity and let H H λ denote the hitting time to H λ . Then we can write We are going to show that the first term in (4.2) decays stretched exponentially in Lemma 4.1 and that the second term obeys the right power law decay in Lemma 4.3.
, there exists c( ) > 0 such that P-almost surely, for all sufficiently large t, Quenched tail estimate for RWRS and RCM II Proof. We use some estimates from [16]. Let t (·) = t 0 1 Su (·) du be the occupation time measure for the simple random walk. Then we have the following obvious bound as in [16, eq. (49)]: The second term on the right-hand side is easily seen to decay stretched exponentially in t by the reflection principle and Lemma 3.1. For the first term on the right-hand side, we can drop the summands with k < (ρ − 1)/ − 1 since the total mass of t is t. In order to control the other summands, we introduce η = 1 − ρ + (k + 3) . Then we can verify that the condition η/α < k in [16, Lemma 2] holds for all k ≤ ρ/ − 4 and ≤ (1 − αρ)/3 as follows: , which holds true for the largest k = ρ/ − 4; • when α ≥ 1, this is equivalent to ρ > 1 + (1 − α)k + 3 and it holds for any k ∈ N under our assumption < −1 3 . Once the condition of [16, Lemma 2] is verified, we can follow the same argument as in [16, eq. (74)] to show that We can prove the following version by almost the same argument.
For any ∈ (0, α−1 20(α+2) ), there exists c( ) > 0 such that P-almost surely, for all sufficiently large t, Proof. We mostly repeat the proof of Lemma 4.1 with ρ replaced by 1. The only difference is that the summands in (4.4) are positive for all k ≥ 0. We instead introduce c ∈ (E[z(0)], c) and write The last term decays stretched exponentially just as before. Next, if we let l be the smallest integer larger than 3 α−1 , then for any k ≥ l, one can check that the probability decays stretched exponentially (see [16, eq. (75)] and the following discussion therein).
Finally, note that our assumption on and the choice of l imply that l < 1 20 . Then [16, eq. (76)] shows that the first term in (4.7) decays stretched exponentially.  for all sufficiently small > 0. When α > 1, this bound remains valid for ρ = 1.
The above three lemmas yield the upper bound in (1.6). Let us discuss how to include the pinning restriction S t = 0. Our starting point is where θ denotes the time shift operator and we have used the time reversal. By using the Markov property at time t/2 and Lemma 3.1, we obtain  Next we proceed to prove the lower bounds. We only prove the lower bound in (1.7) since the argument essentially contains the proof of (1.6). Recall that t denotes the occupation time measure for (S u ) u∈[0,t] and let where we choose > 0 so small that αρ d + < 1 2 . Then it suffices to show that To this end, we first bound the probability that the random walk visits H ρ before t/2, by using the so-called second moment method. We introduce the annulus Then a simple argument using the Borel-Cantelli lemma yields that P-almost surely, for all sufficiently large t > 0.
Second moment: To bound the second moment, we first write  Quenched tail estimate for RWRS and RCM II Substituting this bound into (4.23) and using (4.22), we find the bound Finally, using the strong Markov property and Lemma 3.1 together with |x| ≤ t 1/2 for x ∈ H ρ , we obtain Since > 0 is arbitrary, the desired bound (4.18) follows.

On-diagonal lower bounds
In this section, we prove the lower bounds in Theorem 1.3. Recall the representation (1.24).

Lower bound under E[z(0)] < ∞
We first deal with the simplest case E[z(0)] < ∞. In this case, let us fix M > E[z(0)] and bound the return probability as follows: as t → ∞. The last probability is bounded by just as in (4.15)-(4.16). This right-hand side is o(t −d/2 ) thanks to the law of large numbers for the random walk in random scenery. Coming back to (5.1) and recalling that M > E[z(0)] is arbitrary, we conclude that

Lower bound for d = 1, 2 with E[z(0)] = ∞
In order to prove the lower bounds for d = 1, 2 with E[z(0)] = ∞ (hence α ≤ 1), we recall (1.24) to bound the return probability from below by in this case, the desired lower bound follows.

Lower bound for d ≥ 3 with E[z(0)] = ∞
It remains to deal with the case d ≥ 3 and E[z(0)] = ∞. As before, we write Since the probability in the last line is o(t −d/2 ) by Theorem 1.1, we are done.

On-diagonal upper bounds
In this section, we prove the upper bounds in Theorem 1.3. Assuming Proposition 1.2, we can deduce the desired upper bound as follows:   This is an upper deviation for the random walk in random scenery and (76) in [16] shows that for any > 0, P-almost surely, for all sufficiently large t. Therefore we obtain the desired bound This improves the bound in (6.5) to exp{−ct(log t) −2/d }. We include the proof of above weaker result for the sake of completeness.

Upper bound for α = 1 with E[z(0)] = ∞
Recall that we have s(d, α) = 1. Let us define Bernoulli random variables bỹ z(x) = 1 z(x)≥1 ≤ z(x) (6.6) and letÃ(t) = t 0z (S u ) du. Then we know for some c > 0. This bound is first proved in the continuous setting in [26]. See [9] for a result in more general setting and [21] for a simple argument to derive (6.7) from the result in [17]. By (6.7) and Chebyshev's inequality, we find for sufficiently large t, and Proposition 1.2 follows in this case.

Upper bound for α < 1 and d = 1
In this case, we have s(d, α) = α+1 2α . Let us recall the notation for all sufficiently large t.
In order to estimate the probability (6.10), we introduce the successive times of returns to/departures from H 1/2α− . Set R 0 = D 0 = 0 and for k ≥ 1, See also Figure 2. We are going to show that the number of returns before time t, which we denote by N t = sup{k ≥ 1 : R k < t}, (6.13) cannot be too small. To this end, we need some controls on the structure of H 1/2α− . More precisely, for D k − R k to be not too large, we need a control on the size of connected components of H 1/2α− ; and for R k − D k to be not too large, we need to control the size of vacant intervals. Here and below, the term interval indicates the set of the form {k + l} 1≤l≤m for k ∈ Z and m ∈ N, and we write it as [k, k + m]. The following lemma provides the above-mentioned controls. Lemma 6.3. When > 0 is sufficiently small, P-almost surely, the following hold: any interval I ⊂ H 1/2α− has length at most 3 (6.14) for all sufficiently large t.
Proof. Using the assumption on the tail, we have (1) . (6.16) From this and the union bound, it is easy to see that P ∃an interval I ⊂ H 1/2α− , |I| ≥ 4 ≤ t − 3 2 +c (6.17) and (6.14) follows by the Borel-Cantelli lemma.
Next, for any x ∈ [−t 1 2 + , t by using (6.16), since the event in the first line requires that successive t 1/2− points fail to belong to H 1/2α− . Since α < 1 and there are at most 2t 1/2+ choices of x as the left endpoint of the interval I, the union bound and the Borel-Cantelli lemma yield (6.15).
We use this lemma to derive an upper bound on R 1 and R k+1 − R k = R 2 • θ R k for k ≥ 1 (in the sense of stochastic domination). Since it is useful only when the random walk is started from the interior (6.19) we introduce the stopping time (6.20) and show the following lemma.
Lemma 6.4. Fix > 0 and suppose that (6.15) holds. Then there exists c > 0 such that for all sufficiently large t.
Proof. It follows from (6.15) that −t Then the desired bound is a simple consequence of the reflection principle and Lemma 3.1.

Lemma 6.5. Fix
> 0 and suppose that (6.14) and (6.15) hold. Then there exist constants c, C > 0 such that the following hold for all sufficiently large t: Proof. We first prove (6.24). Let x ∈ H • 1/2α− and denote the left and right neighbors of x in H 1/2α− by x − and x + respectively, that is, x − = max{y ∈ H 1/2α− : y < x}, (6.25) x + = min{y ∈ H 1/2α− : y > x}. (6.26) Then by using the notion of hitting time H y to a point y ∈ Z, the return time R 2 is written as . This allows us to bound the left-hand side of (6.24) By (6.14), the first term is bounded by e −cr and negligible compared with the right-hand side of (6.24). The probability in the second term is asymptotic to (πr) −1/2 (see [19, Lemma 1 and (2.4) in Chapter III] for the discrete time analogue). The probability in the third term is for the random walk to stay in (x − , x + ) for a time duration r, which decays Combined with (6.15), this concludes the proof of (6.24).
Finally, the first assertion (6.23) follows in the same way as (6.28).
Proof of Proposition 6.2. In view of Lemma 6.3, we assume that (6.15) holds throughout the proof. Then Lemma 6.4 yields that P 0 N t < t Recalling R 0 = 0, we rewrite the first term as The first term is bounded by exp{−ct } by (6.23). By the strong Markov property and Lemma 6.5, the summands in the right-hand side are stochastically dominated by a family of independent and identically distributed random variables {µ k } k<t 1/2+ +1 whose tail distribution is given by the right-hand side of (6.24). We assume that this family is defined on the same probability space as the random walk with the measure P 0 . Then, the above right-hand side is bounded by In order to bound the first term, we use a large deviation principle for truncated sums proved in [13]. We summarize the statement for the reader's convenience. Let ({H k } k∈N , P) be R d -valued independent and identically distributed random variables with power law tail (see [13, (1.1)] for the definition), and let M n > 0 be a sequence satisfying lim n→∞ nP(|H 1 | > M n ) = ∞. Then the truncated sum 1 nM n P(|H 1 | > M n ) n k=1 H k · 1 {|H k |≤Mn} (6.32) satisfies a large deviation principle with speed nP(|H 1 | > M n ) whose rate function vanishes only at zero. We apply this theorem with the choice n = t 1 2 + , M n = t 1− , and P(H k > r) = Cr − 1 2 ∧ 1, (6.33) so that H k · 1 {|H k |≤Mn} has the same law as µ k · 1 {µ k ≤t 1− } . Then [13, Theorem 3.1] yields Therefore we conclude that when > 0 is small, for all sufficiently large t. Finally, on the event {N t ≥ t 1/2+ }, we have 36) whose right-hand side is bounded from below by a sum of the independent exponential random variables with rate one. Therefore, a simple large deviation bound leads us to and we are done.

Upper bound for α < 1 and d ≥ 2
In this case, we have s(d, α) = 1/α. To prove Proposition 1.2 in this case, we introduce the relevant level set: This is effectively the same as H 1/α− used before, but we will take intersection with [−t, t] d instead of [−t 1/2+ , t 1/2+ ] d to simplify the notation below. Then it suffice to prove the following proposition: Proposition 6.6. Let d ≥ 2. For sufficiently small > 0 depending on α, there exists c( ) > 0 such that P-almost surely, for all sufficiently large t.
We first estimate the probability that the random walk avoids H * while staying inside a large box [−t, t] d . Let us denote the exit time from this box by for all sufficiently large t.
Proof. We use the "method of enlargement of obstacles" in [27] with a slight modification made in [20]. Strictly speaking, the method in [27,20] is developed in the continuum setting but it can be adapted to the discrete setting as is done in [7].
Let us choose the parameters in [20]. First, let us define the scale r = t 1/2−δ with δ = α /(4 + d) and divide [−t, t] d into the boxes of the form r(q + [0, 1) d ) (q ∈ Z d ). Then we scale the space by r −1 so that we have the box [−t/r, t/r] d divided into unit cubes. Next, let γ = (d − 2)/d + δ and divide the unit cubes into dyadic boxes (which we call mesoscopic cells) with side length 2 −nγ ∈ [r −γ , 2r −γ ). it follows that for all sufficiently large t. Since there are only polynomially many mesoscopic cells intersecting [−t/r, t/r] d , the assertion follows by the union bound and the Borel-Cantelli lemma.
If all the mesoscopic cells in a unit cube q + [0, 1) d contains a point of r −1 H * , then since the volume of each cell ( r −(d−2)−dδ ) is much smaller than the capacity of a point in the scaled lattice ( r −(d−2) ), the unit cube q + [0, 1) d is more crowded than the so-called "constant capacity regime" in the crushed ice problem, see [27, p.116]. Roughly speaking, the method of enlargement of obstacles in [27] allows us to solidify such a unit cube when we consider the principal eigenvalue In our setting, Lemma 6.8 and [20,Proposition 2.7] imply that every q + [0, 1) d intersecting [−t/r, t/r] d belongs to D r (H * ) and hence [−t/r, t/r] d \ D r (H * ) = ∅. Therefore (6.44) implies that λ 1 ([−t/r, t/r] d \r −1 H * )∧M is arbitrarily close to M > 0. Reverting the scaling, we conclude that for any M > 0, P-almost surely, ≤ exp −t c( ) . (6.46) This completes the proof of Lemma 6.7.
We also need the following sparsity result which can be proved in the same way as (6.14). Let us define the times of returns to/departures from H * : R 0 = D 0 = 0 and for k ≥ 1, We have the following bound on the number of returns before time t defined by N t = sup{k ≥ 1 : R k < t}. ≤ exp −t c( ) . (6.50) Proof. By the reflection principle and Lemma 3.1, it follows that P 0 (T * ≤ t) ≤ exp{−ct} (6.51) for some c > 0. Thus it suffices to show that Observe that if N t ≤ t α 16d and T * > t, then there exists k ≤ t α 16d + 1 such that (6.53) and this event further implies that Then by the union bound and the strong Markov property, we obtain By Lemma 6.9, D 1 under P x for x ∈ H * has an exponential tail and hence the first probability in the last line decays exponentially in t 1−α /8d . The second probability is bounded by exp{−t c( ) } by (6.46).
Proof of Proposition 6.6. On the event {N t ≥ t α /8d }, we have 56) whose right-hand side is bounded from below by a sum of the independent exponential random variables with rate one. Therefore, it follows from Lemma 6.10 that and we are done.

Local central limit theorem or failure of thereof
In this section, we prove that the local central limit theorem holds when α > d 2 ∨ 1, and fails when α < d 2 ∨ 1. For the latter, since the case α < 1 is covered by Theorem 1.3, we focus on d ≥ 3 and 1 ≤ α < d 2 . For simplicity, we set E[z(0)] = 1 so that P Proof of Proposition 1.5. Let us first fix α > d 2 ∨ 1, > 0, R > 0, and (x 1 , x 2 ) ∈ Z 1+d with |(x 1 , x 2 )| ≤ Rt 1/2 and consider the transition probability  (7.2) which is negligible for the purpose of the local central limit theorem. Next, for the first term on the right-hand side of (7.1), note first that on the event {|t 1 A(t) − 1| ≤ } for all sufficiently large t. Moreover, by using (7.2), we have that as t → ∞. Substituting these controls into (7.1), we arrive at for all sufficiently large t. Since > 0 was arbitrary, we are done. Next, let d ≥ 3 and 1 ≤ α < d 2 . Then for any R > 0 and ∈ (0, 1), there is an where D 1 is the first jump time of the random walk. By Proposition 1.2 and Lemma 3.1, the first term on the right-hand side is bounded by By using the strong Markov property at D 1 and Lemma 3.1 again, we get and hence the first term on the right-hand side of (7.6) is bounded by ct − 1+d This is bounded by t − 1+d 2 − 2 for sufficiently small > 0 since α < d 2 . Therefore we conclude that P ω 0 (X t = (x 1 , x 2 )) ≤ ct − 1+d 2 − 2 . (7.10)
The second term on the right-hand side is stretched exponentially small and hence negligible. Let > 0 be a small constant such that 2δ − > 1 α ∨ 1. If A(t) ≤ t 2δ− , then the exponential factor in the first term decays stretched exponentially and hence we can drop this event. In this way, we arrive at the upper bound where we have used the integration by parts in the last equality. We can also obtain a similar lower bound on P ω 0 (X t = t δ e 1 ) with t 2δ− replaced by t 2δ+ . Since the following argument is insensitive to this change, we will only estimate the right-hand side of (8.2). The first term gives the desired asymptotics by (1.7). As for the second term, the contribution from the region u > t Hence the relevant region is u as t → ∞. We divide the above integral into those on intervals of the form [t k , t (k+1) ] and apply Theorem 1.1 to obtain as t → ∞. As > 0 is arbitrary, this yields the desired upper bound. We can get the lower bound in a similar way and complete the proof in the case δ > 1 2α ∨ 1 2 .
Next we consider δ ≤ as t → ∞ as before. The first term gives the desired asymptotics since P 0 (S t = 0) t −d/2 and P 0 (A(t) < t s(d,α)− ) decays stretched exponentially. As for the second term, note first that for sufficiently large t and the right-hand side is almost the same order as the first term.
for sufficiently large t. The last line is of small order compared with the desired asymptotics and the upper bound follows. To show the lower bound, note that (1.7) implies P 0 A(t) ∈ t s(d,α) , t s(d,α)+ , S t = 0 ≥ t −c P 0 (S t = 0) (8.8) for all sufficiently large t. Thus we can replace the event {S t = 0} in (8.1) by the one on the above left-hand side to obtain P ω 0 (X t = t δ e 1 ) ≥ E 0 A(t) −1/2 ; A(t) ∈ t s(d,α) , t s(d,α)+ , S t = 0 ≥ ct − s(d,α) 2 −c P 0 (S t = 0), (8.9) which yields the desired lower bound.

Asymptotics of the Green function
In this section, we prove Theorem 1.7. Essentially, it is a consequence of the heat kernel estimates that have been proved so far. However, since we only know those estimates in the long time asymptotics, we need an a priori bound to deal with the first small time interval in the integral (1.20).  P ω 0 (X t = ne 1 ) dt (9.2) as n → ∞. To this end, we recall the representation X t = (S 1 A 2 (t) , S 2 t ) and argue as P ω 0 (X t = ne 1 ) ≤ P 0 (A(t) ≥ n 2− ) + sup s≤n 2− p s (0, n) ≤ P 0 A(n c( ) ) ≥ n 2− + sup for any t ≤ n c( ) . Now by Theorem A, the first term on the right-hand side decays stretched exponentially. The second term also decays stretched exponentially in n by as n → ∞ followed by → 0. Note that the asymptotics in (9.5) are determined by the upper limit of the integral when d = 3 or 4, and by the lower limit when d ≥ 5. When d ≥ 5 and α < d singular in e 1 -direction and the elliptic Harnack inequality fails. Indeed, it was (9.5) that caused the anomalous behavior but when e = 0, we have n 2− n c( ) P ω 0 (X t = n(e 1 + e)) dt ≤ n 2− n 4α/d−2 P 0 (S 2 (t) = ne) dt,  for any > 0 as n → ∞. (ii) Suppose d ≥ 3 and 1 ≤ α < d 2 . If we fix > 0 sufficiently small, then just as in (7.10), we can find a sequence {x 2 (n)} n∈N ⊂ {0} × Z d such that n ≤ |x 2 (n)| ≤ 2n and P ω 0 (X t = ne 1 + x 2 (n)) ≤ n −1−d−2 for all t ∈ [n 2− , n 2+ ]. (9.10) Using this bound in (9.9) for t ∈ [n 2− , n 2+ ] instead of Lemma 3.1, we obtain g ω (0, ne 1 + x 2 (n)) ≤ cn 1−d− . (9.11) Since this decays faster than g ω (0, ne 1 ), it follows that the elliptic Harnack inequality fails in this case.