On two-dimensional random walk among heavy-tailed conductances

We consider a random walk among unbounded random conductances on the two-dimensional integer lattice. When the distribution of the conductances has an inﬁnite expectation and a polynomial tail, we show that the scaling limit of this process is the fractional kinetics process. This extends the results of the paper [ BˇC10 ] where a similar limit statement was proved in dimension d ≥ 3. To make this extension possible, we prove several estimates on the Green function of the process killed on exiting large balls.


Introduction and main results
The main purpose of the present paper is to extend the validity of the quenched non-Gaussian functional limit theorem for random walk among heavy-tailed random conductances on Z d to dimension d = 2. Analogous limit theorem for d ≥ 3 was recently obtained in [BČ10].
We recall the model first. Let E d be the set of all non-oriented nearest-neighbour edges in Z d and let Ω = (0, ∞) E d .
On Ω we consider the product probability measure P under which the canonical coordinates (µ e , e ∈ E d ), interpreted as conductances, are positive i.i.d. random variables. Writing x ∼ y if x, y are neighbours in Z d , and denoting by x y the edge connecting x and y, we set For a given realisation µ = (µ e , e ∈ E d ) of the conductances, we consider the continuous-time Markov chain with transition rates p x y . We use X = (X (t), t ≥ 0) and P µ x to denote this chain and its law on the space D d := D([0, ∞), R d ) equipped with the standard Skorokhod J 1 -topology. The total transition rate of X from a vertex x ∈ Z d is independent of x: y∼x p x y = 1. Therefore, as in [BD10,BČ10], we call this process the constant-speed random walk (CSRW) in the configuration of conductances µ. The CSRW is reversible and µ x is its reversible measure.
In this paper we assume that the distribution of the conductances is heavy-tailed and bounded from below: Our main result is the following quenched non-Gaussian functional limit theorem.
The limiting fractional kinetics process FK α is defined as a time change of a Brownian motion by an inverse of a stable subordinator. More precisely, let BM be a standard two-dimensional Brownian motion started at 0, V α an α-stable subordinator independent of BM determined by E[e −λV α (t) ] = e −tλ α , and let V −1 α be the right-continuous inverse of V α . Then (1.6) The quenched limit behaviour of the CSRW among unbounded conductances 1 on Z d was for the first time investigated in [BD10]. It is proved there that, for all d ≥ 1 and all distribution of the conductances satisfying (1.4), the CSRW converges after the normalisation n −1 X (n 2 ·) to a multiple of the d-dimensional Brownian motion, σ BM d , P-a.s. The constant σ might be 0, but [BD10] shows that it is positive iff µ e has a finite P-expectation.
When σ = 0, that is when E[µ e ] = ∞, the above scaling is not the right one and the Brownian motion might not be the right scaling limit. In the case when (1.3), (1.4) are satisfied and d ≥ 3, the paper [BČ10] identifies the fractional kinetics process as the correct scaling limit; the normalisation is as in Theorem 1.1 without the logarithmic correction. The case d ≥ 3 and α = 1 is considered in [BZ10]. Here the Brownian motion is still the scaling limit, however with a normalisation different to [BD10]. Both [BČ10] and [BZ10] do not consider the case d ≤ 2. Our Theorem 1.1 fills this gap for d = 2 and α ∈ (0, 1).
The non-Gaussian limit behaviour of the CSRW is due to trapping that occurs on edges with large conductances: roughly said, the CSRW typically spends at x a time proportional to µ x before leaving it for a long time. The heavy-tailed distributions of conductances makes the trapping important. The trapping mechanism is very similar to the one considered in the so-called trap models, see [BČ06] and the references therein. Actually, the scaling limit results of [BČ10] and of the present paper are analogous (including the normalisation) to the known scaling behaviour of the trap models [BČ07].
We would also like to point out that the dimension d = 1 is rather special for the CSRW (as well as for the trap models). It is not possible to prove any non-degenerated quenched limit theorem when (1.3) holds. The annealed scaling limit is a singular diffusion in a random environment which was defined by Fontes, Isopi and Newman in [FIN02]. As this claim has never appeared in the literature, we prove it in the appendix, adapting the techniques used for the trap models, [BČ05] or Section 3.2 of [BČ06].
The paper [BČ10] considers not only the CSRW but also another important process in a random environment, so-called Bouchaud's trap model (BTM) with the asymmetric dynamics. It shows that for d ≥ 3 this model has the same scaling behaviour as the CSRW.
The BTM can be briefly defined as follows (for more motivation see [BČ10] again, and [BČ06]). Let τ = (τ x : x ∈ Z 2 ) be a collection of i.i.d. positive random variables on a probability space (Ω,P). Given τ and a ∈ [0, 1], letP τ x be the law of the continuous-time Markov chainX with transition rates w x y = τ a−1 x τ a y started at x. This process is naturally associated with the random walk among (not i.i.d.) random conductances given byμ x y = τ a x τ a y . The methods of the present paper can be used with minimal modification (cf. Section 9 of [BČ10]) to show the following scaling limit statement for the two-dimensional BTM. The case a = 0 was treated already in [BČM06,BČ07].
be the rescaled BTM. Then there exists a constant˜ ∈ (0, ∞) such thatP-a.s., underP τ 0 , the sequence of processesX n converges as n → ∞ in law on the space D 2 equipped with the Skorokhod J 1 -topology to a multiple of the two-dimensional fractional kinetics process˜ FK α . Remark 1.3. We emphasize that the topology used in Theorems 1.1 and 1.2 is the usual Skorokhod J 1 -topology and not the uniform topology as in [BČ10]. Actually, as pointed out in [Mou10] (see also [Bil68, Chapter 18]), subtle measurability reasons prevent to define the distribution of the processes X n andX n on the (non-separable) space D d equipped with the uniform topology. It is therefore not possible to replace the J 1 -topology by the stronger uniform one in our results. The results of [BČ10] should be corrected accordingly.
Let us now give more details on the proof of Theorem 1.1 and, in particular, on the new ingredients which do not appear in [BČ10]. As in [BČ10], we use the fact that the CSRW can be expressed as a time change of another process for which the usual functional limit theorem holds and which can be well controlled. This process, called variable speed random walk (VSRW), is a continuoustime Markov chain with transition rates µ x y . We use Y = (Y (t) : t ≥ 0) and (with a slight abuse of notation) P µ x to denote this process and its law. The reversible measure of Y is the counting measure on Z d .
The time change is as follows. Let the clock process S be defined by Then, X can be constructed on the same probability space as Y , setting X (t) = Y (S −1 (t)). Since the behaviour of Y is known (see Proposition 2.1 below), to control the CSRW X we need to know the properties of the clock process S. (1.9) Then, under the assumptions of Theorem 1.1, there exists constant S ∈ (0, ∞) such that P-a.s., under P µ 0 , S n converges as n → ∞ to S V α weakly on the space D 1 equipped with the Skorokhod M 1 -topology.
Theorem 1.1 follows from this proposition by the same reasoning as in [BČ10]: The asymptotic independence of the VSRW and the clock process can be proved as in [BČ10] Lemma 6.8. The convergence of the CSRW can be deduced from the joint convergence of the clock process and the VSRW as in Section 8 of [BČ10] or in Section 11 of [Mou10]. (The measurability problems mentioned in Remark 1.3 do not play substantial role here as can be seen by comparing the arguments of [Mou10] and [BČ10].) Therefore in this paper we concentrate on the proof of Proposition 1.4.
To show the convergence of the clock process, the paper [BČ10] uses substantially two properties of the Green function of the VSRW Y killed on exit from a set A ⊂ Z d , where τ A denotes the exit time of Y from A.
The first property concerns the off-diagonal Green function in balls B(x, r) centred at x with radius r. It roughly states that as r diverges g µ B(x,r) (x, y) behaves (up to a constant factor) as the Green function of the simple random walk, for many centres x and for all y with distance at least r to x and to the boundary of B(x, r) (see Proposition 4.3 in [BČ10], cf. also Lemma 3.5 below). This is shown using a combination of the functional limit theorem for the VSRW and the elliptic Harnack inequality which were both proved in [BD10].
In d = 2 we need finer estimates. We need to consider y with distance of order r ξ , ξ ∈ (0, 1), from x. We will show in Lemma 3.6 that for such y the function g µ B(x,r) (x, y) also behaves as in the simple random walk case, at least if x = 0. The reasoning based on the functional limit theorem and the Harnack inequality does not apply here, since g µ B(x,r) (x, y) is not harmonic at y = x. It turns out, however, that by 'patching' together g µ B(x,r) for many different r's one can control the Green function up to distance r ξ to x (Lemma 3.6).
The second property of the Green function needed in [BČ10] concerns its diagonal behaviour. In rough terms again, we used the fact that for d ≥ 3 the Green function in balls converges to the infinite volume Green function, g , that the random quantity g µ (x, x) has a distribution independent of x, and that g µ (x, x) and g µ ( y, y) are essentially independent when x and y are not too close.
Such reasoning is rather impossible when d = 2. First, of course, the CSRW is recurrent and the infinite volume Green function does not exist. We should thus study the killed Green functions exclusively. We will first show that P-a.s.
with some non-random constant C 0 , see Proposition 3.1. This is proved essentially by integrating the local limit theorem for the VSRW, which can be proved using the same techniques as the local limit theorem for the random walk on a percolation cluster [BH09], see [BD10] Theorem 5.14 .
The next important issue is to extend (1.11) from the origin, x = 0, to many different centres x. While we believe that (1.11) holds true uniformly for x in B(0, K r), say, we were not able to show this. The main obstacle is the fact that the speed of the convergence in the local limit theorem is not known, and therefore we cannot extend the local limit theorem to hold uniformly for many different starting points. Note also that the method based on the integration of the functional limit theorem used in [BČ10] to get estimates for the off-diagonal Green function that are uniform over a large ball does not work. This is due to the fact that the principal contribution to the diagonal Green function in balls of radius r comes from visits that occur at (spatial) scales much smaller than r. These scales are not under control in the usual functional limit theorem.
The impossibility to extend (1.11) uniformly to x ∈ B(0, K n) appeared to be critical for the techniques of [BČ10]. It however turns out that we do not need to consider so many centres x in (1.11). Inspecting the proof of the convergence of the clock process in [BČ10] (see also [BČM06] for convergence of the two-dimensional trap model), we find out that it is sufficient to have (1.11) for O(log r) points x in B(0, K r) only. Moreover these points are typically at distance at least r = 2r/ log 2 r (Lemma 4.1). Since g µ B(x,r) (x, x) is well approximated by the Green function in smaller balls, g µ B(x,r /2) (x, x) (Lemma 3.4), and the smaller balls are disjoint for the centres of interest, we recover enough independence to proceed similarly as in d ≥ 3.
Finaly, we would like to draw reader's attention to the recent paper of J.-C. Mourrat [Mou10], where another very nice proof of the scaling limit for the asymmetric (a = 0) BTM is given for d ≥ 5. The techniques used in [Mou10] differ considerably from those used here and in [BČ10]. They explore the fact that the clock process (1.8) is an additive functional of the environment viewed by the particle which is a stationary ergodic process under the annealed measureP ×P τ . Moreover, the variance estimates of [Mou09] imply that this process is sufficiently mixing for d ≥ 5. This allows to deduce that the clock process must be a subordinator under the annealed measure. An argument in the spirit of [BS02] is then applied to deduce the quenched result; this also requires d ≥ 4 at least.
The present paper is organised as follows. In Section 2, we recall some known results on the VSRW. Section 3, in some sense the most important part of this paper, gives all necessary estimates on the killed Green function of the VSRW. In Section 4, we prove Theorem 1.1. Since this proof follows the lines of [BČ10], we decided not to give all details here. Instead of this, we will state a sequence of lemmas and propositions corresponding to the main steps of the proof of [BČ10]. The formulation of these lemmas is adapted to the two-dimensional situation. We provide proofs only in the cases when they substantially differ from [BČ10]. In the appendix we discuss the CSRW among the heavy-tailed conductances on the one-dimensional lattice Z.
Acknowledgement. The author thanks the anonymous referee for carefully reading the manuscript and providing many valuable comments.

Preliminaries
We begin by introducing some further notation. Let B(x, R) be the Euclidean ball centred at x of radius R and let Q(x, R) be a cube centred at x with side length R whose edges are parallel to the coordinate axes. Both balls and cubes can be understood either as subsets of We use the convention that all large values appearing below are rounded above to the closest integer, if necessary. It allows us to write that, e.g., nZ d ⊂ Z d for ∈ (0, 1) and n large. We use c, c to denote arbitrary positive and finite constants whose values may change in the computations.
We recall some known facts about the VSRW and its transition density Proposition 2.1. Assuming (1.4) and d = 2, the following holds.
(i) (Functional limit theorem) There exists Y ∈ (0, ∞) such that P-a.s., under P µ 0 , the sequence Y n (·) = n −1 Y (n 2 ·) converges as n → ∞ in law on D 2 to a multiple of a standard two-dimensional Brownian motion, Y BM.

(ii) (Heat-kernel estimates) There exist a family of random variables
where [x n] ∈ Z 2 is the point with coordinates x 1 n , x 2 n .
Proof. Claims (i) and (ii) are parts of Theorems 1.1 and 1.2 of [BD10] specialised to d = 2. Claim (iii) is a simple consequence of Theorem 5.14 of [BD10], cf. also Theorems 1.1 and 4.6 of [BH09] where the local limit theorem is shown for the random walk on the super-critical percolation cluster.
Remark 2.2. Proposition 2.1 is the only place in the proof of Theorem 1.1 where the assumption (1.4) is used explicitly. Hence, if Proposition 2.1 is proved with more general assumptions, then Theorem 1.1 will hold under the same assumptions.

Estimates on the Green function
This section contains all estimates on the Green function of the VSRW that we need in the sequel. These estimates might be of independent interest.

Diagonal estimates
We control the diagonal Green function at the origin first.
Proposition 3.1. Let C 0 = (π 2 Y ) −1 , with Y as in Proposition 2.1(i). Then, for P-a.e. environment µ, Proof. We use the local limit theorem (2.6) and the heat-kernel estimates to prove this claim. With (3. 2) The dominant contribution comes from the term I 1 . By the local limit theorem, for every there exists t 0 = t 0 (µ, ) such that for all t > t 0 Therefore, for r large enough, (3.4) A lower bound on I 1 is obtained analogically, yielding lim r→∞ I 1 / log r = C 0 .
Using the strong Markov property at τ B and the symmetry of q t (·, ·), it is possible to estimate I 3 , By splitting the last integral on V 2 0 and on | y| = r(1 + o(1)), using the estimates (2.3) and (2.4), it follows that, as r → ∞, To bound I 2 we need the following lemma. (3.7) and, in consequence, Proof. An easy consequence of (2.1) is the existence of C < ∞ such that (see Lemma 3.3 of [BČ10]) s. for all large r. (3.8) Using the heat-kernel lower bound (2.5) together with (3.8), we obtain for all large r Writing t = kr 2 + s for k ∈ N and s ∈ [0, r 2 ), and using (2.2) together with q µ, (3.10) Summing over z k−1 , z k−2 ,. . . z 1 , using (3.9), this yields q µ,B t (x, y) ≤ c 2 r −2 c k−1 , which is equivalent to the right-hand side of (3.7). The second claim of the lemma follows by summing (3.7) over y ∈ B.
It is now easy to finish the proof of Proposition 3.1. By the previous lemma, (3.11) Therefore, I 2 and I 3 are o(log r), and the proof is completed.
We need rougher estimates on the diagonal Green function, uniform in a large ball.
Lemma 3.3. There exist c 1 , c 2 ∈ (0, ∞) such that for every ∈ (0, 1), K > 1, P-a.s., (3.12) Proof. This can be proved exactly by the same argument as Proposition 3.1, replacing the local limit theorem used in (3.4) by the heat-kernel upper and lower bounds from Proposition 2.1(ii), using again the fact (3.8) to control the random variables V x .

Approximation of the diagonal Green function
As discussed in the introduction, to recover some independence required to show Theorem 1.1, we need to approximate the diagonal Green function in large sets by smaller ones. (3.13) Proof. By the monotonicity of g µ A in A and the strong Markov property g (3.14) where for the last term we used the fact that (3.15) By Lemmas 3.2 and 3.3, the second term on the right-hand side of (3.14) is O(n −c log n) = o(1). The first term can be controlled using the heat-kernel estimates again: Observing that, by (3.8), (3.16) The first term is clearly o(1). After the substitution t/|x − y| 2 = u, the second term is smaller than

Off-diagonal estimates
Next, we need off-diagonal estimates on g µ B(x,r) (x, y). The following lemma provides them for y not too close to x and the boundary of the ball (cf. Proposition 4.3 of [BČ10]).
Lemma 3.5. Let K > 1, 0 < 3 o < g < K/3, δ > 0 and r ∈ [ g , g + o /2). Then, P-a.s. for all but finitely many n, for all points x ∈ B(0, (K − g )n), and y ∈ B(x, Proof. This lemma can be proved in the same way as Proposition 4.3 of [BČ10], using a suitable integration of the functional limit theorem and the elliptic Harnack inequality. For d = 2, one should only replace the formula for the Green function g B(x,r) (x, y) of the d-dimensional Brownian motion Y BM d killed on exiting B(x, r) ((4.5) in [BČ10]) with its two-dimensional analogue g B(x,r) (x, y) = C 0 (log r − log |x − y|). Remark also that the condition g < 1/2 appearing in the statement of Proposition 4.3 of [BČ10] is not necessary for the proof, so we omitted it here.
The previous lemma does not give any estimate on the Green function near to the centre of the ball. When x = 0, we can improve it.
For x = 0 we have the following upper estimate.

Proof of the main theorem
We now have all estimates required to prove Theorem 1.1. As we have already remarked, it is sufficient to show Proposition 1.4 only. Theorem 1.1 follows from it as in [BČ10]. Moreover, since the proof of Proposition 1.4 mostly follows the lines of [BČ10], we focus on the difficulties appearing for d = 2 and we explain the modifications needed to resolve them.
The proof explores the fact that the stable subordinator V α at time T is well approximated by the sum of a large but finite number of its largest jumps before T . These jumps of the limiting process corresponds in S n to visits of the VSRW Y to sites with µ x ∼ n 2/α log −1/α n.
To understand this scale heuristically, observe that a fixed time T for S n corresponds to the time T n 2 for Y , see (1.9). At this time Y typically visits N ∼ n 2 / log n different sites, similarly to the two-dimensional simple random walk. The maximum of N independent variables with the same distribution as µ e is then of order N 1/α ∼ n 2/α log −1/α n.

We thus define (cf. (6.1) and (6.3) of [BČ10])
E n (u, w) = {e ∈ E 2 : µ e ∈ [u, w)n 2/α log −1/α n}, Unlike in [BČ10], it is not necessary to define the 'bad' edges (cf. (6.2) of [BČ10]). This is the consequence of the next lemma that shows that the edges of the set E n (u, ∞) are well separated in d = 2. This lemma might appear technical, but it is crucial for the applied technique. It implies the independence of g µ B(x,n/ log 2 n) (x, x) for x ∈ T n (u, ∞) not sharing the same edge. Proof. Observe first that for k ≥ 2 and 2 k−1 ≤ n ≤ 2 k , E n (u, ∞) ⊂ E 2 k (2 −2/α u, ∞). To prove (4.2) it is thus sufficient to show that for all u > 0, P-a.s. for all k ∈ N large, The probability of the complement of this event is bounded from above by We now investigate the rescaled clock process S n . As in [BČ10], we fix s small and treat separately the contributions of vertices from T n (0, s ), T n ( s , −1 s ) and T n ( −1 s , ∞) to this process. We first show that the contribution of the visits to the set T n (0, s ) can be neglected, cf. Proposition 5.1 of [BČ10].

Proposition 4.2.
For every δ > 0 there exists s such that for all K ≥ 1 and B n = B(0, K n), P-a.s. for all but finitely many n, (4.7) Proof. The proof resembles to the proof of Proposition 5.1 of [BČ10], but there are some differences caused by the recurrence of the CSRW in d = 2. We will show that P-a.s. for all large n, (t) ∈ T n (0, s )} dt ≤ K 2 n 2/α log 1− 1 α nδ 2 . (4.8) The proposition then follows by using the Markov inequality.
We set i max = min{i : 2 −i s n 2/α log − 1 α n ≤ c} = O(log n), and H n (i) = {e ∈ B n : µ e ∈ s n 2/α log − 1 α n(2 −i , 2 −i+1 ]}. Observe that by Proposition 3.1 and Lemma 3.6, P-a.s. for all n large, g µ n (e) ≤ḡ n (e) for all e ∈ B n . Using this notation, we have P-a.s. for all n large (4.11) Setting p n,i = P[µ e ∈ H n (i)] ≤ C −α s 2 iα n −2 log n, we have for a C > 0 and λ > 0 (1 + p n,i (e λḡ n (e) − 1)). (4.12) The logarithm of the last product is bounded by p n,i e∈B n e λḡ n (e) . To bound this expression we use the fact that for a small enough λ there is a constant c independent of n such that e∈B n e λḡ n (e) ≤ cK 2 n 2 , (4.13) which can be proved as Lemma A.2 of [BČM06]. Therefore, (4.12) is bounded from above by (4.14) Taking C large, this is summable (even after multiplication by log n). The Borel-Cantelli lemma then implies that the complement of the event on the left-hand side of (4.12) holds for all i ≤ i max , P-a.s. for all large n. Therefore, the left-hand side of (4.8) is bounded by i max i=1 C 1−α 2 i(α−1) K 2 n 2/α log 1− 1 α n, which is smaller than the right-hand side, if is small enough.
To treat the dominant contribution of T n ( s , −1 s ), we apply the same coarse-graining construction as in [BČ10]. In this construction we observe the VSRW before the exit from the large ball B n . The VSRW spends a time of order K 2 n 2 in this ball and visits only finitely many pairs of sites from T n ( s , −1 s ). In every pair it spends a time of order log n. This logarithmic factor explains the additional power of the logarithm appearing in the normalisation of S n and not in the definition of E n (u, v).
We now start the construction. Let ν n = n/ log 2 n. For e ∈ E d , z ∈ Z d we set where C eff denotes the effective conductance between two sets, see e.g. [BČ10] (3.8) for the usual definition. We have the following analogue of Lemma 6.2 of [BČ10]. (1 + c )γ n (z) ≥ γ n (e) ≥ γ n (z). Proof. Claims (i), (ii) are proved as in [BČ10]. Claim (iii) follows using the identity C eff (z, B(z, A c )) = g µ A (z, z) −1 valid for any A ⊂ Z 2 , Proposition 3.1, the translation invariance of P, and claim (ii).
We now split the sets E n (u, v) according to the value of γ n (e). To this end we choose a sequence h n so that h n 0, b n := P[|C 0 γ n (e) log n − 1| > h n ] 0, and b n log n log 1/2 n. This is possible by Lemma 4.3(iii). We define (4.18) We also set T 0 Similarly to Lemma 6.3 of [BČ10] (cf. also (6.41) there), these sets are homogeneously spread over B n .
Lemma 4.4. Let 0 < u < v, δ, b > 0 be fixed and set p n (u, v) = n −2 log n(u −α − v −α ). Then, P-a.s. for all but finitely many n, for all x ∈ b nZ 2 ∩ B n , and all i ∈ {0, . . . , 2 log 2 log n} Proof. The proof is a concentration argument for binomial random variables. However, since p n (u, v)n 2 = O(log n), we need to work with subsequences in order to apply the Borel-Cantelli lemma.
As the first step, we disregard γ n (e) and show (4.19) for E n instead of E 0 n : We start by proving the upper bound. For n ∈ N we define k = k(n) ∈ N and s = s(n) ∈ [1, 2) by n = s2 k . We set for some y = y(x, n) ∈ 1 20 δ b 2 k Z 2 . Moreover, by definition of p n (u, v) and (1.3), (4.24) Hence, to prove the upper bound of (4.22), it is sufficient to show that P-a.s. for all k large, for all i 1 , i 2 , i 3 ∈ {0, . . . , i max }, and for all y ∈ 1 20 δ b 2 k Z 2 ∩ B 2 k , (4.25) The number of y's in consideration and i max are finite. The probability of the complement of (4.25) for given i 1 , i 2 , i 3 , can be bounded using the exponential Chebyshev inequality, using the independence of µ e 's, by exp{−c2 2k p 2 k (·, ·)} ∼ exp(−ck). As this is summable, the upper bound follows. The proof of the lower bound in (4.22) is analogous.
The proof of (4.21) is very similar to the proof of the upper bound of (4.22). In the upper bound on the probability of the complementary event, it is in addition necessary to sum over 0 ≤ i ≤ c log log 2 k and consider O(log 2 2 k ) possible values for y. On the other hand, the term log 1/2 n on the right-hand side of (4.21) assures that the Chebyshev inequality gives at least a factor exp{−c log 2 k }, which assures the summability.
It remains to show (4.20), since, as b n → 0, (4.19) follows from (4.20) and (4.22). From the proofs of Lemma 4.1 and of (4.22), we know that (4.22) and (4.2) hold out of events whose probabilities are summable along the subsequence 2 k , k ∈ N. Moreover, if (4.2) holds then (γ n (e) : e ∈ E n (u, v) ∩ B n ) are independent since γ n (e) depends only on the environment restricted to B(e, ν n ). We can now use once more the concentration for binomial random variables as before. The fact that b n log n log 1/2 n and thus exp{−c b 2 k p 2 k (u, v)2 2k } exp{−c k} assures the summability again.
As in [BČ10] Lemma 6.4, P-a.s. for all but finitely many n we can define a family of approximate balls n (x, r) ⊂ Z 2 with the following properties: For all x ∈ B n and r ∈ (0, K n) The existence of these sets follows easily from Lemma 4.1.
We now adapt the definition of the coarse graining from [BČ10]. We use the sets n (·, g n) to cut the trajectory of Y to several parts whose contribution to S n we treat separately: Let g > 0, t n (0) = 0, y n (0) = 0 and for i ≥ 1 let (4.26) We define this is the increment of the (normalised) clock process between times t n (i) and t n (i + 1) caused by sites in T 0 n (u, v). The behaviour of the sequence t n (i) is the same as in [BČ10], Lemma 6.5. The distribution of s 0 n is characterised by the next proposition, cf. Proposition 6.7 of [BČ10].
Proposition 4.5. Let T , s , g > 0. Define s 0 n (i) = s 0 n (i, s , −1 s ) and let n i be the σ-algebra generated by (Y s , s ≤ t n (i)). Then, there is a constant κ > 0 independent of s , g and T , such that P-a.s. for all n large enough, for all i ≤ −2 g T , and for all intervals A = (0, a] with a ∈ (0, ∞) where c s = π( −α s − α s ) and ν s is the measure on (0, ∞) given by Remark 4.6. There is an important difference between Proposition 4.5 and Proposition 6.7 of [BČ10]. In [BČ10], it is claimed that the sequence s n (i) converges to an i.i.d. sequence s ∞ (i) for every fixed g . However, [BČ10] does not show this claim, but only equivalents of our statements (4.28), (4.29).
The i.i.d. sequence s ∞ (i) is used in [BČ10] to prove Lemma 6.8. Its proof can be however modified easily to use (4.28), (4.29) only. We use this occasion to present the correct proof below, see Lemma 4.8 Proof of Proposition 4.5. The proof of this proposition is very similar to the proof of Proposition 6.7 in [BČ10]. If Y visits a site from T 0 n ( s , −1 s ), it spends there an asymptotically exponentially distributed time, as stated in the next lemma. Its proof is exactly the same as the proof of Lemma 6.6 in [BČ10].
Lemma 4.7. Let z ∈ B 0 n = B(0, (K − g )n), e = x y ∈ E n ( s , −1 s ) ∩ B n (z, g n) be such that µ e = un 2/α and γ n (e) log n = v. Then, P-a.s., the distribution of under P µ x and P µ y converges as n → ∞ to the exponential distribution with mean 2u/v.
With this lemma at disposition, we need to estimate the probability that a site x ∈ T 0 n ( s , −1 s ) is visited between t n (i) and t n (i + 1). This probability can be written using the Green functions as g n ( y n (i), g n) ( y n (i), x) g n ( y n (i), g n) (x, x) . (4.32) The denominator (4.32) is C 0 log( g n)(1 + o(1)) by the definition of E 0 n , using also the definition of the approximate balls n ( y n (i), g n) and Lemma 3.4. The numerator can be estimated using Lemma 3.5 when |x − y n (i)| ≥ o n. Such x's give the principal contribution. For the remaining x's one uses Lemma 3.7 as the upper bound.
The proof of Proposition 4.5 then continues exactly as the proof of Proposition 6.7 of [BČ10] by estimating the probability that T n (u, v), s ≤ u < v ≤ −1 s , is visited between t n (i) and t n (i+1) by the sum of probabilities that e ∈ E n (u, v) are visited. Due to the homogeneity of E n (u, v) (Lemma 4.4), this summation can be replaced by an integration with respect to 2p n (u, v) times Lebesgue measure. For the principal contribution coming from e's with d(e, y n (i)) ≥ o n, this leads to the integral The error terms of (4.28), (4.29) which do not appear in [BČ10] (see Remark 4.6) are explained as follows. With one exception, all the errors in the proof of Proposition 6.7 of [BČ10] can be made arbitrarily small with respect to 2 g . The only exception is the error comming from the estimate on the probability that more than two edges from E 0 n ( s , −1 s ) are visited. This probability can be bounded by the right-hand side of (4.28). The error term in (4.29) then corresponds to the event that the first visited edge of E 0 n ( s , −1 s ) gives (after the normalisation) a contribution belonging to A = (0, a], and then another edge from E 0 n ( s , −1 s ) is visited.
The consequence of Proposition 4.5, is the following lemma replacing Lemma 6.8 of [BČ10]. Since, as explained in Remark 4.6, its proof in [BČ10] uses the incorrectly stated Proposition 6.7, we present the corrected proof here.
Lemma 4.8. Let T, g , s > 0, ∈ N, λ 1 , . . . , λ > 0, ξ 1 , . . . , ξ ∈ R 2 , and 0 ≤ i 1 < · · · < i ≤ T −2 g . Define r n (i) = n −1 ( y n (i + 1) − y n (i)). Then P-a.s. for all large n , and the reminder term satisfies for all g small enough Proof. We first control the joint Laplace transform of the pair (s 0 n (i), r n (i)), i ≤ −2 g T , given n i = σ(Y s , s ≤ t n (i)). The calculation very similar to (6.50)-(6.51) of [BČ10], with the conditional expectations replacing the usual ones, gives (4.35) where O ξ ( g ) denotes an error whose absolute value is bounded by c(ξ) g for all n large and g small; this estimate comes from e −2 g |ξ| ≤ e −ξ·r n (i) ≤ e 2 g |ξ| . The first term on the right-hand side of (4.35) can be rewritten using Proposition 4.5 and the fact that, by the functional central limit theorem, the distribution of r n (i) converges to the uniform distribution on the sphere with radius g , (4.36) The second term on the right-hand side of (4.35) can be estimated using Proposition 4.5 again: Itegrating by parts, for all large n, (4.37) Putting these two computation together we get for all n large enough where ξ,λ is random but its absolute value is bouned by c ξ,λ (1 + c 2 s ) 3 g for all g small enough. Using this statement and the fact that e −λs 0 n (i)+ξ·r n (i) ≤ C(ξ) < ∞ for all possible values of s 0 n (i) and r n (i), we can write (4.39) Iterating the last statement and taking the error terms out of the product (using again g small enough) gives the upper bound for (4.34). The lower bound is obtained analogously.
To finish the control of the behaviour of S n we need to estimate the contribution of the sets T n ( −1 s , ∞) and T 1 n ( s , −1 s ) (see below (4.18) for the notation). From the next lemma, which is proved in the same way as Lemmas 7.1, 7.2 of [BČ10], we see that their contribution is zero with a large probability. Lemma 4.9. For every δ, K > 0 there exists s > 0 such that, P-a.s., for all but finitely many n, Propositions 4.2, 4.5 and Lemma 4.9 characterise the contributions of various sites in B n to the clock process S n . Using these results the proof of Proposition 1.4, and consequently of Theorem 1.1, can be completed as in Section 8 of [BČ10].

A The CSRW on the one-dimensional lattice
In this appendix, we study the CSRW among heavy-tailed random conductances on the onedimensional lattice. We show that the scaling limit of this process is the singular diffusion in random environment. This diffusion, which is also the scaling limit of the one-dimensional trap model, see [FIN02,BČ05], is defined as follows.
Definition A.1 (Fontes-Isopi-Newman diffusion). Let (Ω,P) be a probability space on which we define a standard one-dimensional Brownian motion BM and an inhomogeneous Poisson point process (x i , v i ) on R × (0, ∞) with intensity measure dx αv −1−α dv. Let ρ be the random discrete measure ρ = i v i δ x i . Conditionally on ρ, we define the FIN-diffusion (Z(s), s ≥ 0) as a diffusion process (with Z(0) = 0) that can be expressed as a time change of BM with the speed measure ρ: Denoting by (t, y) the local time of BM, we set φ ρ (t) = R (t, y)ρ(d y) and Z(s) = BM(φ −1 ρ (s)).
The following theorem describes the scaling behaviour of the one-dimensional CSRW. converges in distribution to Z(t).
Proof. The proof follows the lines of [BČ05] and Section 3.2 of [BČ06]. We will construct copies X n of X n , n ≥ 1, on the same probability space (Ω,P) as the FIN diffusion. On this probability space we then show thatX n convergesP-a.s. To this end we expressX n as a time-scale change of BM and show that the speed measures ofX n converge to ρ and the scale change is asymptotically negligible. Let us introduce our notation for the time-scale change first. Consider a locally-finite deterministic discrete measure ν(dx) = i∈Z w i δ y i (dx). The measure ν is referred to as the speed measure. Let S be a strictly increasing function defined on the set { y i : i ∈ Z}. We call such S the scaling function. Let us introduce slightly non-standard notation S • ν for the "scaled measure" With (t, y) denoting the local time of the Brownian motion BM, we define φ ν,S (t) = R (t, y)(S • ν)(d y). Then, the time-scale change of Brownian motion with the speed measure ν and the scale function S is a process X ν,S defined by From the definition of the measure ρ, it follows that ρ((0, 1]) law = t −1/α ρ((0, t]). Therefore, for all n, the sequence (μ n x,x+1 : x ∈ Z) has the same distribution as (µ x,x+1 : x ∈ Z). For all n we define a measure ν n on R and a piece-wise constant function S n : R → R by ν n (du) = 1 2c n x∈Z (μ n x,x−1 +μ n x,x+1 )δ x/n (du). We define processesX n = X ν n ,S n . From Lemma A.3 it follows directly thatX n has the same distribution as X n . The key step in the proof of Theorem A.2 is the following lemma Lemma A.4.P-a.s., as n → ∞, where Id denotes the identity map and v → the vague convergence of measures.
Proof. The first clam can be shown as in [BČ05]; it follows from the law of large numbers for triangular arrays and uses only fact that E[µ −1 e ] < ∞ which is a consequence of (1.4). The second claim is proved in [FIN02,BČ05] for slightly different measures, namely forν n = c −1 n x∈Zμ n x,x+1 δ x/n . However, this small difference does not play any role for the vague convergence. The third claim is a direct consequence of the first two.
As a consequence of this lemma we obtain as in [BČ05,BČ06] the following, slightly stronger, convergence result.