Introduction and main results

The model

Denote by \(e_i\) the ith unit vector of \({\mathbb {Z}}^d\). We let \({\mathcal {P}}\) be the set of probability distributions on \(\{\pm e_i : i = 1,\dots ,d\}\) and put \({\varOmega }={\mathcal {P}}^{{\mathbb {Z}}^d}\). Denote by \(\mathcal {F}\) the natural product \(\sigma \)-field on \({\varOmega }\) and by \(\mathbb {P}= \mu ^{\otimes {\mathbb {Z}}^d}\) the product probability measure on \(({\varOmega },\mathcal {F})\).

Given an element (or environment) \(\omega \in {\varOmega }\), let \((X_n)_{n\ge 0}\) be the canonical nearest neighbor Markov chain on \({\mathbb {Z}}^d\) with transition probabilities

$$\begin{aligned} p_\omega (x,x+e) = \omega _x(e),\quad e\in \{\pm e_i : i = 1,\dots ,d\}, \end{aligned}$$

the random walk in random environment (RWRE for short). We write \({\text {P}}_{x,\omega }\) for the “quenched” law of \((X_n)_{n\ge 0}\) starting at \(x\in {\mathbb {Z}}^d\).

We are concerned with RWRE in dimensions \(d\ge 3\) which is an \(\varepsilon \)-perturbation of simple random walk. To fix a perturbative regime, we shall assume the following condition.

  • Let \(0<\varepsilon <1/(2d)\). We say that \(\mathbf{A0}(\varepsilon )\) holds if \(\mu ({\mathcal {P}}_\varepsilon ) = 1\), where

    $$\begin{aligned} {\mathcal {P}}_\varepsilon = \left\{ q\in {\mathcal {P}} : \left| q(\pm e_i) - 1/(2d)\right| \le \varepsilon \text{ for } \text{ all } i=1,\dots ,d\right\} . \end{aligned}$$

Furthermore, we work under two centering conditions on the measure \(\mu \). The first rules out ballistic behavior, while the second guarantees that the RWRE is balanced in direction \(e_1\).

  • We say that \(\mathbf{A1}\) holds if \(\mu \) is invariant under all d reflections \(O_i:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) mapping the unit vector \(e_i\) to its inverse, i.e. \(O_ie_i=-e_i\) and \(O_ie_j=e_j\) for \(j\ne i\). In other words, the laws of \((\omega _0(O_ie))_{|e|=1}\) and \((\omega _0(e))_{|e|=1}\) coincide, for each \(i=1,\dots ,d\).

  • We say that \(\mathbf{B}\) holds if \(\mu (\mathcal {P}^{\text {s},1})=1\), where

    $$\begin{aligned} \mathcal {P}^{\text {s},1}= \{p\in {\mathcal {P}} : p(e_1)=p(-e_1)\}. \end{aligned}$$

We now state our results. Then we discuss them together with our conditions in the context of known results from the literature.

Our main results

Our first statement shows \(\mathbb {P}\)-almost sure convergence of the (normalized) mean sojourn time of the RWRE in a ball when its radius gets larger and larger. Let \(V_L=\{y\in {\mathbb {Z}}^d:|y|\le L\}\) denote the discrete ball of radius L, and \(V_L(x) = x +V_L\). Denote by \(\tau _{V_L(x)}=\inf \{n\ge 0:X_n\notin V_L(x)\}\) the first exit time of the RWRE from \(V_L(x)\). We write \({\text {E}}_{x,\omega }\) for the expectation with respect to \({\text {P}}_{x,\omega }\). We always assume \(d\ge 3\).

Theorem 1

(Quenched mean sojourn times, \(d\ge 3\)) Assume \(\mathbf{A1}\) and \(\mathbf{B}\). Given \(0<\eta < 1\), one can find \(\varepsilon _0=\varepsilon _0(\eta ) > 0\) such that if \(\mathbf{A0}(\varepsilon )\) is satisfied for some \(\varepsilon \le \varepsilon _0\), then the following holds: There exists \(D\in [1-\eta , 1+\eta ]\) such that for \(\mathbb {P}\)-almost all \(\omega \in {\varOmega }\),

$$\begin{aligned} \lim _{L\rightarrow \infty }\left( {\text {E}}_{0,\omega } \left[ \tau _{V_L}\right] /L^2\right) = D. \end{aligned}$$

Moreover, one has for each \(k\in {\mathbb {N}}\), for \(\mathbb {P}\)-almost all \(\omega \),

$$\begin{aligned} \lim _{L\rightarrow \infty }\left( \inf _{x: |x| \le L^k}{\text {E}}_{x,\omega }\left[ \tau _{V_L(x)}\right] /L^2 \right) = \lim _{L\rightarrow \infty }\left( \sup _{x: |x| \le L^k}{\text {E}}_{x,\omega }\left[ \tau _{V_L(x)}\right] /L^2 \right) = D. \end{aligned}$$

Standard arguments then imply the following bound on the moments.

Corollary 1

(Quenched moments) In the setting of Theorem 1, for each k, \(m\in {\mathbb {N}}\) and \(\mathbb {P}\)-almost all \(\omega \),

$$\begin{aligned} \limsup _{L\rightarrow \infty }\left( \sup _{x: |x| \le L^k}{\text {E}}_{x,\omega }\left[ \tau ^m_{V_L(x)}\right] /L^{2m} \right) \le 2^mm!\,. \end{aligned}$$

Combined with results on the spatial behavior of the RWRE from [2], we prove a functional central limit theorem under the quenched measure. In [2], it was shown that under \(\mathbf{A1}\) and \(\mathbf{A0}(\varepsilon )\) for \(\varepsilon \) small, the limit

$$\begin{aligned} 2p_\infty (\pm e_i) = \lim _{L\rightarrow \infty }\sum _{y\in {\mathbb {Z}}^d}\mathbb {E}\left[ {\text {P}}_{0,\omega }\left( X_{\tau _{V_L}}=y\right) \right] \frac{y_i^2}{|y|^2} \end{aligned}$$
(1)

exists for \(i=1,\dots ,d\), and \(|p_\infty (e_i)-1/(2d)|\rightarrow 0\) as \(\varepsilon \downarrow 0\). Let

$$\begin{aligned} \varvec{\Lambda }=\left( 2\,p_\infty (e_i)\delta _{i}(j)\right) _{i,j=1}^d\in {\mathbb {R}}^{d\times d}, \end{aligned}$$
(2)

and define the linear interpolation

$$\begin{aligned} X_t^{n}= X_{\lfloor tn\rfloor } + (tn-\lfloor tn\rfloor )\left( X_{\lfloor tn\rfloor +1}-X_{\lfloor tn\rfloor }\right) ,\quad t\ge 0. \end{aligned}$$

The sequence \((X_t^{n}, t\ge 0)\) takes values in the space \(C({\mathbb {R}}_+,{\mathbb {R}}^d)\) of \({\mathbb {R}}^d\)-valued continuous functions on \({\mathbb {R}}_+\). The set \(C({\mathbb {R}}_+,{\mathbb {R}}^d)\) is tacitly endowed with the uniform topology and its Borel \(\sigma \)-field.

Theorem 2

(Quenched invariance principle, \(d\ge 3\)) Assume \(\mathbf{A0}(\varepsilon )\) for small \(\varepsilon > 0\), \(\mathbf{A1}\) and \(\mathbf{B}\). Then for \(\mathbb {P}\)-a.e. \(\omega \in {\varOmega }\), under \({\text {P}}_{0,\omega }\),

\(X_\cdot ^{n}/\sqrt{n}\) converges in law to a d-dimensional Brownian motion with diffusion matrix \(D^{-1}\varvec{\Lambda }\), where D is the constant from Theorem 1 and \( \varvec{\Lambda }\) is given by (2).

Under Conditions \(\mathbf{A0}(\varepsilon )\) and \(\mathbf{A1}\), a local limit law for RWRE exit measures was proved in [2] for dimensions three and higher, generalizing the results of Bolthausen and Zeitouni [9] and Baur [1] for the case of isotropic perturbative RWRE. While the results of [2, 9] do already imply transience for the random walks under consideration, they do not prove diffusive behavior, since there is no control over time. This was already mentioned in [9]: “In future work we hope to combine our exit law approach with suitable exit time estimates in order to deduce a (quenched) CLT for the RWRE.” Under the additional Condition \(\mathbf{B}\) which shall be discussed below, we fulfill here their hope.

In dimensions \(d>1\), the RWRE under the quenched measure is an irreversible (inhomogeneous) Markov chain. A major difficulty in its analysis comes from the presence of so-called traps, i.e. regions where the random walk can hardly escape and therefore spends a lot of time. A lot of effort has been made to understand the ballistic regime where the limit velocity \(\lim _{n\rightarrow \infty }X_n/n\) is an almost sure constant different from zero. Ballisticity conditions like Sznitman’s condition \((T)_{\gamma },(T'),(T)\) or their polynomial analogs \(({\mathcal {P}})_M,({\mathcal {P}})_0\) have been established, under which law of large numbers and limit theorems can be proved, see e.g. Sznitman [1820], Bolthausen and Sznitman [7], Berger [3], Berger et al. [5] or Drewitz and Ramírez [12]. The lecture notes of Sznitman [21] and Drewitz and Ramírez [13] provide further references. The methods often involve the construction of certain regeneration times, where, roughly speaking, the walker does not move “backward” anymore.

For the non-ballistic case, techniques based on renormalization schemes have been developed. In the small disorder regime, results can be found under the classical isotropy condition on \(\mu \), which is stronger than our condition \(\mathbf{A1}\). It requires that for any orthogonal map \({\mathcal {O}}\) acting on \({\mathbb {R}}^d\) which fixes the lattice \({\mathbb {Z}}^d\), the law of \((\omega _0({\mathcal {O}}e))_{|e|=1}\) and \((\omega _0(e))_{|e|=1}\) coincide. Under this condition, Bricmont and Kupiainen [10] provide a functional central limit theorem under the quenched measure for dimensions \(d\ge 3\). However, it is of a certain interest to find a new self-contained proof of their result. A continuous counterpart is studied in Sznitman and Zeitouni [22]. For \(d\ge 3\), they prove a quenched invariance principle for diffusions in a random environment which are small isotropic perturbations of Brownian motion. Invariance under all lattice isometries is also assumed in the aforementioned work of Bolthausen and Zeitouni [9].

In a non-perturbative fashion, Bolthausen et al. [8] use so-called cut times as a replacement of regeneration times. At such times, past and future of the path do not intersect. However, in order to ensure that there are infinitely many cut times, it is assumed in [8] that the projection of the RWRE onto at least \(d_1\ge 5\) components behaves as a standard random walk. Among other things, a quenched invariance principle is proved when \(d_1\ge 7\) and the law of the environment is invariant under the antipodal transformation sending the unit vectors to their inverses.

Our Condition \(\mathbf{B}\) requires only that the environment is balanced in one fixed coordinate direction (\(e_1\) for definiteness). Then the projection of the RWRE onto the \(e_1\)-axis is a martingale under the quenched measure, which implies a priori bounds on the sojourn times, see the discussion in Sect. 4. Clearly, assuming just Conditions \(\mathbf{A0}(\varepsilon )\) and \(\mathbf{B}\) could still result in ballistic behavior, but the combination of \(\mathbf{A0}(\varepsilon )\), \(\mathbf{A1}\) and \(\mathbf{B}\) provides a natural framework to investigate non-ballistic behavior of “partly-balanced” RWRE in the perturbative regime.

To our knowledge, we are the first who study random walks in random environment which is balanced in only one coordinate direction. The study of fully balanced RWRE when \(\mathbb {P}(\omega _0(e_i)=\omega _0(-e_i)\ \hbox {for all } i=1,\dots ,d)=1\) goes back to Lawler [16]. He proves a quenched invariance principle for ergodic and elliptic environments in all dimensions. Extensions within the i.i.d. setting to the mere elliptic case were obtained by Guo and Zeitouni [14] and to the non-elliptic case by Berger and Deuschel [4]. Recently, Deuschel et al. [11] proved a quenched invariance principle for balanced random walks in time-dependent ergodic random environments. All the mentioned results for fully balanced RWRE construct an invariant measure for the environment viewed from the particle, using a discrete maximum principle à la Kuo–Trudinger [15] (or versions thereof). For such a principle to apply the assumption of a balanced environment is essential. Here we follow a completely different account, where in turn the perturbative setting is essential.

Since the results of [2] do also provide local estimates, we believe that with some more effort, Theorem 2 could be improved to a local central limit theorem. Furthermore, we expect that our results remain true without assuming Condition \(\mathbf{B}\) (in particular in view of the results [10, 22]). Getting rid of this condition would however require a full control over large sojourn times, which remains a major open problem.

Organization of the paper and rough strategy of the proofs

We first introduce the most important notation. For ease of readability, we recapitulate in Sect. 2 those concepts and results from [2] which play a major role here. In Sect. 3 we provide the necessary control over Green’s functions. To a large extend, we can rely on the results from [2], but we need additional difference estimates for our results on mean sojourn times.

In Sect. 4, we prove Theorem 1. In this regard, we shall first show that with high probability, the quenched (normalized) mean times \({\text {E}}_{0,\omega }[\tau _L]/L^2\) lie for large L in a small interval \([1-\eta ,1+\eta ]\) around 1 (Proposition 3). This involves the propagation of a technical Condition \(\mathbf{C2}\) (see Sect. 4.1). Once we have established this, we prove convergence of the (non-random) sequence \(\mathbb {E}[{\text {E}}_{0,\omega }[\tau _L]]/L^2\) towards a constant \(D\in [1-\eta ,1+\eta ]\) (Proposition 4), where \(\mathbb {E}\) denotes the expectation with respect to \(\mathbb {P}\). Finally, a concentration argument shows that with high probability, \({\text {E}}_{0,\omega }[\tau _L]/L^2\) is close to its mean \(\mathbb {E}[{\text {E}}_{0,\omega }[\tau _L]]/L^2\) (Lemma 13). This will allow us to deduce Theorem 1.

In the last part of this paper starting with Sect. 5, we show how Theorem 1 can be combined with the results on exit laws from [2] to obtain Theorem 2. A strategy of proof of this statement can be found at the beginning of Sect. 5.

Some notation

We collect here some notation that is frequently used in this text.

Sets and distances

We put \({\mathbb {N}} = {\mathbb {N}}_0= \{0,1,2,3,\dots \}\). For \(x\in {\mathbb {R}}^d\), |x| is the Euclidean norm of x. The distance between \(A, B \subset {\mathbb {R}}^d\) is denoted \({\text {d}}(A,B) = \inf \{|x-y| : x\in A,\; y\in B\}\). Given \(L > 0\), we let \(V_L=\{x\in {\mathbb {Z}}^d : |x| \le L\}\), and for \(x\in {\mathbb {Z}}^d\), \(V_L(x) = x + V_L\). Similarly, put \(C_L=\{x\in {\mathbb {R}}^d : |x| < L\}\). The outer boundary of \(V\subset {\mathbb {Z}}^d\) is given by \(\partial V =\{x\in {\mathbb {Z}}^d\backslash V: {\text {d}}(\{x\},V) = 1\}\). For \(x\in \overline{C}_L\), we let \({\text {d}}_L(x) = L-|x|\). Note that for \(x\in V_L\), \({\text {d}}_L(x)\le {\text {d}}(\{x\},\partial V_L)\). Finally, for \(0\le a<b\le L\), put

$$\begin{aligned} {\text {Sh}}_L(a,b)=\{x\in V_L : a\le {\text {d}}_L(x) < b\},\quad {\text {Sh}}_L(b) = {\text {Sh}}_L(0,b). \end{aligned}$$

Functions

We use the usual notation \(a\wedge b = \min \{a,b\}\) for reals ab. We further write \(\log \) for the logarithm to the base e. Given two functions \(F,G : {\mathbb {Z}}^d\times {\mathbb {Z}}^d\rightarrow {\mathbb {R}}\), we write FG for the (matrix) product \(FG(x,y) = \sum _{u\in {\mathbb {Z}}^d}F(x,u)G(u,y)\), provided the right-hand side is absolutely summable. \(F^k\) is the kth power defined in this way, and \(F^0(x,y) = \delta _x(y)\). F can also operate on functions \(f:{\mathbb {Z}}^d\rightarrow {\mathbb {R}}\) from the left via \(Ff(x) = \sum _{y\in {\mathbb {Z}}^d}F(x,y)f(y)\).

As usual, \(1_W\) stands for the indicator function of the set W, but we will also write \(1_W\) for the kernel \((x,y)\mapsto 1_W(x)\delta _x(y)\), where the Delta function \(\delta _x(y)\) is equal to one if \(y=x\) and zero otherwise. If \(f:{\mathbb {Z}}^d\rightarrow {\mathbb {R}}\), \(\Vert f\Vert _1 = \sum _{x\in {\mathbb {Z}}^d}|f(x)| \in [0,\infty ]\) denotes its \(L^1\)-norm. For a (signed) measure \(\nu : {\mathbb {Z}}^d\rightarrow {\mathbb {R}}\), we write \(\Vert \nu \Vert _1\) for its total variation norm.

Transition kernels, exit times and exit measures

Denote by \(\mathcal {G}\) the \(\sigma \)-algebra on \(({\mathbb {Z}}^d)^{{\mathbb {N}}}\) generated by the cylinder functions. If \(p = {(p(x,y))}_{x,y\in {\mathbb {Z}}^ d}\) is a family of (not necessarily nearest neighbor) transition probabilities, we write \({\text {P}}_{x,p}\) for the law of the canonical random walk \({(X_n)}_{n\ge 0}\) on \(({({\mathbb {Z}}^d)}^{{\mathbb {N}}},\mathcal {G})\) started from \(X_0=x\) \({\text {P}}_{x,p}\)-a.s. and evolving according to the kernel p.

The simple random walk kernel is denoted \(p_o(x,x\pm e_i) = 1/(2d)\), and we write \({\text {P}}_{x}\) instead of \({\text {P}}_{x,p_o}\). For transition probabilities \(p_\omega \) defined in terms of an environment \(\omega \), we use the notation \({\text {P}}_{x,\omega }\). The corresponding expectation operators are denoted by \({\text {E}}_{x,p}\), \({\text {E}}_x\) and \({\text {E}}_{x,\omega }\), respectively. Every \(p\in {\mathcal {P}}\) gives in an obvious way rise to a homogeneous nearest neighbor transition kernel on \({\mathbb {Z}}^d\), which we again denote by p.

For a subset \(V\subset {\mathbb {Z}}^d\), we let \(\tau _V = \inf \{n\ge 0 : X_n\notin V\}\) be the first exit time from V, with \(\inf \emptyset = \infty \).

Given \(x,z\in {\mathbb {Z}}^d\), \(p\in {\mathcal {P}}\) and a subset \(V\subset {\mathbb {Z}}^d\), we define

$$\begin{aligned} \pi ^{(p)}_V(x,z)= {\text {P}}_{x,p}\left( X_{\tau _V}=z\right) . \end{aligned}$$

For an environment \(\omega \in {\varOmega }\), we set

$$\begin{aligned} \varPi _{V,\omega }(x,z)= {\text {P}}_{x,\omega }\left( X_{\tau _V}=z\right) . \end{aligned}$$

We mostly drop \(\omega \) in the notation and interpret \(\varPi _V(x,\cdot )\) as a random measure.

Recall the definitions of the sets \(\mathcal {P}\), \(\mathcal {P}^{\text {s},1}\) and \(\mathcal {P}_\varepsilon \) from the introduction. For \(0<\kappa <1/(2d)\), let

$$\begin{aligned} \mathcal {P}^{\text {s}}_\kappa =\{p\in \mathcal {P}_\kappa : p(e_i)=p(-e_i),\,i=1,\dots ,d\}, \end{aligned}$$

i.e. \(\mathcal {P}^{\text {s}}_\kappa \) is the subset of \(\mathcal {P}_{\kappa }\) which contains all symmetric probability distributions on \(\{\pm e_i : i = 1,\dots ,d\}\). The parameter \(\kappa \) was introduced in [2] to bound the range of the symmetric transition kernels under consideration. We can think of \(\kappa \) as a fixed but arbitrarily small number (the perturbation parameter \(\varepsilon \) is chosen afterward).

Miscellaneous comments about notation

Our constants are positive and depend only on the dimension \(d\ge 3\) unless stated otherwise. In particular, they do not depend on L, \(p\in \mathcal {P}^{\text {s}}_\kappa \), \(\omega \) or on any point \(x\in {\mathbb {Z}}^d\).

By C and c we denote generic positive constants whose values can change even in the same line. For constants whose values are fixed throughout a proof we often use the symbols \(K, C_1, c_1\).

Many of our quantities, e.g. the transition kernels \(\hat{{\varPi }}_L\), \(\hat{\pi }_L\) or the kernel \(\mathrm {\Gamma }_L\), are indexed by L. We normally drop the index in the proofs. In contrast to [2], these kernels do here not depend on an additional parameter r.

We often drop the superscript (p) from notation and write \(\pi _V\) for \(\pi ^{(p)}_V\). If \(V=V_L\) is the ball around zero of radius L, we write \(\pi _L\) instead of \(\pi _V\), \(\varPi _L\) for \(\varPi _V\) and \(\tau _L\) for \(\tau _{V}\).

By \(\text {P}\) we denote sometimes a generic probability measure, and by \(\text {E}\) its corresponding expectation. If A and B are two events, we often write \(\text {P}(A;\, B)\) for \(\text {P}(A\cap B)\).

If we write that a statement holds for “L large (enough)”, we implicitly mean that there exists some \(L_0>0\) depending only on the dimension such that the statement is true for all \(L\ge L_0\). This applies also to phrases like “\(\delta \) (or \(\varepsilon \), or \(\kappa \)) small (enough)”.

Some of our statements are only valid for large L and \(\varepsilon \) (or \(\delta \), or \(\kappa \)) sufficiently small, but we do not mention this every time.

Results and concepts from the study of exit laws

Our approach uses results and constructions from [2], where exit measures from large balls under \(\mathbf{A0}\) and \(\mathbf{A1}(\varepsilon )\) are studied. We adapt in this section those parts which will be frequently used in this paper. Some auxiliary statements from [2], which play only a minor role here, will simply be cited when they are applied.

The overall idea of [2] is to use a renormalization scheme to transport estimates on exit measures inductively from one scale to the next, via a perturbation expansion for the Green’s function, which we recall first.

A perturbation expansion

Let \(p = \left( p(x,y)\right) _{x,y\in {\mathbb {Z}}^d}\) be a family of finite range transition probabilities on \({\mathbb {Z}}^d\), and let \(V\subset {\mathbb {Z}}^d\) be a finite set. The corresponding Green’s kernel or Green’s function for V is defined by

$$\begin{aligned} g_V(p)(x,y) = \sum _{k=0}^\infty \left( 1_Vp\right) ^k(x,y). \end{aligned}$$
(3)

We now write g for \(g_V(p)\). Let P be another transition kernel with corresponding Green’s function G for V. With \(\mathrm {\Delta }= 1_V\left( P-p\right) \), the resolvent equation gives

$$\begin{aligned} G -g = g\mathrm {\Delta }G = G\mathrm {\Delta }g. \end{aligned}$$
(4)

An iteration of (4) leads to further expansions. Namely, first one has

$$\begin{aligned} G -g = \sum _{k=1}^\infty \left( g\mathrm {\Delta }\right) ^kg. \end{aligned}$$
(5)

Of course, for this to make sense the right hand side has to converge, but this will never be problem in our setting. Next, if we replace the rightmost g by \(g(x,\cdot ) = \delta _x(\cdot )+1_Vpg(x,\cdot )\) and put \(R= \sum _{k=1}^\infty \mathrm {\Delta }^kp\), we obtain

$$\begin{aligned} G = g\sum _{m=0}^\infty {\left( Rg\right) }^m\sum _{k=0}^\infty \mathrm {\Delta }^k. \end{aligned}$$
(6)

Remark 1

In our case, G will mostly play the role of a RWRE Green’s function in a ball V, while a small g will represent a Green’s function in V stemming from some homogeneous random walk. One should note that for \(x\in V\), \(z\notin V\), g(xz) and G(xz) are given by the probability that the corresponding random walk started at x exits V through z.

Coarse grained transition kernels

We fix once for all a probability density \(\varphi \in C^\infty ({\mathbb {R}}_+,{\mathbb {R}}_+)\) with compact support in (1, 2). Given a transition kernel \(p\in {\mathcal {P}}\) and a strictly positive function \(\psi = (m_x)_{x\in W}\), where \(W\subset {\mathbb {R}}^d\), we define coarse grained transition kernels on \(W\cap {\mathbb {Z}}^d\) associated to \((\psi ,\,p)\),

$$\begin{aligned} \hat{\pi }^{(p)}_{\psi }(x,\cdot ) = \frac{1}{m_x}\int _{{\mathbb {R}}_+} \varphi \left( \frac{t}{m_x}\right) \pi _{V_t(x)}^{(p)}(x,\cdot )\text {d}t,\quad x\in W\cap {\mathbb {Z}}^d. \end{aligned}$$
(7)

Often \(\psi \equiv m>0\) will be a constant, and then (7) makes sense for all \(x\in {\mathbb {Z}}^ d\) and therefore gives coarse grained transition kernels on the whole grid \({\mathbb {Z}}^d\).

Note that a Markov chain with transition kernel given by (7) performs a step of a size between \(m_x\) and \(2m_x\) when located at x. To put it differently, its transition probabilities are given by averaged exit distributions from balls \(V_t(x)\), \(t\in [m_x,2m_x]\). The density \(\varphi \) appearing in the definition makes the kernel smooth enough.

We now introduce particular coarse grained transition kernels for walking inside the ball \(V_L\), for both symmetric random walk and RWRE. We set up a coarse graining scheme which will make the link between different scales and allows us to transport estimates on mean sojourn times from one level to the next, see Sect. 4 and in particular Lemma 6 there. A similar scheme depending on an additional parameter r was used in [2] to lift estimates on exit measures to the next level.

Let

$$\begin{aligned} s_L = \frac{L}{(\log L)^3} \quad \text{ and }\quad r_L= \frac{L}{(\log L)^{15}}. \end{aligned}$$

We fix a smooth function \(h : {\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) satisfying

$$\begin{aligned} h(x)= \left\{ \begin{array}{l@{\quad }l} x &{}\text{ for } x\le 1/2\\ 1 &{}\text{ for } x\ge 2 \end{array}\right. , \end{aligned}$$

such that h is concave and increasing on (1 / 2, 2). Define \(h_L : \overline{C}_L \rightarrow {\mathbb {R}}_+\) by

$$\begin{aligned} h_L(x) = \frac{1}{20}\max \left\{ s_L h\left( \frac{{\text {d}}_L(x)}{s_L}\right) ,\, r_L\right\} . \end{aligned}$$
(8)

We write \(\hat{{\varPi }}_L=\hat{{\varPi }}_{L,\omega }\) for the coarse grained RWRE transition kernel inside \(V_L\) associated to \((\psi = \left( h_L(x)\right) _{x\in V_L},p_\omega )\),

$$\begin{aligned} \hat{{\varPi }}_L(x,\cdot ) = \frac{1}{h_L(x)}\int _{{\mathbb {R}}_+} \varphi \left( \frac{t}{h_L(x)}\right) \varPi _{V_t(x)\cap V_L}(x,\cdot )\text {d}t, \end{aligned}$$
(9)

and \(\hat{\pi }^{(p)}_L\) for the corresponding kernel coming from symmetric random walk with transition kernel \(p\in \mathcal {P}^{\text {s}}_\kappa \), where in the definition (9) the random RWRE exit measure \(\varPi \) is replaced by \(\pi ^{(p)}\). For points \(x\in {\mathbb {Z}}^d\backslash V_L\), we set \(\hat{{\varPi }}_L(x,\cdot ) =\hat{\pi }^{(p)}_L(x,\cdot ) = \delta _x(\cdot )\). See Fig. 1.

Fig. 1
figure 1

The coarse graining scheme in \(V_L\): in \(V_L\backslash {\text {Sh}}_L(2s_L)\), the exit distributions are taken from balls of radii between \((1/20)s_L\) and \((1/10)s_L\). When entering \({\text {Sh}}_L(2s_L)\), the coarse graining radii start to shrink, up to the boundary layer \({\text {Sh}}_L(r_L)\), where the exit distributions are taken from intersected balls \(V_t(x)\cap V_L\), \(t\in [(1/20)r_L,\,(1/10)r_L]\). The dotted lines indicate the one-step path which underlies the coarse grained walk

Note our small abuse of notation: \(\hat{\pi }^{(p)}_L\) is always defined in this way and does never denote the coarse grained kernel (7) associated to the constant function \(\psi \equiv L\). Also note that \(\hat{{\varPi }}_L\) was denoted \(\hat{{\varPi }}_{L,r_L}\) in [2], and similarly \(\hat{\pi }_L\) was denoted \(\hat{\pi }_{L,r_L}\). The kernel \(\hat{{\varPi }}_L\) is a random transition kernel depending on \(\omega \). However, when we consider \(\hat{{\varPi }}_L\) under \({\text {P}}_{x,\omega }\), then \(\omega \) is fixed, but also in this case we usually write \(\hat{{\varPi }}_L\) instead of \(\hat{{\varPi }}_{L,\omega }\).

Two Green’s function will play a crucial role (cf. (3)).

  • \(\hat{G}_L\) denotes the (coarse grained) RWRE Green’s function corresponding to \(\hat{{\varPi }}_L\).

  • \(\hat{g}^{(p)}_L\) denotes the Green’s function corresponding to \(\hat{\pi }^{(p)}_L\).

The “goodified” version \(\hat{G}^g_L\) of \(\hat{G}_L\) will be introduced in Sect. 2.5.

Propagation of Condition C1

We recapitulate in this part the technical Condition \(\mathbf{C1}(\delta ,L_0,L_1)\), which is propagated in [2] from level \(L_1\) to level \(L_1(\log L_1)^2\). Roughly, Condition \(\mathbf{C1}(\delta ,L_0,L_1)\) allows to control the RWRE exit measure from balls of radii \(\le L_1\) in terms of the exit measure coming from a homogeneous random walk with a certain symmetric nearest-neighbor kernel \(p_L\). We first define the family of kernels \((p_L)_L\).

Assignment of transition kernels

Let \(L_0>0\) (\(L_0\) shall play the role of a large constant). We define L-dependent symmetric transition kernels by

$$\begin{aligned} p_L(\pm e_i) =\left\{ \begin{array}{l@{\quad }l} p_o(\pm e_i)=1/(2d)&{}\text{ for } 0<L\le L_0\\ \frac{1}{2}\sum _{y\in {\mathbb {Z}}^d}\mathbb {E}\left[ \hat{{\varPi }}_L(0,y)\right] \frac{y_i^2}{|y|^2}&{} \text{ for } L>L_0 \end{array}\right. . \end{aligned}$$
(10)

To be in position to formulate Condition \(\mathbf{C1}\), we introduce some notation.

Let \({\mathcal {U}}_t=\{x\in {\mathbb {R}}^d: t/2<|x|<2t\}\). We write \({\mathcal {M}}_t\) for the set of smooth functions \(\psi :W\rightarrow {\mathbb {R}}_+\) defined on some open set W with \({\mathcal {U}}_t\subset W\subset {\mathbb {R}}^d\), which have the property that their first four derivatives are bounded uniformly by 10 and

$$\begin{aligned} \psi \left( {\mathcal {U}}_t\right) \subset (t/10, 5t). \end{aligned}$$

Coarse grained transition kernels associated to functions in \({\mathcal {M}}_t\) via (7) turn out to have good smoothing properties, see [2] for more on this. For \(p,q\in {\mathcal {P}}\) and \(\psi \in {\mathcal {M}}_t\), define

$$\begin{aligned} D_{t,p,\psi ,q}^{*}&=\sup _{x\in V_{t/5}}\left\| \left( \varPi _{V_t}-\pi ^{(p)}_{V_t}\right) \hat{\pi }_{\psi }^{(q)}(x,\cdot )\right\| _1,\\ D_{t,p}^*&= \sup _{x\in V_{t/5}}\left\| \left( \varPi _{V_t}-\pi ^{(p)}_{V_t}\right) (x,\cdot )\right\| _1. \end{aligned}$$

The first quantity \(D_{t,p,\psi ,q}^{*}\) is a smoothed difference of exit laws measured in the total variation distance, one coming from RWRE and one coming from a homogeneous random walk with kernel p. The smoothing kernel \(\hat{\pi }_{\psi }^{(q)}\) is defined in (9). Note that the underlying one-step transition kernel q can differ from p. The second quantity \(D_{t,p}^*\) is a non-smoothed difference of the exit laws. Of course, both \(D_{t,p,\psi ,q}^{*}\) and \(D_{t,p}^*\) are random variables.

Next, with \(\delta > 0\), we put for \(i=1,2,3\)

$$\begin{aligned}&b_i(L,p,\psi ,q,\delta )\\&\quad = \mathbb {P}\left( \left\{ (\log L)^{-9+9(i-1)/4} < D_{L,p,\psi ,q}^*\le (\log L)^{-9 +9i/4}\right\} \cap \left\{ D_{L,p}^*\le \delta \right\} \right) , \end{aligned}$$

and

$$\begin{aligned} b_4(L,p,\psi ,q,\delta )= \mathbb {P}\left( \left\{ D_{L,p,\psi ,q}^*> (\log L)^{-3+3/4}\right\} \cup \left\{ D_{L,p}^*> \delta \right\} \right) . \end{aligned}$$

We further let \(\iota = (\log L_0)^{-7}\). This constant will bound the range of the kernels q which may be used for the smoothing step. Condition \(\mathbf{C1}\) is now given as follows.

Condition C1

Let \(\delta > 0\) and \(L_1\ge L_0\ge 3\). We say that \(\mathbf{C1}(\delta ,L_0,L_1)\) holds if

  • For all \(3\le L\le 2L_0\), all \(\psi \in {\mathcal {M}}_L\) and all \(q\in \mathcal {P}^{\text {s}}_\iota \),

    $$\begin{aligned} \mathbb {P}\left( \left\{ D_{L,p_o,\psi ,q}^*> (\log L)^{-9}\right\} \cup \left\{ D_{L,p_o}^*> \delta \right\} \right) \le \exp \left( -(\log (2L_0))^2\right) . \end{aligned}$$
  • For all \(L_0< L\le L_1\), \(L'\in [L,2L]\), \(\psi \in {\mathcal {M}}_{L'}\) and \(q\in \mathcal {P}^{\text {s}}_\iota \),

    $$\begin{aligned} b_i(L',p_L,\psi ,q,\delta )\le \frac{1}{4}\exp \left( -\left( (3+i)/4\right) (\log L')^2\right) \quad \text{ for } i=1,2,3,4. \end{aligned}$$

Let us give some explanations. Under \(\mathbf{C1}(\delta ,L_0,L_1)\), we deduce from the first point that both the smoothed and non-smoothed distance between the exit measures of RWRE and simple random walk from balls of radii \(L\le 2L_0\) is small in total variation (recall that \(L_0\) should be thought of as a large constant).

The second point gives a similar control for larger radii \(L_0<L\le L_1\). However, simple random walk is now replaced by a homogeneous random walk with kernel \(p_L\), and the control is more precise since it distinguishes four different ranges of the distance, see the definition of the \(b_i\)’s. This allowed a propagation of estimates in [2], but it will play no role here; the important consequence for us is the bound (11), see the explanation below.

The main statement for \(\mathbf{C1}\)

The first part of [2, Proposition 2.1] implies the following statement.

Proposition 1

Assume \(\mathbf{A1}\). For \(\delta >0\) small enough, there exist \(L_0=L_0(\delta )\) large and \(\varepsilon _0 = \varepsilon _0(\delta ) > 0\) small with the following property: If \(\varepsilon \le \varepsilon _0\) and \(\mathbf{A0}(\varepsilon )\) is satisfied, then \(\mathbf{C1}\left( \delta ,L_0,L\right) \) holds for all \(L\ge L_0\).

Note that for every choice of \(\delta \) and \(L_0\), it is easy to make sure that Condition \(\mathbf{C1}(\delta ,L_0,L_0)\) is fulfilled: We just have to choose the perturbation strength \(\varepsilon \) small enough. This was used to trigger the inductive proof of the proposition in [2]: it was shown there (for suitable choices of \(\delta \) and \(L_0\)) that if \(\mathbf{C1}(\delta ,L_0,L_1)\) holds for some \(L_1\ge L_0\), then also \(\mathbf{C1}(\delta ,L_0,L_1(\log L_1)^2)\) holds. Now one can choose \(\varepsilon \) so small such that \(\mathbf{C1}(\delta ,L_0,L_0)\) is fulfilled, and then \(\mathbf{C1}\left( \delta ,L_0,L\right) \) holds for all \(L\ge L_0\).

For us, the important implication is that if \(\mathbf{C1}(\delta ,L_0,L_1)\) is satisfied, then for any \(3\le L\le L_1\) and for all \(L'\in [L,2L]\), all \(\psi \in {\mathcal {M}}_{L'}\) and all \(q\in \mathcal {P}^{\text {s}}_\iota \),

$$\begin{aligned} \mathbb {P}\left( \left\{ D_{L',p_L,\psi ,q}^*> (\log L')^{-9}\right\} \cup \left\{ D_{L',p_L}^*> \delta \right\} \right) \le \exp \left( -(\log L')^2\right) . \end{aligned}$$
(11)

Indeed, note that if \(3\le L\le L_0\), \(L'\in [L,2L]\), \(\psi \in {\mathcal {M}}_{L'}\) and \(q\in \mathcal {P}^{\text {s}}_\iota \), the first point of Condition \(\mathbf{C1}\) gives the right bound for the probability on the left hand side of (11), whereas in the case \(L_0< L\le L_1\), \(L'\in [L,2L]\), \(\psi \in {\mathcal {M}}_{L'}\) and \(q\in \mathcal {P}^{\text {s}}_\iota \), we bound the left hand side of (11) with the second point of \(\mathbf{C1}\) by

$$\begin{aligned} \sum _{i=1}^4 b_i(L',p_L,\psi ,q,\delta ) \le \exp \left( -(\log L')^2\right) . \end{aligned}$$

In [2, Lemma 3.2] it is shown that the transition kernels \(p_L\) defined in (10) form a Cauchy sequence. Their limit is given by the kernel \(p_\infty \) defined in (1), i.e.

$$\begin{aligned} \lim _{L\rightarrow \infty }p_L(e_i)=p_{\infty }(e_i)\quad \text{ for } i=1,\dots ,d. \end{aligned}$$
(12)

From this fact and (11), one can deduce that the difference in total variation of the exit laws \(\varPi _L\) and \(\pi _L^{(p_\infty )}\) is small when L is large, in both a smoothed and non-smoothed way. See Theorems 1.1 and 1.2 of [2] for precise statements. For us, it will be sufficient to keep in mind (11) and (12).

In the sequel, we write “assume \(\mathbf{C1}(\delta ,L_0,L_1)\)”, if we assume \(\mathbf{C1}(\delta ,L_0,L_1)\) for some \(\delta >0\) and some \(L_1\ge L_0\), where \(\delta \) can be chosen arbitrarily small and \(L_0\) arbitrarily large.

Good and bad points

In [9] and [2, Section 3.2], the concept of good and bad points inside \(V_L\) was introduced. It played a major role in propagating Condition \(\mathbf{C1}(\delta ,L_0,L)\) from one level to the next. It turns out that for controlling mean sojourn times, we need a stronger notion of “goodness”, see Sect. 4.3. It is however more convenient to first recall the original classification.

Recall Assignment (10). A point \(x\in V_L\) is good (with respect to L and \(\delta >0\)), if

  • For all \(t\in [h_L(x),2h_L(x)]\), with \(q=p_{h_L(x)}\),

    $$\begin{aligned} \left\| \left( \varPi _{V_t(x)}-\pi ^{(q)}_{V_t(x)}\right) (x,\cdot )\right\| _1 \le \delta . \end{aligned}$$
  • If \({\text {d}}_L(x) > 2r_L\), then additionally

    $$\begin{aligned} \left\| \left( \hat{{\varPi }}_L - \hat{\pi }^{(q)}_L\right) \hat{\pi }^{(q)}_L(x,\cdot )\right\| _1 \le \left( \log h_L(x)\right) ^{-9}. \end{aligned}$$

The set of environments where all points \(x\in V_L\) are good is denoted \({\text {Good}}_L\). A point \(x\in V_L\) which is not good is called bad, and the set of all bad points inside \(V_L\) is denoted by \(\mathcal {B}_L=\mathcal {B}_L(\omega )\).

Goodified transition kernels and Green’s function

By replacing the coarse grained RWRE transition kernel at bad points \(x\in V_L\) by that of a homogeneous symmetric random walk, we obtain what we call goodified transition kernels inside \(V_L\).

More specifically, write p for \(p_{s_L/20}\) stemming from Assignment (10). The goodified transition kernels are defined as follows.

$$\begin{aligned} \hat{{\varPi }}^g_L(x,\cdot ) = \left\{ \begin{array}{l@{\quad }l} \hat{{\varPi }}_L(x,\cdot ) &{} \text{ for } x \in V_L\backslash \mathcal {B}_L\\ \hat{\pi }^{(p)}_L(x,\cdot ) &{} \text{ for } x \in \mathcal {B}_L \end{array}\right. . \end{aligned}$$
(13)

We write \(\hat{G}^g_L\) for the corresponding (random) Green’s function defined via (3).

Remark 2

Proposition 1 will allow us to concentrate on environments \(\omega \in {\text {Good}}_L\), where \(\hat{{\varPi }}_L=\hat{{\varPi }}^g_L\) and therefore also \(\hat{G}_L=\hat{G}^g_L\). Indeed, we note that for L large and \(x\in V_L\) with \({\text {d}}_L(x) > 2r_L\), the function \(h_{L}(x+\cdot )\) lies in \({\mathcal {M}}_t\) for each \(t\in [h_{L}(x),2h_{L}(x)]\). Thus, for all \(x\ \in V_L\), we can use \(\mathbf{C1}(\delta ,L_0,L_1)\) to control the event \(\{x\in \mathcal {B}_{L}\}\), provided \(2h_{L}(x) \le L_1\). With Proposition 1 at hand, an application of Borel-Cantelli then shows that

See Lemma 8, where the necessary arguments are given.

In the next section, we provide the necessary estimates for the goodified coarse grained RWRE Green’s function \(\hat{G}^g_L\).

Control on Green’s functions

We first recapitulate some estimates on Green’s functions. Then we establish difference estimates for these functions, which will be used to control differences of (quenched) mean sojourn times from balls \(V_t(x)\) and \(V_t(y)\) that have a sufficiently large intersection.

Bounds on Green’s functions

Recall that \(\mathcal {P}^{\text {s}}_\kappa \) denotes the set of kernels which are symmetric in every coordinate direction and \(\kappa \)-perturbations of the simple random walk kernel. The statements in this section are valid for small \(\kappa \), in the sense that there exists \(0<\kappa _0<1/(2d)\) such that for \(0<\kappa \le \kappa _0\), the statements hold true for \(p\in \mathcal {P}^{\text {s}}_\kappa \), with constants that are uniform in \(p\in \mathcal {P}^{\text {s}}_\kappa \).

Let \(p\in \mathcal {P}^{\text {s}}_\kappa \) and \(m\ge 1\). Denote by the coarse grained transition probabilities on \({\mathbb {Z}}^d\) associated to \(\psi _m = {(m_x)}_{x\in {\mathbb {Z}}^d}\), where \(m_x= m\) is chosen constant in x, cf. (7). We constantly drop p from notation. The kernel \(\hat{\pi }_{\psi _m}\) is centered, with covariances

$$\begin{aligned} \sum _{y\in {\mathbb {Z}}^d}(y_i-x_i)(y_j-x_j)\hat{\pi }_{\psi _m}(x,y) = \lambda _{m,i}\delta _{i}(j), \end{aligned}$$

where for large m, \(C^{-1}<\lambda _{m,i}/m^2< C\) for some \(C>0\). We set

$$\begin{aligned} \varvec{\Lambda }_m= \left( \lambda _{m,i}\delta _i(j)\right) _{i,j=1}^d,\quad \mathcal {J}_m(x) = |\varvec{\Lambda }_m^{-1/2}x|\quad \hbox {for } x\in {\mathbb {Z}}^d, \end{aligned}$$

and denote by

$$\begin{aligned} \hat{g}_{m,{{\mathbb {Z}}^d}}(x,y) = \sum _{n=0}^\infty (\hat{\pi }_{\psi _m})^n(x,y) \end{aligned}$$

the Green’s function corresponding to \(\hat{\pi }_{\psi _m}\). In [2], the following behavior of \(\hat{g}_{m,{{\mathbb {Z}}^d}}\) was established. The proof is standard and based on a local central limit theorem for \(\hat{\pi }_{\psi _m}\), which we do not restate here.

Proposition 2

Let \(p\in \mathcal {P}^{\text {s}}_\kappa \). Let \(x,y\in {\mathbb {Z}}^d\), and assume \(m \ge m_0 > 0\) large enough.

  1. (i)

    For \(|x-y| < 3m\),

    $$\begin{aligned} \hat{g}_{m,{{\mathbb {Z}}^d}}(x,y) = \delta _{x}(y) + O(m^{-d}). \end{aligned}$$
  2. (ii)

    For \(|x-y| \ge 3m\), there exists a constant \(c(d) > 0\) such that

    $$\begin{aligned} \hat{g}_{m,{{\mathbb {Z}}^d}}(x,y)= \frac{c(d)\det \varvec{\Lambda }_m^{-1/2}}{\mathcal {J}_m(x-y)^{d-2}} + O\left( \frac{1}{|x-y|^{d}}\left( \log \frac{|x-y|}{m}\right) ^d\right) . \end{aligned}$$

Recall that \(\tau _L=\tau _{V_L}\) denotes the first exit time from \(V_L\). The proposition can be used to estimate the corresponding Green’s function for \(V_L\),

$$\begin{aligned} \hat{g}_{m, V_L}(x,y) = \sum _{n=0}^\infty \left( 1_{V_L}\hat{\pi }_{\psi _m}\right) ^n(x,y). \end{aligned}$$

Indeed, \(\hat{g}_{m, V_L}\) is bounded from above by \(\hat{g}_{m,{{\mathbb {Z}}^d}}\), and more precisely, the strong Markov property shows

$$\begin{aligned} \hat{g}_{m,V_L}(x,y) = {\text {E}}_{x,\hat{\pi }_{\psi _m}}\left[ \sum _{k=0}^{\tau _L-1}1_{\{X_k = y\}}\right] = \hat{g}_{m,{\mathbb {Z}}^d}(x,y) - {\text {E}}_{x,\hat{\pi }_{\psi _m}}\left[ \hat{g}_{m,{\mathbb {Z}}^d}\left( X_{\tau _L},y\right) \right] .\nonumber \\ \end{aligned}$$
(14)

Here, according to our notational convention, \({\text {E}}_{x,\hat{\pi }_{\psi _m}}\) is the expectation with respect to \({\text {P}}_{x,\hat{\pi }_{\psi _m}}\), the law of a random walk started at x and running with kernel \(\hat{\pi }_{\psi _m}\).

We next recall the definition of the (deterministic) kernel \(\mathrm {\Gamma }_L\), which was introduced in [2] to dominate coarse grained Green’s functions from above.

We formulate our definitions and results in terms of the larger ball \(V_{L+r_L}\), so that we can refer to the proofs given in [2]. Note that we always work with \(r=r_L\) here. For \(x\in V_{L+r_L}\), let

$$\begin{aligned} {\tilde{{\text {d}}}}(x) = \max \left( \frac{{\text {d}}_{L+r_L}(x)}{2},3r_L\right) ,\quad a(x) =\min \left( {\tilde{{\text {d}}}}(x),s_L\right) . \end{aligned}$$

For \(x,y\in V_{L+r_L}\), the kernel \(\mathrm {\Gamma }_L\) is now defined by

$$\begin{aligned} \mathrm {\Gamma }_L(x,y) = \min \left\{ \frac{\tilde{d}(x)\tilde{d}(y)}{a(y)^2(a(y) + |x-y|)^d},\,\frac{1}{a(y)^2(a(y) + |x-y|)^{d-2}}\right\} . \end{aligned}$$
(15)

In the proofs, we mostly write \(\varGamma \) instead of \(\varGamma _L\).

For \(x\in V_{L+r_L}\), we let \(U(x) = V_{a(x)}(x)\cap V_{L+r_L}\) denote the a(x)-neighborhood around x. Given two positive functions \(F, G: V_{L+r_L}\times V_{L+r_L}\rightarrow {\mathbb {R}}_+\), we write \(F\preceq G\) if for all \(x,y\in V_{L+r_L}\),

$$\begin{aligned} F(x,U(y)) \le G(x,U(y)), \end{aligned}$$

where F(xU) stands for \(\sum _{y\in U\cap {\mathbb {Z}}^d}F(x,y)\). We write \(F\asymp 1\), if there is a constant \(C>0\) such that for all \(x,y\in V_{L+r_L}\),

$$\begin{aligned} \frac{1}{C}F(x,y)\le F(\cdot ,\cdot )\le CF(x,y)\quad \text{ on } U(x)\times U(y). \end{aligned}$$

We now formulate one of the key estimates, which shows how both \(\hat{g}^{(q)}_L\) and \(\hat{G}^g_L\) can be dominated from above by the deterministic kernel \(\mathrm {\Gamma }_L\).

Lemma 1

  1. (i)

    There exists a constant \(C>0\) such that for all \(q\in \mathcal {P}^{\text {s}}_\kappa \),

    $$\begin{aligned} \hat{g}^{(q)}_L\preceq C\mathrm {\Gamma }_L. \end{aligned}$$
  2. (ii)

    Assume \(\mathbf{C1}(\delta ,L_0,L_1)\), and let \(L_1\le L\le L_1(\log L_1)^2\). There exists a constant \(C>0\) such that for \(\delta >0\) small,

    $$\begin{aligned} \hat{G}^g_L \preceq C\mathrm {\Gamma }_L. \end{aligned}$$

Let us recall the ideas behind the proof which is given in [2, Lemma 5.2]. For (i), one has to show that \(\hat{g}^{(q)}_L(x,U(y))\le C\varGamma _L(x,U(y))\) for all \(x,y\in V_{L+r_L}\). Roughly, this follows from hitting estimates of symmetric random walk killed upon exiting the ball \(V_{L+r_L}\), see [2, Lemma 4.2], and from the bound on the lattice Green’s function stated in Proposition 2.

For (ii), the idea is to use the perturbation expansion (6) to express \(\hat{G}^g_L\) as an infinite series in terms of \(\hat{g}^{(p)}_L\), where \(p=p_{s_L/20}\), and differences \(\varDelta =1_{V_L}(\hat{{\varPi }}^g_L-\hat{\pi }^{(p)}_L)\) and their smoothed versions \(\varDelta \hat{\pi }^{(p)}_L\). The deterministic Green’s function \(\hat{g}^{(p)}_L\) is controlled by means of part (i). The differences \(\varDelta \) and \(\varDelta \hat{\pi }^{(p)}_L\) involve exit measures on a smaller scale which are under good control by definition of the goodified transition kernel \(\hat{{\varPi }}^g_L\), cf. (13). One has then to show that these kernels behave well under convolution, see [2, Proposition 5.4]. Similar arguments will also be used later in the proof of Lemma 3 (ii).

The importance of Lemma 1 comes from the following fact: Proposition 1 states that Condition\(~\mathbf{C1}(\delta ,L_0,L)\) is fulfilled for all large L, which implies that we only have to deal with environments from the set \({\text {Good}}_L\), see Remark 2. On such environments, the coarse grained transition kernel \(\hat{{\varPi }}_L\) agrees with \(\hat{{\varPi }}^g_L\), and so do their corresponding Green’s functions. Now the second part of Lemma 1 essentially tells us that for upper bounds, we can replace the (random) Green’s function \(\hat{G}^g_L\) by the (non-random) kernel \(\varGamma _L\). The latter is explicitly given and therefore easy to control. This allowed to bound differences of exit measures in [2] and will be used here to bound differences of mean sojourn times in Sect. 4.

Thus it is essential to establish good bounds for \(\mathrm {\Gamma }_L\). More specifically, we will use the following properties of \(\mathrm {\Gamma }_L\), which form part of [2, Lemma 5.4]

Lemma 2

(Properties of \(\mathrm {\Gamma }_L\))

  1. (i)

    \(\mathrm {\Gamma }_L\asymp 1\).

  2. (ii)

    For \(1\le j \le \frac{1}{3r}s_L\), with \({\mathcal {E}}_j=\{y\in V_{L+r_L} : {\tilde{{\text {d}}}}(y)\le 3jr_L\}\),

    $$\begin{aligned} \sup _{x\in V_{L+r_L}}\mathrm {\Gamma }_L(x,{\mathcal {E}}_j) \le C\log (j+1), \end{aligned}$$

    and for \(0\le \alpha < 3\),

    $$\begin{aligned} \sup _{x\in V_{L+r_L}}\mathrm {\Gamma }_L\left( x,{\text {Sh}}_L\left( s_L, L/(\log L)^\alpha \right) \right) \le C(\log \log L)(\log L)^{6-2\alpha }. \end{aligned}$$
  3. (iii)

    For \(x\in V_{L+r_L}\),

    $$\begin{aligned} \mathrm {\Gamma }_L(x,V_L) \le C\max \left\{ \frac{{\tilde{{\text {d}}}}(x)}{L}(\log L)^6,\, \left( \frac{{\tilde{{\text {d}}}}(x)}{r_L}\wedge \log \log L\right) \right\} . \end{aligned}$$

These bounds follow in a straightforward way from the definition of \(\mathrm {\Gamma }_L\). Details are given in [2].

Difference estimates

For controlling mean sojourn times, we will need difference estimates for the coarse grained Green’s functions \(\hat{g}^{(q)}_L\) and \(\hat{G}^g_L\). We first recall our notation:

  • For \(q\in \mathcal {P}^{\text {s}}_\kappa \), \(\hat{g}^{(q)}_L\) is the Green’s function in \(V_L\) corresponding to \(\hat{\pi }^{(q)}_L\).

  • \(\hat{G}^g_L\) is the Green’s function in \(V_L\) corresponding to \(\hat{{\varPi }}^g_L\), cf. (13).

  • For \(m>0\), \(\hat{g}^{(q)}_{m,V_L}\) is the Green’s function in \(V_L\) corresponding to \(1_{V_L}\hat{\pi }^{(q)}_{\psi _m}\), where \(\psi _m\equiv m\).

  • \(\hat{g}^{(q)}_{m,{\mathbb {Z}}^d}\) is the Green’s function on \({\mathbb {Z}}^d\) corresponding to \(\hat{\pi }^{(q)}_{\psi _m}\), where \(\psi _m\equiv m\).

Lemma 3

There exists a constant \(C >0\) such that for all \(q\in \mathcal {P}^{\text {s}}_\kappa \),

  1. (i)
    $$\begin{aligned} \sup _{x,x'\in V_L: |x-x'|\le s_L}\sum _{y\in V_L}\big |\hat{g}^{(q)}_L(x,y)-\hat{g}^{(q)}_L(x',y)\big | \le C(\log \log L)(\log L)^3. \end{aligned}$$
  2. (ii)

    Assume \(\mathbf{C1}(\delta ,L_0,L_1)\), and let \(L_1\le L\le L_1(\log L_1)^2\). There exists a constant \(C>0\) such that for \(\delta >0\) small,

    $$\begin{aligned} \sup _{x,x'\in V_L: |x-x'|\le s_L}\sum _{y\in V_L}\big |\hat{G}^g_L(x,y)-\hat{G}^g_L(x',y)\big | \le C(\log \log L)(\log L)^3. \end{aligned}$$

Proof

(i) The underlying one-step transition kernel is always given by \(q\in \mathcal {P}^{\text {s}}_\kappa \), which we constantly omit in this proof, i.e. \(\hat{\pi }_{\psi _m} = \hat{\pi }^{(q)}_{\psi _m}\), \(\hat{g}_{m,V_L} = \hat{g}^{(q)}_{m,V_L}\), \(\hat{g}_{m,{\mathbb {Z}}^d} = \hat{g}^{(q)}_{m,{\mathbb {Z}}^d}\), or \({\text {P}}_{x} = {\text {P}}_{x,q}\), and so on. Also, we suppress the subscript L, i.e. \(\hat{g}=\hat{g}_L\). Set \(m = s_L/20\). We write

$$\begin{aligned}&\sum _{y\in V_L}\left| \hat{g}(x,y)-\hat{g}(x',y)\right| \le \sum _{y\in V_L}\left| \left( \hat{g}-\hat{g}_{m,V_L}\right) (x,y)\right| \nonumber \\&\quad + \sum _{y\in V_L}\left| \hat{g}_{m,V_L}(x,y)-\hat{g}_{m,V_L}(x',y)\right| + \sum _{y\in V_L}\left| \left( \hat{g}_{m,V_L}-\hat{g}\right) (x',y)\right| . \end{aligned}$$
(16)

If \(x\in V_L\backslash {\text {Sh}}_L(2s_L)\), we have \(\hat{\pi }(x,\cdot )= \hat{\pi }_{\psi _m}(x,\cdot )\). Moreover, since \(m=s_L/20\), \(\sup _{x\in V_L}\hat{g}_{m,V_L}(x,{\text {Sh}}_L(2s_L)) \le C\). Thus, with \(\mathrm {\Delta }= 1_{V_L}\left( \hat{\pi }_{\psi _m}-\hat{\pi }\right) \), the perturbation expansion (4) and Lemma 2 yield (remember \(\hat{g}\preceq C\mathrm {\Gamma }\) by Lemma 1)

$$\begin{aligned}&{\sum _{y\in V_L}\left| (\hat{g}_{m,V_L}-\hat{g})(x,y)\right| = \sum _{y\in V_L}|\hat{g}_{m,V_L}\mathrm {\Delta }\hat{g}(x,y)|}\\&\quad \le 2\,\hat{g}_{m,V_L}(x,{\text {Sh}}_L(2s_L))\sup _{v\in {\text {Sh}}_L(3s_L)}\hat{g}(v,V_L) \le C(\log L)^3. \end{aligned}$$

It remains to handle the middle term of (16). By (14),

$$\begin{aligned}&\hat{g}_{m,V_L}(x,y)-\hat{g}_{m,V_L}(x',y) \\&\quad =\hat{g}_{m,{\mathbb {Z}}^d}(x,y)- \hat{g}_{m,{\mathbb {Z}}^d}(x',y) + {\text {E}}_{x',\hat{\pi }_m}\left[ \hat{g}_{m,{\mathbb {Z}}^d}(X_{\tau _L},y)\right] - {\text {E}}_{x,\hat{\pi }_m}\left[ \hat{g}_{m,{\mathbb {Z}}^d}(X_{\tau _L},y)\right] . \end{aligned}$$

Using Proposition 2, it follows that for \(|x-x'| \le s_L\),

$$\begin{aligned} \sum _{y\in V_L}\left| \hat{g}_{m,{\mathbb {Z}}^d}(x,y)-\hat{g}_{m,{\mathbb {Z}}^d}(x',y)\right| \le C(\log L)^3. \end{aligned}$$

At last, we claim that

$$\begin{aligned}&\sum _{y\in V_L}\left| {\text {E}}_{x',\hat{\pi }_{\psi _m}}\left[ \hat{g}_{m,{\mathbb {Z}}^d}(X_{\tau _L},y)\right] - {\text {E}}_{x,\hat{\pi }_{\psi _m}}\left[ \hat{g}_{m,{\mathbb {Z}}^d}(X_{\tau _L},y)\right] \right| \nonumber \\&\quad \le C(\log \log L){(\log L)}^3. \end{aligned}$$
(17)

Since \(|x-x'| \le m\), we can define on the same probability space, whose probability measure we denote by \({\mathbb {Q}}\), a random walk \((Y_n)_{n\ge 0}\) starting from x and a random walk \((\tilde{Y}_n)_{n\ge 0}\) starting from \(x'\), both moving according to \(\hat{\pi }_{\psi _m}\) on \({\mathbb {Z}}^d\), such that for all times n, \(|Y_n - \tilde{Y}_n| \le s_L\). However, with \(\tau = \inf \{n \ge 0 : Y_n \notin V_L\}\), \(\tilde{\tau }\) the same for \(\tilde{Y}_n\), we cannot deduce that \(|Y_{\tau }-\tilde{Y}_{\tilde{\tau }} | \le s_L\), since it is possible that one of the walks, say \(Y_n\), exits \(V_L\) first and then moves far away from the exit point, while the other walk \(\tilde{Y}_n\) might still be inside \(V_L\). In order to show that such an event has a small probability, we argue similarly to [17, Proposition 7.7.1]. Define

$$\begin{aligned} \sigma (s_L) = \inf \left\{ n\ge 0 : Y_n\in {\text {Sh}}_L(s_L)\right\} , \end{aligned}$$

and analogously \(\tilde{\sigma }(s_L)\). Let \(\vartheta = \sigma (s_L)\wedge \tilde{\sigma }(s_L)\). Since \(|Y_{\vartheta }-\tilde{Y}_{\vartheta }| \le s_L\),

$$\begin{aligned} \max \left\{ \sigma (2s_L),\,\tilde{\sigma }(2s_L)\right\} \le \vartheta . \end{aligned}$$

For \(k\ge 1\), we introduce the events

$$\begin{aligned} B_k&=\left\{ \big |Y_i-Y_{\sigma (2s_L)}\big | > ks_L \text{ for } \text{ all } i=\sigma (2s_L),\dots ,\tau \right\} ,\\ \tilde{B}_k&=\left\{ \big |\tilde{Y}_i-\tilde{Y}_{\tilde{\sigma }(2s_L)} \big | > ks_L \text{ for } \text{ all } i=\tilde{\sigma }(2s_L),\dots ,\tilde{\tau }\right\} . \end{aligned}$$

By the strong Markov property and the gambler’s ruin estimate of [17], p. 223 (7.26), there exists a constant \(C_1>0\) independent of k such that

$$\begin{aligned} {\mathbb {Q}}\left( B_k\cup \tilde{B}_k\right) \le C_1/k \end{aligned}$$

for some \(C_1>0\) independent of k. Applying the triangle inequality to

$$\begin{aligned} Y_{\tau } - \tilde{Y}_{\tilde{\tau }} = (Y_{\tau }-Y_{\vartheta }) + (Y_{\vartheta }-\tilde{Y}_{\vartheta }) + (\tilde{Y}_{\vartheta }-\tilde{Y}_{\tilde{\tau }}), \end{aligned}$$

we deduce, for \(k \ge 3\),

$$\begin{aligned} {\mathbb {Q}}\left( \big |Y_\tau - \tilde{Y}_{\tilde{\tau }}\big |\ge ks_L\right) \le 2C_1/(k-1). \end{aligned}$$

Since \(|Y_\tau -\tilde{Y}_{\tilde{\tau }}| \le 2(L +s_L) \le 3L \), it follows that

$$\begin{aligned} {\mathbb {E}}_{{\mathbb {Q}}}\left[ \big |Y_\tau -\tilde{Y}_{\tilde{\tau }} \big |\right] \le \sum _{k=1}^{3L}{\mathbb {Q}}\left( \big |Y_\tau -\tilde{Y}_{\tilde{\tau }} \big | \ge k\right) \le C(\log \log L)s_L. \end{aligned}$$

Also, for vw outside and y inside \(V_L\),

$$\begin{aligned} \left| \frac{1}{|v-y|^{d-2}}-\frac{1}{|w-y|^{d-2}}\right| \le C\frac{|v-w|}{(L+1-|y|)^{d-1}}. \end{aligned}$$

Applying Proposition 2, (17) now follows from the last two displays and a summation over \(y\in V_L\).

(ii) We take \(p=p_{s_L/20}\) stemming from Assignment (10) and work with p as the underlying one-step transition kernel, which will be suppressed from the notation, i.e. \(\hat{\pi }=\hat{\pi }^{(p)}\) and \(\hat{g}=\hat{g}^{(p)}\).

Let \(x, x'\in V_L\) with \(|x-x'|\le s_L\) and set \(\mathrm {\Delta }= 1_{V_L}(\hat{{\varPi }}^g-\hat{\pi })\). With \(B = V_L\backslash {\text {Sh}}_L(2r_L)\),

$$\begin{aligned} \hat{G}^g= \hat{g}1_{B}\mathrm {\Delta }\hat{G}^g+\hat{g}1_{V_L\backslash B}\mathrm {\Delta }\hat{G}^g+\hat{g}. \end{aligned}$$

Replacing successively \(\hat{G}^g\) in the first summand on the right-hand side,

$$\begin{aligned} \hat{G}^g= \sum _{k=0}^\infty {\left( \hat{g}1_B\mathrm {\Delta }\right) }^k\hat{g}+ \sum _{k=0}^\infty {\left( \hat{g}1_B\mathrm {\Delta }\right) }^k\hat{g}1_{V_L\backslash B}\mathrm {\Delta }\hat{G}^g= F+F1_{V_L\backslash B}\mathrm {\Delta }\hat{G}^g, \end{aligned}$$

where we have set \(F=\sum _{k=0}^\infty {\left( \hat{g}1_B\mathrm {\Delta }\right) }^k\hat{g}\). With \(R = \sum _{k=1}^{\infty }(1_B\mathrm {\Delta })^k\hat{\pi }\), expansion (6) gives

$$\begin{aligned} F = \hat{g}\sum _{m=0}^\infty (R\hat{g})^m\sum _{k=0}^\infty \left( 1_B\mathrm {\Delta }\right) ^k. \end{aligned}$$
(18)

From this representation, we conclude that the kernel |F| is dominated from above by \(\varGamma \). Indeed, a term of exactly this form is bounded in the proof of [2, Lemma 5.2 (ii)]. Noting that \(\Vert 1_B\mathrm {\Delta }\Vert _1\le \delta \), and \(\Vert 1_B\mathrm {\Delta }\hat{\pi }\Vert _1\le C(\log L)^{-9}\), we can follow the arguments there to deduce \(|F|\preceq C\mathrm {\Gamma }\), provided \(\delta \) is sufficiently small. Next, by Lemma 2 (ii) and (iii), we then see that for large L, uniformly in \(x\in V_L\),

$$\begin{aligned} \Vert F1_{V_L\backslash B}\mathrm {\Delta }\hat{G}^g(x,\cdot )\Vert _1 \le C\mathrm {\Gamma }(x,{\text {Sh}}_L(2r_L))\sup _{v\in {\text {Sh}}_L(3r_L)}\mathrm {\Gamma }(v,V_L) \le C\log \log L. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{y\in V_L}\big |\hat{G}^g(x,y)-\hat{G}^g(x',y)\big | \le C\log \log L + \sum _{y\in V_L}\left| F(x,y)-F(x',y)\right| . \end{aligned}$$

Using (18) and twice part (i),

$$\begin{aligned} \sum _{y\in V_L}\left| F(x,y)-F(x',y)\right|&\le \sum _{y\in V_L}\Big |\hat{g}\sum _{k=0}^\infty \left( 1_B\mathrm {\Delta }\right) ^k(x,y) -\hat{g}\sum _{k=0}^\infty \left( 1_B\mathrm {\Delta }\right) ^k(x',y)\Big |\nonumber \\&\quad + \sum _{y\in V_L}\left| \hat{g}RF(x,y) - \hat{g}RF(x',y)\right| . \end{aligned}$$
(19)

The first expression on the right is estimated by

$$\begin{aligned} \sum _{y\in V_L}\Big |\sum _{w\in V_L}\left( \hat{g}(x,w)-\hat{g}(x',w)\right) \sum _{k=0}^\infty \left( 1_B\mathrm {\Delta }\right) ^k(w,y)\Big | \le C(\log \log L)(\log L)^3, \end{aligned}$$

where we have used part (i) and \(\Vert 1_B\mathrm {\Delta }(w,\cdot )\Vert _1\le \delta \). The second factor of (19) is again bounded by (i) and the fact that for \(u \in V_L\),

$$\begin{aligned} \sum _{y\in V_L}|RF(u,y)|&= \sum _{y\in V_L}\Big |\sum _{k=1}^\infty \left( 1_B\mathrm {\Delta }\right) ^k\hat{\pi }F(u,y)\Big |\\&\le \sum _{k=0}^\infty \left\| 1_B\mathrm {\Delta }(u,\cdot )\right\| _1^k\sup _{v\in B}\left\| 1_B\mathrm {\Delta }\hat{\pi }(v,\cdot )\right\| _1\sup _{w\in V_L} \sum _{y\in V_L}|F(w,y)|\\&\le C(\log L)^{-9+6} = C (\log L)^{-3}. \end{aligned}$$

Altogether, this proves part (ii). \(\square \)

Mean sojourn times in the ball

Using the results about the exit measures from Proposition 1 and the estimates for the Green’s functions from Sect. 3, we prove in this part our main results on mean sojourn times in large balls.

Condition C2 and the main technical statement

Similarly to Condition \(\mathbf{C1}(\delta ,L_0,L_1)\), cf. Sect. 2.3, we formulate a condition on the mean sojourn times which we propagate from one level to the next.

We first introduce a monotone increasing function which will upper and lower bound the normalized mean sojourn time in the ball. Let \(0<\eta <1\), and define \(f_\eta : {\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) by setting

$$\begin{aligned} f_\eta (L)= \frac{\eta }{3}\sum _{k=1}^{\lceil \log L\rceil }k^{-3/2}. \end{aligned}$$

Note \(\eta /3\le f_\eta (L)< \eta \) and therefore \(\lim _{\eta \downarrow 0}\lim _{L\rightarrow \infty }f_\eta (L) = 0\).

Recall that \({\text {E}}_x={\text {E}}_{x,p_o}\) is the expectation with respect to simple random walk starting at \(x\in {\mathbb {Z}}^d\), and \(\tau _L\) is the first exit time from \(V_L\).

Condition C2

We say that \(\mathbf{C2}(\eta ,L_1)\) holds, if for all \(3\le L \le L_1\),

$$\begin{aligned} \mathbb {P}\left( {\text {E}}_{0,\omega }\left[ \tau _L\right] \notin \left[ 1-f_\eta (L),\, 1+f_\eta (L)\right] \cdot {\text {E}}_0\left[ \tau _L\right] \right) \le \exp \left( -(1/2)(\log L)^{2}\right) . \end{aligned}$$

Our main technical result for the mean sojourn times is

Proposition 3

Assume \(\mathbf{A1}\) and \(\mathbf{B}\), and let \(0<\eta <1\). There exists \(\varepsilon _0=\varepsilon _0(\eta )> 0\) with the following property: If \(\varepsilon \le \varepsilon _0\) and \(\mathbf{A0}(\varepsilon )\) holds, then there exists \(L_0=L_0(\eta )>0\) such that for \(L_1\ge L_0\),

$$\begin{aligned} \mathbf{C2}(\eta ,L_1)\Rightarrow \mathbf{C2}(\eta ,L_1(\log L_1)^2). \end{aligned}$$

Remark 3

Given \(\eta \) and \(L_0\), we can always guarantee (by making \(\varepsilon \) smaller if necessary) that \(\mathbf{A0}(\varepsilon )\) implies \(\mathbf{C2}(\eta ,L_0)\).

The proof of this statement is deferred to Sect. 4.4.

Some preliminary results

We begin with an elementary statement about the mean time a symmetric random walk with kernel \(p\in \mathcal {P}^{\text {s}}_\kappa \) spends in the ball \(V_L\).

Lemma 4

Let \(p\in \mathcal {P}^{\text {s}}_\kappa \), and let \(x\in V_L\). Then

$$\begin{aligned} L^2-|x|^2\le {\text {E}}_{x,p}\left[ \tau _L\right] \le (L+1)^2-|x|^2. \end{aligned}$$

The proof of this standard lemma (see e.g. [17, Proposition 6.2.6]) uses the fact that \(|X_{n\wedge \tau _L}|^2-n\wedge \tau _L\) is a martingale, which leads by optional stopping to \({\text {E}}_{x,p}[\tau _L] = {\text {E}}_{x,p}[|X_{\tau _L}|^2] -|x|^2\). In particular, for different \(p,q\in \mathcal {P}^{\text {s}}_\kappa \), the corresponding mean sojourn times satisfy

$$\begin{aligned} {\text {E}}_{0,p}\left[ \tau _L\right] = {\text {E}}_{0,q}\left[ \tau _L\right] (1+O(L^{-1})). \end{aligned}$$

We will compare the RWRE sojourn times on all scales with \({\text {E}}_{0}[\tau _L]\), the corresponding quantity for simple random walk. This is somewhat in contrast to our comparison of the exit measure in [2], where we use the scale-dependent kernels \(p_L\) given by (10).

Using that \(\mu \) is supported on transition probabilities which are balanced in the first coordinate direction, we obtain a similar upper bound for the mean sojourn time of the RWRE.

Lemma 5

For \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\),

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ \tau _L\right] \le \frac{d}{1-2\varepsilon d}(L+1)^2-(x\cdot e_1)^2. \end{aligned}$$

Proof

For \(\omega \in (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\), \(\omega _x(e_1) = \omega _x(-e_1)\) for each \(x\in {\mathbb {Z}}^d\). Then

$$\begin{aligned} M_n = (X_n\cdot e_1)^2 - \sum _{k=0}^{n-1}\left( \omega _{X_k}(e_1) + \omega _{X_k}(-e_1)\right) \end{aligned}$$

is a \({\text {P}}_{x,\omega }\)-martingale with respect to the filtration generated by the walk \((X_n)_{n\ge 0}\). By the optional stopping theorem, \({\text {E}}_{x,\omega }\left[ M_{n\wedge \tau _L}\right] = (x\cdot e_1)^2\). Since for \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\),

$$\begin{aligned} \omega _{X_k}(e_1)+\omega _{X_k}(-e_1) \ge 1/d -2\varepsilon , \end{aligned}$$

it follows that

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ n\wedge \tau _L\right] \le {\left( 1/d-2\varepsilon \right) }^{-1}{\text {E}}_{x,\omega }\left[ (X_{n\wedge \tau _L}\cdot e_1)^2\right] - (x\cdot e_1)^2. \end{aligned}$$

Letting \(n\rightarrow \infty \) proves the statement. \(\square \)

Remark 4

Conditions \(\mathbf{A0}(\varepsilon )\) and \(\mathbf{B}\) guarantee that the event \(({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\) has full \(\mathbb {P}\)-measure. The a priori fact that \({\text {E}}_{0,\omega }[\tau _L]\le CL^2\) for almost all environments will be crucial to obtain more precise bounds on these times.

We will now express the mean sojourn time of the RWRE in \(V_L\) in terms of mean sojourn times in smaller balls \(V_t(x)\subset V_L\), for \(t\in [h_L(x),2h_L(x)]\), see Fig. 2. Recall the definition of \(h_L\) and the corresponding coarse graining scheme inside \(V_L\). As in Sect. 2.2, we put

Fig. 2
figure 2

We express the mean sojourn time \({\text {E}}_{0,\omega }[\tau _L]\) as a convolution of the coarse grained RWRE Green’s function \(\hat{G}_L\) with mean sojourn times in smaller balls \(V_t(x)\cap V_L\), where \(t\in [h_L(x),2h_L(x)]\) (see Lemma 6). Inductive control over the sojourn times on smaller scales \(\le s_L\) and over the Green’s function then allow us to obtain the right estimate for \(V_L\)

$$\begin{aligned} s_t = \frac{t}{(\log t)^3} \quad {\hbox {and}}\quad r_t= \frac{t}{(\log t)^{15}}. \end{aligned}$$

Let \(h_t^x(\cdot )= h_{t}(\cdot -x)\), where \(h_{t}(\cdot -x)\) is defined in (8) (with L replaced by t). By translating the origin into x, we transfer the coarse graining schemes on \(V_L\) in the obvious way to \(V_t(x)\), using \(h_t^x\) instead of \(h_L\). We write \(\hat{{\varPi }}_t^x\) for the coarse grained transition probabilities in \(V_t(x)\) associated to \(((h_t^x(y))_{y\in V_t(x)},p_\omega )\), cf. (9). Given \(p\in \mathcal {P}^{\text {s}}_\kappa \), the kernel \(\hat{\pi }_t^{(p),x}\) is defined similarly, with \(p_\omega \) replaced by p.

For the corresponding Green’s functions we use the expressions \(\hat{G}_t^x\) and \(\hat{g}_t^{(p),x}\). If we do not keep x as an index, we always mean \(x=0\) as before. If it is clear with which p we are working, we drop the superscript (p). Notice that for \(y,z \in V_t(x)\) and \(p\in \mathcal {P}^{\text {s}}_\kappa \), we have \(\hat{\pi }_t^{(p),x}(y,z) = \hat{\pi }^{(p)}_t(y-x,z-x)\) and \(\hat{g}^{(p)}_t(y,z) = \hat{g}_t^{(p)}(y-x,z-x)\). Since \(p_\omega \) is in general not homogeneous in space, this is not true for \(\hat{{\varPi }}_t^x\) and \(\hat{G}_t^x\).

Define the averaged RWRE sojourn time

$$\begin{aligned} \varLambda _L(x) = 1_{V_L}(x)\, \frac{1}{h_L(x)}\int \limits _{{\mathbb {R}}_+}\varphi \left( \frac{t}{h_L(x)}\right) {\text {E}}_{x,\omega }\left[ \tau _{V_t(x)\cap V_L}\right] \text {d}t, \end{aligned}$$

and the analog for random walk with kernel \(p\in \mathcal {P}^{\text {s}}_\kappa \),

$$\begin{aligned} \lambda ^{(p)}_L(x) = 1_{V_L}(x)\, \frac{1}{h_L(x)}\int \limits _{{\mathbb {R}}_+}\varphi \left( \frac{t}{h_L(x)}\right) {\text {E}}_{x,p}\left[ \tau _{V_t(x)\cap V_L}\right] \text {d}t. \end{aligned}$$

We also consider the corresponding quantities \(\varLambda _t^x\) and \(\lambda _t^{(p),x}\) for balls \(V_t(x)\). For example,

$$\begin{aligned} \varLambda _t^x(y) = 1_{V_t(x)}(y)\,\frac{1}{h_t^x(y)}\int \limits _{{\mathbb {R}}_+}\varphi \left( \frac{s}{h_t^x(y)}\right) {\text {E}}_{y,\omega }\left[ \tau _{V_s(y)\cap V_t(x)}\right] \text {d}s. \end{aligned}$$

Note that we should rather write \(\varLambda _{L,\omega }\) and \(\varLambda _{t,\omega }^x\), but we again suppress \(\omega \) in the notation. In the rest of this part, we often let operate kernels on mean sojourn times from the left. As an example,

$$\begin{aligned} \hat{G}_L\varLambda _L(x) = \sum _{y\in V_L}\hat{G}_L(x,y)\varLambda _L(y). \end{aligned}$$

Both \(\varLambda _L\) and \(\hat{G}_L\) should be understood as random functions, but sometimes (for example in the proof of the next statement) the environment \(\omega \) is fixed.

The following lemma connects sojourn times on different scales and forms the basic statement for our renormalization scheme.

Lemma 6

For environments \(\omega \in \left( {\mathcal {P}}_\varepsilon \right) ^{{\mathbb {Z}}^d}\), \(x\in {\mathbb {Z}}^d\),

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ \tau _L\right] = \hat{G}_L\varLambda _L(x). \end{aligned}$$

In particular, for \(\omega \) the homogeneous environment with transition probabilities given by \(p\in \mathcal {P}^{\text {s}}_\kappa \),

$$\begin{aligned} {\text {E}}_{x,p}\left[ \tau _L\right] = \hat{g}^{(p)}_L\lambda ^{(p)}_L(x). \end{aligned}$$

Proof

We construct a probability space where we can observe in \(V_L\) both the random walk running with transition kernel \(p_\omega \) and its coarse grained version running with kernel \(\hat{{\varPi }}_L(\omega )\). For this reason, we take a probability space \((\varXi , {\mathcal {A}}, \mathbb {Q})\) carrying a family of i.i.d. [1, 2]-valued random variables \((\xi _n: n\in {\mathbb {N}})\) distributed according to \(\varphi (t)\text {d}t\). We now consider the probability space \((({\mathbb {Z}}^d)^{{\mathbb {N}}}\times \varXi , \mathcal {G}\otimes {\mathcal {A}}, {\text {P}}_{x,\omega }\otimes \mathbb {Q})\). By a small abuse of notation, we denote here by \(X_n\) the projection on the nth component of the first factor of \(({\mathbb {Z}}^d)^{{\mathbb {N}}}\times \varXi \), so that under \(\widetilde{{\text {P}}}_{x,\omega }={\text {P}}_{x,\omega }\otimes \mathbb {Q}\), \((X_n)_{n\ge 0}\) evolves as the canonical Markov chain under \({\text {P}}_{x,\omega }\).

Set \(T_0 = 0\) and define the “randomized” stopping times

$$\begin{aligned} T_{n+1} = \inf \left\{ m> T_n: X_m\notin V_{\xi _{T_n}\cdot h_L(X_{T_n})}\left( X_{T_n}\right) \right\} \wedge \tau _L. \end{aligned}$$

Then the coarse grained Markov chain in \(V_L\) running with transition kernel \(\hat{{\varPi }}_{L,\omega }\) can be obtained by observing \(X_n\) at the times \(T_n\), that is by considering \((X_{T_n})_{n\ge 0}\). Moreover, the Markov property of \(X_n\) and the i.i.d. property of the \(\xi _n\) ensure that under \(\widetilde{{\text {P}}}_{x,\omega }\), conditionally on \(X_{T_n}\), the random vector \(((X_{T_n},X_{T_n+1},\dots ),T_{n+1}-T_n)\) is distributed as \(((X_{0},X_{1},\dots ),T_{1})\) under \(\widetilde{{\text {P}}}_{X_{T_n},\omega }\). Indeed, defining the filtration \(\mathcal {G}_n = \sigma (X_0,\dots ,X_n,\xi _0, \dots , \xi _{n-1})\), \((X_n)_{n\ge 0}\) is also a \(\mathcal {G}_n\)-Markov chain. By induction, one sees that \(T_n\) is a \(\mathcal {G}_n\)-stopping time, and the strong Markov property gives the stated equality in law. Writing \(\widetilde{{\text {E}}}_{x,\omega }\) for the expectation with respect to \(\widetilde{{\text {P}}}_{x,\omega }\), we obtain

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ \tau _L\right]&= \sum _{z\in V_L}\widetilde{{\text {E}}}_{x,\omega }\left[ \sum _{n=0}^\infty 1_{\{z\}}(X_n)1_{\{n < \tau _L\}}\right] \\&=\sum _{z\in V_L}\widetilde{{\text {E}}}_{x,\omega }\left[ \sum _{n=0}^\infty \sum _{k=T_n}^{T_{n+1}-1} 1_{\{z\}}(X_k)\right] \\&=\sum _{z\in V_L}\widetilde{{\text {E}}}_{x,\omega }\left[ \sum _{n=0}^\infty \left( \sum _{y\in V_L}1_{\{y\}}(X_{T_n})\right) \sum _{k=T_n}^{T_{n+1}-1} 1_{\{z\}}(X_k)\right] \\&=\sum _{y\in V_L}\sum _{n=0}^{\infty }\widetilde{{\text {E}}}_{x,\omega }\left[ 1_{\{y\}}\left( X_{T_n}\right) \widetilde{{\text {E}}}_{x,\omega }\left[ \sum _{z\in V_L}\sum _{k=T_n}^{T_{n+1}-1}1_{\{z\}}(X_k)\bigm |X_{T_n}\right] \right] \\&= \sum _{y\in V_L}\sum _{n=0}^\infty \widetilde{{\text {E}}}_{x,\omega }\left[ 1_{\{y\}}\left( X_{T_n}\right) \widetilde{{\text {E}}}_{X_{T_n,\omega }}\left[ \sum _{z\in V_L}\sum _{k=0}^{T_1-1}1_{\{z\}}\left( X_k\right) \right] \right] \\&= \sum _{y\in V_L}\sum _{n=0}^\infty \widetilde{{\text {E}}}_{x,\omega }\left[ 1_{\{y\}}\left( X_{T_n}\right) \right] \varLambda _L(y) = \sum _{y\in V_L}\hat{G}_L(x,y)\varLambda _L(y)\\&= \hat{G}_L\varLambda _L(x). \end{aligned}$$

\(\square \)

Space-good/bad and time-good/bad points

We classify the grid points inside \(V_L\) into good and bad points, with respect to both space and time. We start by defining space-good and space-bad points. Unlike in [2], we need simultaneous control over two scales. This suggests the following stronger notion of “goodness”.

Space-good and space-bad points

Recall Assignment (10). We say that \(x\in V_L\) is space-good (with respect to L and \(\delta >0\)), if

  • \(x\in V_L\backslash \mathcal {B}_L\), that is x is good in the sense of Sect. 2.4.

  • If \({\text {d}}_L(x) > 2s_L\), then additionally for all \(t\in [h_L(x),2h_L(x)]\) and for all \(y\in V_t(x)\),

    • For all \(t'\in [h_t^x(y), 2h_t^x(y)]\), with \(\tilde{q}=p_{h_t^x(y)}\),

      $$\begin{aligned} \left\| (\varPi _{V_{t'}(y)}-\pi _{V_{t'}(y)}^{(\tilde{q})})(y,\cdot )\right\| _1 \le \delta . \end{aligned}$$
    • If \(t-|y-x| > 2r_t\), then additionally (with the same \(\tilde{q}\))

      $$\begin{aligned} \left\| (\hat{{\varPi }}_t^x - \hat{\pi }_t^{(\tilde{q}),x})\hat{\pi }_t^{(\tilde{q}),x}(y,\cdot )\right\| _1 \le (\log h_t^x(y))^{-9}. \end{aligned}$$

In other words, for a point \(x\in V_L\) with \({\text {d}}_L(x)>2s_L\) to be space-good, we do not only require that x is good in the sense of Sect. 2.4, but also that all points \(y\in V_t(x)\) for every \(t\in [h_L(x),2h_L(x)]\) are good. In this way, we obtain control over the exit distributions from smaller balls in the bulk of \(V_L\), whose radii are on a preceding scale. An illustration is given by Fig. 3.

A point \(x\in V_L\) which is not space-good is called space-bad. The (random) set of all space-bad points inside \(V_L\) is denoted by \(\mathcal {B}_L^{\mathrm{sp}}=\mathcal {B}_L^{\mathrm{sp}}(\omega )\). We write \({\text {Good}}_L^{\mathrm{sp}}= \{\mathcal {B}_L^{\mathrm{sp}}= \emptyset \}\) for the set of environments which contain no bad points, and \({\text {Bad}}_L^{\mathrm{sp}}= \left\{ \mathcal {B}_L^{\mathrm{sp}}\ne \emptyset \right\} \) for its complement. Notice that in the notation of Sect. 2.4, \(\mathcal {B}_L \subset \mathcal {B}_L^{\mathrm{sp}}\) and \({\text {Good}}_L^{\mathrm{sp}}\subset {\text {Good}}_L\).

Fig. 3
figure 3

Definition of space-good points: For \(x\in V_L\) with \({\text {d}}_L(x)>2s_L\), we do not only require that exit laws from balls \(V_t(x)\) of radii \(t\in [h_L(x),2h_L(x)]\) are close to those of a symmetric random walk (in both a smoothed and non-smoothed way), but also that exit laws from smaller balls \(V_{t'}(y)\), \(y\in V_t(x)\), behave well. For points \(v\in V_L\) with \({\text {d}}_L(v)\le 2s_L\), we only require control over exit laws on the next lower scale. A similar two-scale control is needed for sojourn times, see Sect. 4.3.2

On the event \({\text {Good}}_L^{\mathrm{sp}}\), we have control over the coarse grained Green’s functions \(\hat{G}_L\) and \(\hat{G}_t^x\) in terms of the deterministic kernel \(\mathrm {\Gamma }\).

Lemma 7

There exist \(\delta >0\) small and \(L_0\) large such that if \(\mathbf{C1}(\delta , L_0,L_1)\) is satisfied for some \(L_1\ge L_0\), then we have for \(L_1\le L\le L_1(\log L_1)^2\) on \({\text {Good}}_L^{\mathrm{sp}}\) and for some constant \(C>0\),

  1. (i)

    \(\hat{G}_L \preceq C\mathrm {\Gamma }_L\).

  2. (ii)

    If \(x\in V_L\) with \({\text {d}}_L(x) > 2s_L\), then for all \(t\in [h_L(x), 2h_L(x)]\),

    $$\begin{aligned} \hat{G}_t^x \preceq C\mathrm {\Gamma }_{t}(\cdot -x,\cdot -x). \end{aligned}$$

Proof

(i) Since \({\text {Good}}_L^{\mathrm{sp}}\subset {\text {Good}}_L\), we have \(\hat{G}= \hat{G}^g\) on \({\text {Good}}_L^{\mathrm{sp}}\), and Lemma 1 applies. For (ii), with x and t as in the statement, there are no bad points within \(V_t(x)\) on \({\text {Good}}_L^{\mathrm{sp}}\). Therefore, also the kernel \(\hat{G}_t^x\) coincides with its goodified version, and the claim follows again from Lemma 1. \(\square \)

Lemma 8

Assume \(\mathbf{C1}(\delta ,L_0,L_1)\). Then for L with \(L_1\le L \le L_1(\log L_1)^2\),

$$\begin{aligned} \mathbb {P}({\text {Bad}}_L^{\mathrm{sp}}) \le \exp \left( -(2/3)(\log L)^{2}\right) . \end{aligned}$$

Proof

Let us first compute the probability \(\mathbb {P}(x\in \mathcal {B}_L)\) for \(x\in V_L\) with \({\text {d}}_L(x) > 2r_L\). Set \(q=p_{h_L(x)}\) and \(\varDelta = 1_{V_L}(\hat{{\varPi }}_{L} - \hat{\pi }^{(q)}_{L})\). Put \(D_{t,q,h_L,q}(x)=\Vert (\varPi _{V_t(x)}-\pi ^{(q)}_{V_t(x)})\hat{\pi }^{(q)}_{h_L}(x,\cdot )\Vert _1\), \(D_{t,q}(x)=\Vert (\varPi _{V_t(x)}-\pi ^{(q)}_{V_t(x)})(x,\cdot )\Vert _1\). Note that \(\frac{1}{20} r_L \le h_L(x) \le s_L\le L_1/2\). Then, under \(\mathbf{C1}(\delta ,L_0,L_1)\),

$$\begin{aligned}&\mathbb {P}\left( x\in \mathcal {B}_L\right) \\&\quad \le \mathbb {P}\left( \cup _{t\in \left[ h_L(x),2h_L(x)\right] }\left\{ D_{t,q,h_L,q}(x)>(\log h_L(x))^{-9}\right\} \cup \left\{ D_{t,q}(x) > \delta \right\} \right) \\&\quad \le Cs_L^{d}\exp \left( -\left( \log (r_L/20)\right) ^{2}\right) , \end{aligned}$$

and the last bound also in the case \({\text {d}}_L(x) \le 2r_L\) with a similar argument. If \(x\in \mathcal {B}_L^{\mathrm{sp}}\backslash \mathcal {B}_L\), then necessarily \({\text {d}}_L(x)>2s_L\), and at least one of the two conditions under the second point in the definition of space-good points must be violated. If \({\text {d}}_L(x) > 2s_L\), then \(h_L(x)= s_L/20\), whence for all \(t\in [h_L(x),\, 2h_L(x)]\), all \(y\in V_t(x)\), it follows that \(h_t^x(y) \ge r_{(s_L/20)}/20\). With a similar argument as above, we obtain under \(\mathbf{C1}(\delta ,L_0,L_1)\)

$$\begin{aligned} \mathbb {P}\left( x\in \mathcal {B}_L^{\mathrm{sp}}\backslash \mathcal {B}_L\right) \le Cs_L^{3d}\exp \left( -\left( \log (r_{(s_L/20)}/20)\right) ^{2}\right) . \end{aligned}$$

Therefore, a union bound gives

$$\begin{aligned} \mathbb {P}({\text {Bad}}_L^{\mathrm{sp}})&= \mathbb {P}(\mathcal {B}_L^{\mathrm{sp}}\ne \emptyset )\\&\le CL^ds_L^{3d}\exp \left( -\left( \log (r_{(s_L/20)}/20)\right) ^{2}\right) \le \exp \left( -(2/3)(\log L)^{2}\right) . \end{aligned}$$

\(\square \)

Time-good and time-bad points

We also classify points inside \(V_L\) according to the mean time the RWRE spends in surrounding balls. Remember Condition \(\mathbf{C2}(\eta ,L_1)\) and the function \(f_\eta \) introduced above. We now fix \(0<\eta <1\).

For points in the bulk of \(V_L\), we need control over two scales, as above. We say that a point \(x\in V_L\) is time-good if the following holds:

  • For all \(t\in [h_L(x),2h_L(x)]\),

    $$\begin{aligned}{\text {E}}_{x,\omega }\left[ \tau _{V_{t}(x)}\right] \in \left[ 1-f_\eta (s_L),\, 1+f_\eta (s_L)\right] \cdot {\text {E}}_{x}\left[ \tau _{V_{t}(x)}\right] . \end{aligned}$$
  • If \({\text {d}}_L(x) > 2s_L\), then additionally for all \(t\in [h_L(x),2h_L(x)]\), for all \(y\in V_t(x)\) and for all \(t'\in [h_t^x(y),2h_t^x(y)]\),

    $$\begin{aligned} {\text {E}}_{y,\omega }\left[ \tau _{V_{t'}(y)}\right] \in \left[ 1-f_\eta (s_t),\, 1+f_\eta (s_t)\right] \cdot {\text {E}}_{y}\left[ \tau _{V_{t'}(y)}\right] . \end{aligned}$$

A point \(x\in V_L\) which is not time-good is called time-bad. We denote by \(\mathcal {B}_L^{\mathrm{tm}}= \mathcal {B}_L^{\mathrm{tm}}(\omega )\) the set of all time-bad points inside \(V_L\). With

$$\begin{aligned} {\mathcal {D}}_L = \left\{ V_{4h_L(x)}(x) : x\in V_L\right\} , \end{aligned}$$

we let \({\text {OneBad}}_L^{\mathrm{tm}}= \left\{ \mathcal {B}_L^{\mathrm{tm}}\subset D \text{ for } \text{ some } D\in {\mathcal {D}}_L\right\} \), \({\text {ManyBad}}_L^{\mathrm{tm}}={\left( {\text {OneBad}}_L^{\mathrm{tm}}\right) }^c\), and \({\text {Good}}_L^{\mathrm{tm}}= \{\mathcal {B}_L^{\mathrm{tm}}= \emptyset \}\subset {\text {OneBad}}_L^{\mathrm{tm}}\).

The next lemma ensures that for propagating Condition \(\mathbf{C2}\), we can forget about environments with space-bad points or widely spread time-bad points.

Lemma 9

Assume \(\mathbf{C1}(\delta ,L_0,L_1)\) and \(\mathbf{C2}(\eta ,L_1)\). For \(L_1\le L \le L_1(\log L_1)^2\),

$$\begin{aligned} \mathbb {P}\left( {\text {Bad}}_L^{\mathrm{sp}}\cup {\text {ManyBad}}_L^{\mathrm{tm}}\right) \le \exp \left( -(1/2)(\log L)^2\right) . \end{aligned}$$

Proof

We have \(\mathbb {P}\left( {\text {Bad}}_L^{\mathrm{sp}}\cup {\text {ManyBad}}_L^{\mathrm{tm}}\right) \le \mathbb {P}\left( {\text {Bad}}_L^{\mathrm{sp}}\right) + \mathbb {P}\left( {\text {ManyBad}}_L^{\mathrm{tm}}\right) \). Lemma 8 bounds the first summand. Now if \(x\in \mathcal {B}_L^{\mathrm{tm}}\), then either

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ \tau _{V_t(x)}\right] \notin \left[ 1-f_\eta (t),\, 1+f_\eta (t)\right] \cdot {\text {E}}_x\left[ \tau _{V_{t}(x)}\right] \end{aligned}$$

for some \(t\in [h_L(x),2h_L(x)]\) (recall that \(f_\eta \) is increasing), or, if \({\text {d}}_L(x)>2s_L\),

$$\begin{aligned} {\text {E}}_{y,\omega }\left[ \tau _{V_{t'}(y)}\right] \notin \left[ 1-f_\eta (t'),\, 1+f_\eta (t')\right] \cdot {\text {E}}_y\left[ \tau _{V_{t'}(y)}\right] \end{aligned}$$

for some \(y\in V_{t}(x)\), \(t\in [h_L(x),2h_L(x)]\), \(t'\in [h_{t}^x(y),2h_{t}^x(y)]\).

For all \(x\in V_L\), we have \(h_L(x) \ge r_L/20\). Moreover, for x with \({\text {d}}_L(x) > 2s_L\), we have \(h_L(x)= s_L/20\), and for all \(t\in [h_L(x),\, 2h_L(x)]\), all \(y\in V_t(x)\), it holds that \(h_t^x(y) \ge r_{(s_L/20)}/20\). Therefore, under \(\mathbf{C2}(\eta ,L_1)\),

$$\begin{aligned} \mathbb {P}\left( x\in \mathcal {B}_L^{\mathrm{tm}}\right)&\le Cs_L^d\exp \left( -(1/2)\left( \log \left( r_L/20\right) \right) ^2\right) \\&\quad + Cs_L^{3d}\exp \left( -(1/2)\left( \log \left( r_{s_L/20}/20\right) \right) ^2\right) . \end{aligned}$$

We now observe that if \(\omega \in {\text {ManyBad}}_L^{\mathrm{tm}}\), then there are at least two time-bad points xy inside \(V_L\) with \(|x-y|>2h_L(x)+2h_L(y)\). For such xy, the events \(\{x\in \mathcal {B}_L^{\mathrm{tm}}\}\) and \(\{y\in \mathcal {B}_L^{\mathrm{tm}}\}\) are independent. With the last display, we therefore conclude that

$$\begin{aligned} \mathbb {P}\left( {\text {ManyBad}}_L^{\mathrm{tm}}\right)&\le CL^{8d}\left[ \exp \left( -(1/2)\left( \log \left( r_{s_L/20}/20\right) \right) ^2\right) \right] ^2\\&\le \exp \left( -(2/3)(\log L)^2\right) . \end{aligned}$$

\(\square \)

Proof of the main technical statement

In this part, we prove Proposition 3. We will always assume that \(\delta \) and L are such that Lemma 7 can be applied. We start with two auxiliary statements: Lemma 10 proves a difference estimate for mean sojourn times. Here the difference estimates for the coarse grained Green’s functions from Lemma 3 in Sect. 3.2 play a crucial role. Lemma 11 then provides the key estimate for proving the main propagation step.

Note that due to Lemma 5, we have for \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\), that is for \(\mathbb {P}\)-almost all environments,

$$\begin{aligned} \varLambda _L(x) \le Cs_L^2\le C(\log L)^{-6}L^2\quad \text{ for } \text{ all } x\in V_L. \end{aligned}$$
(20)

Lemma 10

Assume \(\mathbf{A0}(\varepsilon )\), \(\mathbf{B}\), \(\mathbf{C1}(\delta , L_0,L_1)\), and let \(L_1\le L\le L_1(\log L_1)^2\). Let \(0\le \alpha < 3\) and \(x, y\in V_{L-2s_L}\) with \(|x-y| \le (\log s_L)^{-\alpha }\,s_L\). Then for \(\mathbb {P}\)-almost all \(\omega \in {\text {Good}}_L^{\mathrm{sp}}\),

$$\begin{aligned} \left| \varLambda _L(x) - \varLambda _L(y)\right| \le C(\log \log s_L)(\log s_L)^{-\alpha } s_L^2. \end{aligned}$$

Proof

We let \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\cap {\text {Good}}_L^{\mathrm{sp}}\). The statement follows if we show that for all \(t\in \left[ (1/20)s_L,(1/10)s_L\right] \),

$$\begin{aligned} \left| {\text {E}}_{x,\omega }\left[ \tau _{V_t(x)}\right] - {\text {E}}_{y,\omega }\left[ \tau _{V_t(y)}\right] \right| \le C(\log \log t)(\log t)^{-\alpha } t^2. \end{aligned}$$

We fix such a t and set \(t' = \left( 1-20(\log t)^{-\alpha }\right) t\). Then \(V_{t'}(x) \subset V_t(x)\cap V_t(y)\). Now put \(B = V_{t'-2s_t}(x)\). By Lemma 6, we have the representation

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ \tau _{V_t(x)}\right] = \hat{G}_t^x1_B\varLambda _t^x(x) + \hat{G}_t^x1_{V_{t}(x)\backslash B}\varLambda _t^x(x). \end{aligned}$$
(21)

By (20), \(\varLambda _t^x(z)\le C(\log t)^{-6}t^2\), for all \(z\in V_t(x)\). Moreover, since \(\omega \in {\text {Good}}_L^{\mathrm{sp}}\), we have by Lemma 7 \(\hat{G}_t^x\preceq C\mathrm {\Gamma }_t(\cdot -x,\cdot -x)\). Applying Lemma 2 (ii) gives

$$\begin{aligned} \hat{G}_t^x1_{V_{t}(x)\backslash B}\varLambda _t^x(x)\le C\mathrm {\Gamma }_t\left( 0,V_t\backslash V_{t'-2s_t}\right) \,(\log t)^{-6}t^2 \le (\log t)^{-\alpha }t^2 \end{aligned}$$

for L (and therefore also t) sufficiently large. Concerning \({\text {E}}_{y,\omega }\left[ \tau _{V_t(y)}\right] \), we write again

$$\begin{aligned} {\text {E}}_{y,\omega }\left[ \tau _{V_t(y)}\right] = \hat{G}_t^y1_B\varLambda _t^y(y) + \hat{G}_t^y1_{V_t(y)\backslash B}\varLambda _t^y(y). \end{aligned}$$

The second summand is bounded by \((\log t)^{-\alpha }t^2\), as in the display above. For \(z \in B\), we have \(h_t^x(z) = h_t^y(z) = (1/20)s_t\). In particular, \(\hat{{\varPi }}_t^x(z,\cdot ) = \hat{{\varPi }}_t^y(z,\cdot )\), and also \(\varLambda _t^x(z) = \varLambda _t^y(z)\). Since both x and y are contained in \(B\subset V_t(x)\cap V_t(y)\), the strong Markov property gives

$$\begin{aligned} \hat{G}_t^y(y,z) = \hat{G}_t^x(y,z) +b(y,z), \end{aligned}$$

where

$$\begin{aligned} b(y,z) = {\text {E}}_{y,\hat{{\varPi }}_t^y (\omega )}\left[ \hat{G}_t^y(\tau _B,z);\, \tau _B < \infty \right] -{\text {E}}_{y,\hat{{\varPi }}_t^x(\omega )}\left[ \hat{G}_t^x(\tau _B,z);\, \tau _B < \infty \right] . \end{aligned}$$

Therefore,

$$\begin{aligned}&\left| {\text {E}}_{x,\omega }\left[ \tau _{V_t(x)}\right] -{\text {E}}_{y,\omega }\left[ \tau _{V_t(y)}\right] \right| \\&\quad \le 2(\log t)^{-\alpha }t^2 +\sum _{z\in B}\left( \big |\hat{G}_t^x(x,z)-\hat{G}_t^x(y,z)\big | + |b(y,z)|\right) \varLambda _t^x(z). \end{aligned}$$

The quantity \(\varLambda _t^x(z)\) is again bounded by \(C(\log t)^{-6}t^2\). For the part of the sum involving |b(yz)|, we notice that if \(w\in V_t(y)\backslash B\), then \(t-|w-y|\le C(\log t)^{-\alpha }t\) and similarly for \(v\in V_t(x)\). We can use twice Lemma 2 (ii) to get

$$\begin{aligned} \sum _{z\in B}|b(y,z)| \le \sup _{v\in V_t(x)\backslash B}\hat{G}_t^x(v,B) + \sup _{w\in V_t(y)\backslash B}\hat{G}_t^y(w,B) \le C(\log t)^{6-\alpha }. \end{aligned}$$

Finally, for the sum over the Green’s function difference, we recall that \(\hat{G}_{t}^x\) coincides with its goodified version. Applying Lemma 3 \(O\left( (\log t)^{3-\alpha }\right) \) times gives

$$\begin{aligned} \sum _{z\in B}\big |\hat{G}_{t}^x(x,z)-\hat{G}_{t}^x(y,z)\big | \le C(\log \log t)(\log t)^{6-\alpha }. \end{aligned}$$

This proves the statement. \(\square \)

Lemma 11

Assume \(\mathbf{A0}(\varepsilon )\), \(\mathbf{B}\), \(\mathbf{C1}(\delta , L_0,L_1)\), and let \(L_1\le L\le L_1(\log L_1)^2\). Let \(p=p_{s_L/20}\), cf. (10), and set \(\mathrm {\Delta }= 1_{V_L}(\hat{{\varPi }}_L-\hat{\pi }_L^{(p)})\). For \(\mathbb {P}\)-almost all \(\omega \in {\text {Good}}_L^{\mathrm{sp}}\),

$$\begin{aligned} \sup _{x\in V_L}\big |\hat{G}_L\mathrm {\Delta }\hat{g}_L^{(p)}\varLambda _L(x)\big | \le C(\log L)^{-5/3}L^2. \end{aligned}$$

Proof

Again, we consider \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\cap {\text {Good}}_L^{\mathrm{sp}}\). Write \(\hat{g}= \hat{g}^{(p)}\), \(\hat{\pi }=\hat{\pi }^{(p)}\). First,

$$\begin{aligned} \hat{G}\mathrm {\Delta }\hat{g}\varLambda _L(x) = \hat{G}\mathrm {\Delta }\hat{\pi }\hat{g}\varLambda _L(x) + \hat{G}\mathrm {\Delta }\varLambda _L(x)= A_1 + A_2. \end{aligned}$$

By Lemma 7, \(\hat{G}= \hat{G}^g\preceq C\mathrm {\Gamma }\). Therefore, with \(B_1= V_{L-2r_L}\), we bound \(A_1\) by

$$\begin{aligned} \left| A_1\right|&\le \big |\hat{G}1_{B_1}\mathrm {\Delta }\hat{\pi }\hat{g}\varLambda _L(x)\big | + \big |\hat{G}1_{V_L\backslash B_1}\mathrm {\Delta }\hat{\pi }\hat{g}\varLambda _L(x)\big |\\&\le \left| \sum _{v\in B_1, w\in V_L}\hat{G}(x,v)\mathrm {\Delta }\hat{\pi }(v,w)\sum _{y\in V_L}\left( \hat{g}(w,y)-\hat{g}(v,y)\right) \varLambda _L(y)\right| + C(\log L)^{-2}L^2 \\&\le C(\log L)^{-5/3}L^2, \end{aligned}$$

where in the next to last inequality we have used the bound on \(\varLambda _L(y)\) given by (20) and Lemma 2 (ii), (iii) for \(\mathrm {\Gamma }\), and for the last inequality additionally Lemma 3. For the term \(A_2\), we let \(B= V_{L-5s_L}\) and split into

$$\begin{aligned} A_2 = \hat{G}1_{B}\mathrm {\Delta }\varLambda _L(x) + \hat{G}1_{V_L\backslash B}\mathrm {\Delta }\varLambda _L(x). \end{aligned}$$

Lemma 2 (ii) yields

$$\begin{aligned} \hat{G}(x,{\text {Sh}}_L(5s_L))\le C\log \log L. \end{aligned}$$

Since \(\varLambda _L(y)\le (\log L)^{-2}L^2\) by (20), this is good enough for the second summand of \(A_2\). For the first one,

$$\begin{aligned} \hat{G}1_B\mathrm {\Delta }\varLambda _L(x) \le C\mathrm {\Gamma }(x,B)\sup _{v\in B}\left| \mathrm {\Delta }\varLambda _L(v)\right| . \end{aligned}$$

Since \(\mathrm {\Gamma }(x,B) \le C(\log L)^6\), the claim follows if we show that for \(v\in B\),

$$\begin{aligned} \left| \mathrm {\Delta }\varLambda _L(v)\right| \le C(\log L)^{-8}L^2, \end{aligned}$$
(22)

which, by definition of \(\mathrm {\Delta }\), in turn follows if for all \(t\in [h_L(v), 2h_L(v)]\),

$$\begin{aligned} \left| \left( \varPi _{V_t(v)}-\pi _{V_t(v)}\right) \varLambda _L(v)\right| \le C(\log L)^{-8}L^2, \end{aligned}$$

where we have set \(\pi _{V_t(v)} = \pi _{V_t(v)}^{(p)}\). Notice that on B, \(h_L(\cdot ) = (1/20)s_L\). We now fix \(v\in B\) and \(t\in [(1/20)s_L, (1/10)s_L]\). Set \(\mathrm {\Delta }'= 1_{V_t(v)}(\hat{{\varPi }}_t^v-\hat{\pi }_t^{(p),v})\) and \(B'= V_{t-2r_t}(v)\). By expansion (4),

$$\begin{aligned} \left( \varPi _{V_t(v)}-\pi _{V_t(v)}\right) \varLambda _L(v) = \hat{G}_t^v1_{B'}\mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(v) + \hat{G}_t^v1_{V_t(v)\backslash B'}\mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(v).\nonumber \\ \end{aligned}$$
(23)

Since \(\pi _{V_t(v)} = \hat{\pi }_t^{(p),v}\pi _{V_t(v)}\), we get

$$\begin{aligned} \big |\hat{G}_t^v1_{B'}\mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(v)\big |&\le \hat{G}_t^v(v,B')\sup _{w\in B'}\big |\mathrm {\Delta }'\hat{\pi }_t^v(w,\cdot )\big |_1\sup _{y\in \partial V_t(v)}\varLambda _L(y)\\&\le C(\log s_L)^6(\log L)^{-6}L^2\sup _{w\in B'}\big \Vert \mathrm {\Delta }'\hat{\pi }_t^{(p),v}(w,\cdot )\big \Vert _1. \end{aligned}$$

Here, in the last inequality we have used (20) and Lemma 2 (iii). In order to bound \(\Vert \mathrm {\Delta }'\hat{\pi }_t^{(p),v}(w,\cdot )\Vert _1\) for \(w\in B'\), we use the fact that v is space-good and \({\text {d}}_L(v)>2s_L\), which gives also control over the exit distributions from smaller balls inside \(V_t(v)\). Indeed, by definition we first have for \(w\in B'\), with \(\tilde{q}=p_{h_t^x(y)}\),

$$\begin{aligned} \big \Vert 1_{V_t(v)}(\hat{{\varPi }}_t^v-\hat{\pi }_t^{(\tilde{q}),v})\hat{\pi }_t^{(\tilde{q}),v}(w,\cdot )\big \Vert _1 \le (\log h_t^v(w))^{-9}\le C(\log L)^{-9}. \end{aligned}$$

The last inequality follows from the bound \(h_t^v(w) \ge (1/20)r_{s_L/20}\). Furthermore, under \(\mathbf{C1}(\delta ,L_0,L_1)\), the kernel \(\tilde{q}\) is close to p: in fact, one has \(\Vert \tilde{q}-p\Vert _1\le C(\log L)^{-8}\), see [2, Lemma 3.2] and the arguments in the proof there. This bound transfers to the exit measures, so that, arguing as in [2, Lemma 5.1],

$$\begin{aligned} \sup _{w\in B'}\big \Vert \mathrm {\Delta }'\hat{\pi }_t^{(p),v}(w,\cdot )\big \Vert _1 =\sup _{w\in B'}\big \Vert (\hat{{\varPi }}_t^v-\hat{\pi }_t^{(p),v})\hat{\pi }_t^{(p),v}(w,\cdot )\big \Vert _1 \le C(\log L)^{-8}. \end{aligned}$$

Putting the estimates together, we obtain as desired

$$\begin{aligned} \big |\hat{G}_t^v1_{B'}\mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(v)\big | \le C(\log L)^{-8}L^2. \end{aligned}$$

For the second summand of (23), Lemma 2 (ii) gives \(\hat{G}_t^v(v,V_t(v)\backslash B') \le C\), whence

$$\begin{aligned} \big |\hat{G}_t^v1_{V_t(v)\backslash B'}\mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(v)\big | \le C \sup _{w\in V_t(v)\backslash B'}\big |\mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(w)\big |. \end{aligned}$$

Fix \(w\in V_t(v)\backslash B'\). Set \(\eta = {\text {d}}(w,\partial V_t(v))\le 2r_t + \sqrt{d}\) and choose \(y_w\in \partial V_t(v)\) such that \(|w-y_w| = \eta \). With

$$\begin{aligned} I(y_w) = \left\{ y\in \partial V_t(v) : |y-y_w|\le (\log L)^{-5/2}s_L\right\} , \end{aligned}$$

we write

$$\begin{aligned} \mathrm {\Delta }'\pi _{V_t(v)}\varLambda _L(w)&= \sum _{y\in \partial V_t(v)}\mathrm {\Delta }'\pi _{V_t(v)}(w,y)\left( \varLambda _L(y)-\varLambda _L(y_w)\right) \nonumber \\&= \sum _{y \in I(y_w)}\mathrm {\Delta }'\pi _{V_t(v)}(w,y)\left( \varLambda _L(y)-\varLambda _L(y_w)\right) \nonumber \\&\quad + \sum _{y\in \partial V_t(v)\backslash I(y_w)}\mathrm {\Delta }'\pi _{V_t(v)}(w,y)\left( \varLambda _L(y)-\varLambda _L(y_w)\right) . \end{aligned}$$
(24)

For \(y\in I(y_w)\), Lemma 10 yields \(|\varLambda _L(y)-\varLambda _L(y_w)| \le C(\log L)^{-7/3}s_L^2\). Therefore,

$$\begin{aligned} \sum _{y \in I(y_w)}\left| \mathrm {\Delta }'\pi _{V_t(v)}(w,y)\right| \left| \varLambda _L(y)-\varLambda _L(y_w)\right| \le C(\log L)^{-8}L^2. \end{aligned}$$

It remains to handle the second term of (24). To this end, let \(U(w) = \{u\in V_t(v) : |\mathrm {\Delta }'(w,u)|> 0\}\). Using for \(y\in \partial V_t(v)\backslash I(y_w)\) the trivial bound

$$\begin{aligned} \left| \varLambda _L(y)-\varLambda _L(y_w)\right| \le \varLambda _L(y) + \varLambda _L(y_w) \le C(\log L)^{-6}L^2, \end{aligned}$$

see (20), we obtain

$$\begin{aligned}&\sum _{y\in \partial V_t(v)\backslash I(y_w)}\left| \mathrm {\Delta }'\pi _{V_t(v)}(w,y)\left( \varLambda _L(y)-\varLambda _L(y_w)\right) \right| \\&\quad \le C(\log L)^{-6}L^2\sup _{u\in U(w)}\pi _{V_t(v)}\left( u,\partial V_t(v)\backslash I(y_w)\right) . \end{aligned}$$

If \(u\in U(w)\) and \(y\in \partial V_t(v)\backslash I(y_w)\), then

$$\begin{aligned} |u-y| \ge |y-y_w| -|y_w-u| \ge (\log L)^{-5/2}s_L - 3r_t \ge (1/2)(\log L)^{-5/2}s_L. \end{aligned}$$

For such u, we get by standard hitting estimates, see e.g. [2, Lemma 4.2 (ii)],

$$\begin{aligned} \pi _{V_t(v)}\left( u,\partial V_t(v)\backslash I(y_w)\right)&\le C r_t \sum _{y\in \partial V_t(v)\backslash I(y_w)}\frac{1}{|u-y|^d} \\&\le Cr_t(\log L)^{5/2}(s_L)^{-1} \le C(\log L)^{-9}. \end{aligned}$$

The estimate on the sum can be obtained from [2, Lemma 4.6]. This bounds the second term of (24). We have proved (22) and hence the lemma. \(\square \)

Now it is easy to prove the main propagation step.

Lemma 12

Assume \(\mathbf{A0}(\varepsilon )\) and \(\mathbf{B}\). There exists \(L_0=L_0(\eta )\) such that if \(L_1\ge L_0\) and \(\mathbf{C1}(\delta ,L_0,L_1)\) holds, then for \(L_1\le L\le L_1(\log L_1)^2\) and \(\mathbb {P}\)-almost all \(\omega \in {\text {Good}}_L^{\mathrm{sp}}\cap {\text {OneBad}}_L^{\mathrm{tm}}\),

$$\begin{aligned} {\text {E}}_{0,\omega }\left[ \tau _L\right] \in \left[ 1-f_\eta (L),\,1+f_\eta (L)\right] \cdot {\text {E}}_0\left[ \tau _L\right] . \end{aligned}$$

Proof

We let \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\cap {\text {Good}}_L^{\mathrm{sp}}\cap {\text {OneBad}}_L^{\mathrm{tm}}\). Put \(p=p_{s_L/20}\). In this proof, we keep the superscript (p) in \(\hat{g}^{(p)}\). By Lemma 6 and the perturbation expansion (4), with \(\mathrm {\Delta }= 1_{V_L}(\hat{{\varPi }}-\hat{\pi }^{(p)})\),

$$\begin{aligned} {\text {E}}_{0,\omega }\left[ \tau _L\right] = \hat{G}\varLambda _L(0) = \hat{g}^{(p)}\varLambda _L(0) +\hat{G}\mathrm {\Delta }\hat{g}^{(p)}\varLambda _L(0) = A_1 + A_2. \end{aligned}$$

Set \(B= V_L\backslash (\mathcal {B}_L^{\mathrm{tm}}\cup {\text {Sh}}_L(L/(\log L)^2)\). The term \(A_1\) we split into

$$\begin{aligned} A_1 = \hat{g}^{(p)}1_B\varLambda _L(0) + \hat{g}^{(p)}1_{V_L\backslash B}\varLambda _L(0). \end{aligned}$$

Since \(\hat{g}^{(p)}(0,V_L\backslash B) \le C(\log L)^3\) by Lemma 2 (ii) and \(\varLambda _L(x) \le (\log L)^{-6}L^2\), the second summand of \(A_1\) can be bounded by \(O\left( (\log L)^{-3}\right) {\text {E}}_0[\tau _L]\). The main contribution comes from the first summand. First notice that

$$\begin{aligned} \hat{g}^{(p)}1_B\lambda ^{(p)}_L(0) = \hat{g}^{(p)}1_B\lambda _L(0)\left( 1+O\left( s_L^{-1}\right) \right) ={\text {E}}_{0}\left[ \tau _L\right] \left( 1+O\left( (\log L)^{-6}\right) \right) . \end{aligned}$$

Furthermore, we have for \(x\in B\) by definition

$$\begin{aligned} \varLambda _L(x) \in \left[ 1-f_\eta \left( (\log L)^{-3}L\right) ,\, 1+f_\eta \left( (\log L)^{-3}L\right) \right] \cdot \lambda _L(x). \end{aligned}$$

Collecting all terms, we conclude that \(A_1\) is contained in

$$\begin{aligned}&\left[ 1 - O\left( (\log L)^{-3}\right) - f_\eta \left( (\log L)^{-3}L\right) ,\, 1 + O\left( (\log L)^{-3}\right) + f_\eta \left( (\log L)^{-3}L\right) \right] \\&\quad \cdot {\text {E}}_0\left[ \tau _L\right] . \end{aligned}$$

Lemma 11 bounds \(A_2\) by \(O((\log L)^{-5/3}){\text {E}}_0[\tau _L]\). Since for L sufficiently large,

$$\begin{aligned} f_\eta (L) > f_\eta \left( (\log L)^{-3}L\right) +C(\log L)^{-5/3}, \end{aligned}$$

we arrive at

$$\begin{aligned} {\text {E}}_{0,\omega }\left[ \tau _L\right] = A_1 + A_2 \,\in \, \left[ 1-f_\eta (L),\,1+f_\eta (L)\right] \cdot {\text {E}}_0\left[ \tau _L\right] , \end{aligned}$$

as claimed. \(\square \)

Proposition 3 is now an immediate consequence of our estimates.

Proof (of Proposition 3)

From Lemmata 9 and 12 we deduce that for large \(L_0\), if \(L_1\ge L_0\) and \(L_1\le L\le L_1(\log L_1)^2\), we have under \(\mathbf{C1}(\delta ,L_0,L_1)\) and \(\mathbf{C2}(\eta ,L_1)\)

$$\begin{aligned}&\mathbb {P}\left( {\text {E}}_{0,\omega }\left[ \tau _L\right] \notin \left[ 1-f(L),\, 1+f(L)\right] \cdot {\text {E}}_0\left[ \tau _L\right] \right) \\&\quad \le \mathbb {P}\left( {\text {Bad}}_L^{\mathrm{sp}}\cup {\text {ManyBad}}_L^{\mathrm{tm}}\right) \\&\qquad + \mathbb {P}\left( {\text {E}}_{0,\omega }\left[ \tau _L\right] \notin \left[ 1-f(L),\,1+f(L)\right] \cdot {\text {E}}_0[\tau _L];\,{\text {Good}}_L^{\mathrm{sp}}\cap {\text {OneBad}}_L^{\mathrm{tm}}\right) \\&\quad \le \exp \left( -(1/2)(\log L)^2\right) . \end{aligned}$$

By Proposition 1, if \(\delta >0\) is small and \(L_0\) is sufficiently large, \(\mathbf{C1}(\delta ,L_0,L)\) holds under \(\mathbf{A0}(\varepsilon )\) for all \(L\ge L_0\), provided \(\varepsilon \le \varepsilon _0(\delta )\). This proves the proposition. \(\square \)

Proof of the main theorem on sojourn times

We shall first prove convergence of the (non-random) sequence \(\mathbb {E}[{\text {E}}_{0,\omega }[\tau _L]]/L^2\) towards a constant D that lies in a small interval around 1. Note that Proposition 3 together with Lemma 5 already tells us that for any \(0<\eta <1\), under \(\mathbf{A0}\), \(\mathbf{B}\) and \(\mathbf{A1}(\varepsilon )\) for \(\varepsilon (\eta )\) small,

$$\begin{aligned} \mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right] /L^2 \in [1-\eta ,1+\eta ]\quad \hbox {for large }L. \end{aligned}$$

Proposition 4

Assume \(\mathbf{A1}\) and \(\mathbf{B}\). Given \(0<\eta < 1\), one can find \(\varepsilon _0=\varepsilon _0(\eta ) > 0\) such that if \(\mathbf{A0}(\varepsilon )\) is satisfied for some \(\varepsilon \le \varepsilon _0\), then there exists \(D\in [1-\eta , 1+\eta ]\) such that

$$\begin{aligned} \lim _{L\rightarrow \infty }\left( \mathbb {E}\left[ {\text {E}}_{0,\omega } \left[ \tau _L\right] \right] /L^2\right) =D. \end{aligned}$$

Proof

Let \(0<\eta <1\). By choosing first \(\delta \), then \(L_0\) and then \(\varepsilon _0\) small respectively large enough, we know from Propositions 1 and 3 that under \(\mathbf{A1}\) and \(\mathbf{B}\), whenever \(\mathbf{A0}(\varepsilon )\) is satisfied for some \(\varepsilon \le \varepsilon _0\), \(\mathbf{C1}(\delta ,L_0,L)\) and \(\mathbf{C2}(\eta /2,L)\) hold true for all \(L\ge L_0\). We can therefore assume both conditions. We obtain from Lemma 8

$$\begin{aligned} \mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right] = \mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] ;\,{\text {Good}}_L^{\mathrm{sp}}\right] +O\left( L^2\exp \left( -(2/3)(\log L)^2\right) \right) . \end{aligned}$$

Thus it suffices to look at \({\text {E}}_{0,\omega }[\tau _L]\) on the event \(({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d} \cap {\text {Good}}_L^{\mathrm{sp}}\). Setting \(B= V_L\backslash ({\text {Sh}}_L(L/(\log L)^2))\), we see from the proof of Lemma 12 that on this this event,

$$\begin{aligned} {\text {E}}_{0,\omega }\left[ \tau _L\right] = (\hat{g}^{(p)}1_B\varLambda _L)(0) + O\left( (\log L)^{-{5/3}}L^2\right) , \end{aligned}$$

where the constant in the error term does only depend on d (and not on L or the environment). For \(x\in B\), \(h_L(x)= s_L/20\). In particular, this implies on the set B

$$\begin{aligned} \mathbb {E}\left[ \varLambda _L(\cdot )\right] \equiv \mathbb {E}\left[ \varLambda _L(0)\right] ,\quad \hbox {and } \lambda ^{(p)}(\cdot ) \equiv \lambda ^{(p)}(0). \end{aligned}$$

Now put \(c_L = \mathbb {E}\left[ \varLambda _L(0)\right] /\lambda ^{(p)}(0)\). We have

$$\begin{aligned} \mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right]&= \hat{g}^{(p)}(0,B)\cdot \mathbb {E}\left[ \varLambda _L(0)\right] + O\left( (\log L)^{-5/3}L^2\right) \\&= c_L\,\hat{g}^{(p)}(0,B)\cdot \lambda ^{(p)}(0) + O\left( (\log L)^{-5/3}L^2\right) \\&= c_L\,{\text {E}}_{0,p}\left[ \tau _L\right] +O\left( (\log L)^{-5/3}L^2\right) . \end{aligned}$$

Since \({\text {E}}_{0,p}\left[ \tau _L\right] /L^2\) converges to 1 by Lemma 4, convergence of \(\mathbb {E}\left[ {\text {E}}_{0,\omega }[\tau _L]\right] /L^2\) follows if we show that \(\lim _{L\rightarrow \infty }c_L\) exists. Let \(L'\in (L,2L]\). As before,

$$\begin{aligned} \mathbb {E}\left[ {\text {E}}_{0,\omega }[\tau _{L'}]\right] =c_{L'}\,{\text {E}}_{0,p}\left[ \tau _{L'}\right] +O\left( (\log L)^{-5/3}L^2\right) . \end{aligned}$$
(25)

On the other hand, we claim that (25) also holds with \(c_{L'}\) replaced by \(c_L\). To see this, we slightly change the coarse graining scheme inside \(V_{L'}\). More specifically, we define for \(L'\in (L,2L]\) the coarse graining function \(\tilde{h}_{L'} : \overline{C}_{L'} \rightarrow {\mathbb {R}}_+\) by setting

$$\begin{aligned} \tilde{h}_{L'}(x) = \frac{1}{20}\max \left\{ s_L h\left( \frac{\text {d}_{L'}(x)}{s_{L'}}\right) ,\, r_L\right\} . \end{aligned}$$

Then \(\tilde{h}_{L'}(x) = h_L(0)=s_L/20\) for \(x\in V_{L'}\) with \(\text {d}_{L'}(x) \ge 2s_{L'}\). We consider the analogous definition of space-good/bad and time-good/bad points within \(V_{L'}\), which uses the coarse graining function \(\tilde{h}_{L'}\) instead of \(h_{L'}\) and the coarse grained transition kernels \(\tilde{\varPi }\) and \(\tilde{\pi }\) in \(V_{L'}\) defined in terms of \(\tilde{h}_{L'}\), cf. (9). Clearly, all the above statements of this section remain true (at most the constants change), and we can work with the same kernel \(p=p_{s_L/20}\). See Fig. 4.

Fig. 4
figure 4

Inside \(V_L\backslash {\text {Sh}}_L(2s_L)\), the transition kernels \(\tilde{\varPi }\), \(\tilde{\pi }^{(p)}\) of the coarse grained walks in \(V_{L'}\) defined in terms of \(\tilde{h}_{L'}\) agree with the transition kernels \(\hat{{\varPi }}\), \(\hat{\pi }^{(p)}\) of the coarse grained walks in \(V_L\) defined in terms of \(h_L\)

Denoting by \(\tilde{g}^{(p)}\) the Green’s function corresponding to \(\tilde{\pi }^{(p)}\) and by \(B'\) the set \(V_{L'}\backslash {\text {Sh}}_{L'}(L'/(\log L')^2)\), we obtain as above

$$\begin{aligned} \mathbb {E}\left[ {\text {E}}_{0,\omega }[\tau _{L'}]\right]&= \tilde{g}^{(p)}(0,B')\mathbb {E}\left[ \varLambda _L(0)\right] + O\left( (\log L)^{-5/3}L^2\right) \\&= c_L\,\tilde{g}^{(p)}(0,B')\lambda ^{(p)}(0) + O\left( (\log L)^{-5/3}L^2\right) \\&= c_L\,{\text {E}}_{0,p}\left[ \tau _{L'}\right] +O\left( (\log L)^{-5/3}L^2\right) . \end{aligned}$$

Note that since \(\tilde{h}_{L'}(\cdot )\equiv h_L(0)\) on \(B'\), the quantities \(c_L\), \(\mathbb {E}\left[ \varLambda _L(0)\right] \) and \(\lambda ^{(p)}(0)\) do indeed appear in the above display. Comparing with (25), this shows that for some constant \(C>0\)

$$\begin{aligned} |c_L-c_{L'}|\le C(\log L)^{-5/3}, \end{aligned}$$

which readily implies that \(c_L\) is a Cauchy sequence and thus \(\lim _{L\rightarrow \infty }c_L=D\) exists. From Proposition 3 we already know that \(D\in [1-\eta ,1+\eta ]\). This finishes the proof. \(\square \)

We shall now employ Hoeffding’s inequality to show that \({\text {E}}_{0,\omega }[\tau _L]\) is close to its mean.

Lemma 13

Assume \(\mathbf{A1}\) and \(\mathbf{B}\). There exists \(\varepsilon _0 > 0\) such that if \(\mathbf{A0}(\varepsilon )\) holds for some \(\varepsilon \le \varepsilon _0\), then

$$\begin{aligned} \mathbb {P}\left( \frac{1}{L^2}\big |{\text {E}}_{0,\omega }\left[ \tau _L\right] -\mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right] \big |>(\log L)^{-4/3}\right) \le \exp \left( -(1/3)(\log L)^2\right) . \end{aligned}$$

Let us first show how to prove Theorem 1 from this result.

Proof (of Theorem 1)

We know from Proposition 4, D the constant from there,

$$\begin{aligned} \Big |\frac{1}{L^2}{\text {E}}_{0,\omega }\left[ \tau _L\right] -D\Big | \le \frac{1}{L^2}\big |{\text {E}}_{0,\omega }\left[ \tau _L\right] -\mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right] \big | + \alpha (L) \end{aligned}$$

for some (deterministic) sequence \(\alpha (L)\rightarrow 0\) as \(L\rightarrow \infty \). Putting

$$\begin{aligned} \alpha '(L) = \max \left\{ (\log L)^{-4/3}, \alpha (L)\right\} , \end{aligned}$$

we deduce from Lemma 13 that

$$\begin{aligned} \mathbb {P}\left( \Big |\frac{1}{L^2}{\text {E}}_{0,\omega }\left[ \tau _L\right] -D\Big |\ge 2\alpha '(L)\right) \le \exp \left( -(1/3)(\log L)^2\right) . \end{aligned}$$

This implies the first statement of the theorem. For the second, we have

$$\begin{aligned}&\mathbb {P}\left( \Big |\sup _{x: |x| \le L^k}{\text {E}}_{x,\omega }\left[ \tau _{V_L(x)}\right] /L^2 -D\Big | \ge 2\alpha '(L)\right) \\&\quad \le CL^{kd} \mathbb {P}\left( \left| {\text {E}}_{0,\omega }\left[ \tau _L\right] /L^2 -D\right| \ge 2\alpha '(L)\right) \le \exp \left( -(1/4)(\log L)^2\right) \end{aligned}$$

for large L, and the same bound holds with the supremum over x with \(|x|\le L^k\) replaced by the infimum. The second claim of the theorem follows now from the lemma of Borel–Cantelli. \(\square \)

It remains to prove Lemma 13.

Proof (of Lemma 13)

By Proposition 1 and Lemma 8, we can find \(\varepsilon _0>0\) such that under \(\mathbf{A0}\) and \(\mathbf{A1}(\varepsilon )\) for \(\varepsilon \le \varepsilon _0\),

$$\begin{aligned} \mathbb {P}\left( {\text {Bad}}_L^{\mathrm{sp}}\right) \le \exp \left( -(2/3)(\log L)^2\right) \quad \hbox {for }L\hbox { large.} \end{aligned}$$

As in the proof of Proposition 4 (or Lemma 12), we have for \(\omega \in ({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\) in the complement of \({\text {Bad}}_L^{\mathrm{sp}}\), that is \(\mathbb {P}\)-almost surely on the event \({\text {Good}}_L^{\mathrm{sp}}\),

$$\begin{aligned} {\text {E}}_{0,\omega }\left[ \tau _L\right] = (\hat{g}^{(p)}1_B\varLambda _L)(0) + O\left( (\log L)^{-{5/3}}L^2\right) , \end{aligned}$$

where \(B= V_L\backslash ({\text {Sh}}_L(L/(\log L)^2))\). In the proof of Proposition 4 we have also seen that

$$\begin{aligned} \mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right] = \hat{g}^{(p)}(0,B)\mathbb {E}\left[ \varLambda _L(0)\right] + O\left( (\log L)^{-5/3}L^2\right) . \end{aligned}$$

Therefore, on \({\text {Good}}_L^{\mathrm{sp}}\),

$$\begin{aligned}&{\text {E}}_{0,\omega }\left[ \tau _L\right] \\&\quad = \hat{g}^{(p)}(0,B)\mathbb {E}\left[ \varLambda _L(0)\right] + \sum _{y\in B}\hat{g}^{(p)}(0,y)\left( \varLambda _L(y)-\mathbb {E}\left[ \varLambda _L(0)\right] \right) \\&\qquad + O\left( (\log L)^{-5/3}L^2\right) \\&\quad = \mathbb {E}\left[ {\text {E}}_{0,\omega }\left[ \tau _L\right] \right] + \sum _{y\in B}\hat{g}^{(p)}(0,y)\left( \varLambda _L(y)-\mathbb {E}\left[ \varLambda _L(0)\right] \right) + O\left( (\log L)^{-5/3}L^2\right) . \end{aligned}$$

The statement of the lemma will thus follow if we show that

$$\begin{aligned} \mathbb {P}\left( \Big |\sum _{y\in B}\hat{g}^{(p)}(0,y)\left( \varLambda _L(y)-\mathbb {E}\left[ \varLambda _L(0)\right] \right) \Big |>(\log L)^{-3/2}L^2\right) \le \exp \left( -(\log L)^2\right) .\nonumber \\ \end{aligned}$$
(26)

We use a similar strategy as in the proof of [2, Lemma 6.4]. First, define for \(j\in {\mathbb {Z}}\) the interval \(I_j = (js_L,(j+1)s_L]\). Now divide B into subsets \(W_{\mathbf{j}} = B\cap \left( I_{j_1}\times \dots \times I_{j_d}\right) \), where \({\mathbf{j}} = (j_1,\dots ,j_d) \in {\mathbb {Z}}^d\). Let J be the set of those \(\mathbf{j}\) for which \(W_{\mathbf{j}} \ne \emptyset \). Then there exists a constant \(K=K(d)\) and a disjoint partition of J into sets \(J_1,\dots ,J_K\), such that for any \(1\le \ell \le K\),

$$\begin{aligned} \mathbf{j},\mathbf{j}' \in J_\ell ,\ \mathbf{j}\ne \mathbf{j}' \Longrightarrow {\text {d}}(W_{\mathbf{j}},W_{\mathbf{j}'}) > s_L. \end{aligned}$$
(27)

We set

$$\begin{aligned} \xi _{\mathbf{j}} = \sum _{y\in W_{\mathbf{j}}}\hat{g}^{(p)}(0,y)\left( \varLambda _L(y)-\mathbb {E}\left[ \varLambda _L(0)\right] \right) \end{aligned}$$

and \(t = t(d,L) = (\log L)^{-3/2}L^2\). From (27) we see that the random variables \(\xi _{\mathbf{j}}\), \(\mathbf{j}\in J_\ell \), are independent and centered (we recall again that \(\mathbb {E}\left[ \varLambda _L(y)\right] = \mathbb {E}\left[ \varLambda _L(0)\right] \) for \(y\in B\)). Put \({\varOmega }'=({\mathcal {P}}_\varepsilon )^{{\mathbb {Z}}^d}\cap (\mathcal {P}^{\text {s},1})^{{\mathbb {Z}}^d}\). Applying Hoeffding’s inequality, we obtain with \({\Vert \xi _{\mathbf{j}}\Vert }_{\infty } = \sup _{\omega \in {\varOmega }'}|\xi _{\mathbf{j}}(\omega )|\), for some constant \(c>0\),

$$\begin{aligned} \mathbb {P}\left( \Big |\sum _{\mathbf{j}\in J}\xi _{\mathbf{j}}\Big | > t\right) \le K \max _{1\le \ell \le K}\mathbb {P}\left( \Big |\sum _{\mathbf{j}\in J_\ell }\xi _{\mathbf{j}}\Big | > \frac{t}{K}\right) \le 2\exp \left( -c\frac{(\log L)^{-3}L^{4}}{\sum _{\mathbf{j}\in J_\ell }{\Vert \xi _{\mathbf{j}}\Vert }^2_{\infty }}\right) .\nonumber \\ \end{aligned}$$
(28)

It remains to estimate the \(\sup \)-norm of the \(\xi _{\mathbf{j}}\). We have, by Lemmata 1 and 2,

$$\begin{aligned} \hat{g}(x,W_{\mathbf{j}}) \le \frac{Cs_L^d}{s_L^2(s_L + {\text {d}}(x,W_{\mathbf{j}}))^{d-2}} = C\left( 1+\frac{{\text {d}}(x,W_{\mathbf{j}})}{s_L}\right) ^{2-d}. \end{aligned}$$

For \(y\in B\), the a priori bound (20) gives

$$\begin{aligned} \left| \varLambda _L(y)-\mathbb {E}\left[ \varLambda _L(0)\right] \right| \le C (\log L)^{-6}L^2. \end{aligned}$$

Putting the last two estimates together, we obtain, using \(d\ge 3\),

$$\begin{aligned} \sum _{\mathbf{j}\in J_\ell }{\Vert \xi _{\mathbf{j}}\Vert }^2_{\infty } \le C\sum _{r=1}^{C(\log L)^3}r^{-d+3}(\log L)^{-12}L^4\le C(\log L)^{-9}L^4. \end{aligned}$$

Going back to (28), this shows

$$\begin{aligned} \mathbb {P}\left( \Big |\sum _{y\in B}\hat{g}^{(p)}(0,y)\left( \varLambda _L(y)-\mathbb {E}\left[ \varLambda _L(0)\right] \right) \Big |\ge (\log L)^{-3/2}L^2\right) \le 2\exp \left( -c(\log L)^{6}\right) , \end{aligned}$$

which is more than we need, cf. (26). This completes the proof of the lemma. \(\square \)

Proof (of Corollary 1)

Let \(k\in {\mathbb {N}}\), and let first \(m=1\). By Proposition 3, we obtain under our conditions (for \(\varepsilon \) small)

$$\begin{aligned}&\mathbb {P}\left( \sup _{x: |x| \le L^{k}}\sup _{y\in V_L(x)}{\text {E}}_{y,\omega }\left[ \tau _{V_L(x)}\right] /L^2\ge 2\right) \le CL^{kd}\mathbb {P}\left( \sup _{y\in V_L}{\text {E}}_{y,\omega }\left[ \tau _L\right] /L^2\ge 2\right) \\&\quad \le CL^{(k+1)d}\mathbb {P}\left( {\text {E}}_{0,\omega }\left[ \tau _L\right] /L^2\ge 2\right) \le \exp \left( -(1/3)(\log L)^2\right) . \end{aligned}$$

This implies by the lemma of Borel–Cantelli that

$$\begin{aligned} \limsup _{L\rightarrow \infty }\sup _{x: |x| \le L^k}\sup _{y\in V_L(x)}{\text {E}}_{y,\omega }\left[ \tau _{V_L(x)}\right] /L^2\le 2\quad \mathbb {P}\hbox {-almost surely}. \end{aligned}$$
(29)

For the rest of the proof, take an environment \(\omega \) that satisfies (29). Assume \(m \ge 2\). Then

$$\begin{aligned} {\text {E}}_{x,\omega }\left[ \tau ^m_{V_L(x)}\right]&= \sum _{\ell _1,\dots ,\ell _m\ge 0}{\text {P}}_{x,\omega }\left( \tau _{V_L(x)} > \ell _1,\dots ,\tau _{V_L(x)}>\ell _m\right) \\&\le m!\sum _{0\le \ell _1\le \dots \le \ell _m}{\text {P}}_{x,\omega }\left( \tau _{V_L(x)} > \ell _m\right) . \end{aligned}$$

By the Markov property, using the case \(m=1\) and induction in the last step,

$$\begin{aligned}&\sum _{0\le \ell _1\le \dots \le \ell _m}{\text {P}}_{x,\omega }\left( \tau _{V_L(x)} > \ell _m\right) \\&\quad = \sum _{0\le \ell _1\le \dots \le \ell _{m-1}}{\text {E}}_{x,\omega }\left[ \sum _{\ell =0}^\infty {\text {P}}_{X_{\ell _{m-1}},\omega }\left( \tau _{V_L(x)} > \ell \right) ;\, \tau _{V_L(x)}> \ell _{m-1}\right] \\&\quad \le \sup _{z\in V_L(x)}{\text {E}}_{z,\omega }\left[ \tau _{V_L(x)}\right] \sum _{0\le \ell _1\le \dots \le \ell _{m-1}}{\text {P}}_{x,\omega }\left( \tau _{V_L(x)}> \ell _{m-1}\right) \le 2^mL^{2m}, \end{aligned}$$

if \(L = L(\omega )\) is sufficiently large. \(\square \)

A quenched invariance principle

Here we combine the results on the exit distributions from [2] and our results on the mean sojourn times to prove Theorem 2, which provides a functional central limit theorem for the RWRE under the quenched measure. Let us recall the precise statement.

Assume \(\mathbf{A0}(\varepsilon )\) for small \(\varepsilon > 0\), \(\mathbf{A1}\) and \(\mathbf{B}\). Then for \(\mathbb {P}\)-a.e. \(\omega \in {\varOmega }\), under \({\text {P}}_{0,\omega }\), the \(C({\mathbb {R}}_+,{\mathbb {R}}^d)\)-valued sequence \(X_t^{n}/\sqrt{n}\), \(t\ge 0\), converges in law to a d-dimensional Brownian motion with diffusion matrix \(D^{-1}\varvec{\Lambda }\), where D is the constant from Theorem 1, \(\varvec{\Lambda }\) is given by (2), and \(X_t^{n}\) is the linear interpolation \( X_t^{n}= X_{\lfloor tn\rfloor } + {(tn-\lfloor tn\rfloor )}(X_{\lfloor tn\rfloor +1}-X_{\lfloor tn\rfloor }).\)

The statement follows if we show that for each real \(T>0\), weak convergence occurs in \(C([0,T],{\mathbb {R}}^d)\). In order to simplify notation, we will restrict ourselves to \(T=1\), the general case being a simple generalization of this case.

Let us first give a rough (simplified) idea of our proof. We define the step size \(L_n=(\log n)^{-1}\sqrt{n}\). From Theorem 1 we infer that the RWRE should have left about \((\log n)^2/D\) balls of radius \(L_n\) in the first n steps. Proposition 1 tells us that for sufficiently large n, the exit law from each of those balls is close to that of a symmetric random walk with nearest neighbor kernel \(p_{L_n}\). For our limit theorem, this will imply that we can replace the coarse grained RWRE taking steps of size \(L_n\), i.e. the RWRE observed at the successive exit times from balls of radius \(L_n\), by the analogous coarse grained random walk with kernel \(p_{L_n}\). For the latter, we apply the multidimensional Lindeberg-Feller limit theorem. Since we already know that the kernels \(p_{L_n}\) converge to \(p_\infty \), see (12), we obtain in this way the stated convergence of the one-dimensional distributions. Moreover, since our estimates on exit measures and exit times are sufficiently uniform in the starting point, multidimensional convergence and tightness then follow from standard arguments. See Fig. 5 for an illustration.

Fig. 5
figure 5

The coarse grained RWRE \(\hat{X}_{n,i}\), \(i\in {\mathbb {N}}\), which is obtained from observing the RWRE at the successive exit times from balls of radius \(L_n\). Here \(k_n\) denotes the maximal number of such balls which are left by the RWRE in the first n steps

Construction of coarse grained random walks on \({\mathbb {Z}}^d\)

We start with a precise description of the coarse grained random walks. Let

$$\begin{aligned} L_n=(\log n)^{-1}\sqrt{n}. \end{aligned}$$

Similarly to the proof of Lemma 6, given an environment \(\omega \in {\varOmega }\), we introduce a probability space where we can observe both the random walk with kernel \(p_\omega \) and a coarse grained version of it taking steps of a size between \(L_n\) and \(2L_n\).

More specifically, we take a probability space \((\varXi , {\mathcal {A}}, \mathbb {Q})\) that carries a family of i.i.d. random variables \((\xi _{n,i}: i\in {\mathbb {N}})\), with \(\xi _{n,i}\) distributed according to \(\varphi (t)\text {d}t\). We then consider the probability space \((({\mathbb {Z}}^d)^{{\mathbb {N}}}\times \varXi , \mathcal {G}\otimes {\mathcal {A}},\widetilde{{\text {P}}}_{x,\omega })\), where \(\widetilde{{\text {P}}}_{x,\omega } = {\text {P}}_{x,\omega }\otimes \mathbb {Q}\). On this space, \(X_k\) denotes again the projection on the kth component of \(\left( {\mathbb {Z}}^d\right) ^{{\mathbb {N}}}\), so that under \(\widetilde{{\text {P}}}_{x,\omega }\), \(X_k\) has the law of a random walk started from x with transition kernel \(p_{\omega }\).

Set \(T_{n,0} = 0\), and recursively for integers \(i\in {\mathbb {N}}\),

$$\begin{aligned} T_{n,i+1}&= \inf \left\{ m> T_{n,i}: X_m\notin V_{\xi _{n,T_{n,i}}\cdot L_n}\left( X_{T_{n,i}}\right) \right\} ,\\ \hat{X}_{n,i}&= X_{T_{n,i}}. \end{aligned}$$

Under \(\widetilde{{\text {P}}}_{x,\omega }\), for fixed n, \(\hat{X}_{n,i}\) is the coarse grained Markov chain running with transition probabilities

$$\begin{aligned} Q_{n,\omega }(y,\cdot ) = \frac{1}{L_n}\int _{{\mathbb {R}}_+}\varphi \left( \frac{t}{L_n}\right) \varPi _{V_t(y)}(y,\cdot )\text {d}t \end{aligned}$$

and started from x, i.e. \(\widetilde{{\text {P}}}_{x,\omega }(\hat{X}_{n,0}=x)=1\). Note that the steps of the coarse grained walk have a size between \(L_n\) and \(2L_n\) and do not depend on the current location. We shall suppress the environment \(\omega \) in the notation and write \(Q_{n}\) instead of \(Q_{n,\omega }\).

We will compare \(Q_n\) with the coarse grained (non-random) kernel

$$\begin{aligned} q_{n}(y,\cdot ) = \frac{1}{L_n}\int _{{\mathbb {R}}_+}\varphi \left( \frac{t}{L_n}\right) \pi _{V_t}^{(p_{L_n})}(0,\cdot -y)\text {d}t, \end{aligned}$$

where the kernel \(p_{L_n}\) stems from Assignment (10).

Good events

Good behavior in space

We shall now introduce an event \(A_1\) with \(\mathbb {P}(A_1)=1\) on which the RWRE has a “good” behavior in terms of exit distributions. Let

$$\begin{aligned} D_{L,p,\psi ,q}(x) = \left\| \left( \varPi _{V_L(x)}-\pi ^{(p)}_{V_L(x)}\right) \pi ^{(q)}_{\psi }(x,\cdot )\right\| _1. \end{aligned}$$

We require that all smoothed differences of exit measures \(D_{L,p_{L_n},\psi _n,p_{L_n}}(x)\), where \(x\in {\mathbb {Z}}^d\) with \(|x|\le n^3\), \(L\in [L_n,2L_n]\) and \(\psi _n\equiv L_n\), are small when n is large.

In this regard, note that Proposition 1 implies for large n

$$\begin{aligned}&\mathbb {P}\left( \sup _{x:|x|\le n^3}\sup _{L_n\le L\le 2L_n}D_{L,p_{L_n},\psi _n,p_{L_n}}(x) > (\log L_n)^{-9}\right) \\&\quad \le C n^{4d}\sup _{L_n\le L\le 2L_n}\mathbb {P}\left( D_{L,p_{L_n},\psi _n,p_{L_n}}^*> (\log L_n)^{-9}\right) \le \exp \left( -(1/5)(\log n)^2\right) . \end{aligned}$$

An application of Borel–Cantelli’s lemma shows that on a set \(A_1\) of full \(\mathbb {P}\)-measure,

$$\begin{aligned} \limsup _{L\rightarrow \infty }\sup _{x:|x|\le n^3}\sup _{L_n\le L\le 2L_n}D_{L,p_{L_n},\psi _n,p_{L_n}}(x) =0. \end{aligned}$$
(30)

Good behavior in time

We next specify an event \(A_2\) with \(\mathbb {P}(A_2)=1\) on which we have uniform control over mean sojourn times. Let

$$\begin{aligned} c_\varphi =\int _{1}^2t^2\varphi (t)\text {d}t. \end{aligned}$$

Under our usual conditions, we obtain by Theorem 1 and dominated convergence, for \(\mathbb {P}\)-almost all \(\omega \in {\varOmega }\), D the constant from the theorem,

$$\begin{aligned} \lim _{n\rightarrow \infty }\left( \inf _{x: |x|\le n^3} \widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,1}\right] /L_n^2\right) = \lim _{n\rightarrow \infty }\left( \sup _{x: |x|\le n^3} \widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,1}\right] /L_n^2\right) = c_\varphi D.\nonumber \\ \end{aligned}$$
(31)

Moreover, by Corollary 1, for \(\mathbb {P}\)-almost all \(\omega \),

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left( \sup _{x: |x|\le n^3} \widetilde{{\text {E}}}_{x,\omega }\left[ T^2_{n,1}\right] /L_n^4\right) \le 8. \end{aligned}$$
(32)

We denote by \(A_2\) the set of environments of full \(\mathbb {P}\)-measure on which both (31) and (32) hold true.

A law of large numbers

Recall Fig. 4. We shall not merely consider \(k_n=k_{n,1}\), but more generally for \(t\in [0,1]\)

$$\begin{aligned} k_{n,t}= k_{n,t}(\omega )=\max \left\{ i\in {\mathbb {N}} : T_{n,i}\le tn \right\} . \end{aligned}$$

We shall need a (weak) law of large numbers for \(k_{n,t}\) under \(\widetilde{{\text {P}}}_{x,\omega }\), uniformly in \(|x|\le n^2\). In view of (31), it is natural to expect that \(k_{n,t}\) has the same asymptotic behavior as \(t\beta _n\), where

$$\begin{aligned} \beta _n= \left\lfloor \frac{n}{c_\varphi DL_n^2}\right\rfloor = \left\lfloor \frac{(\log n)^2}{c_\varphi D}\right\rfloor \end{aligned}$$

We first establish a bound on the variance of \(T_{n,\ell }\).

Lemma 14

For \(\mathbb {P}\)-almost all environments,

$$\begin{aligned} \sup _{\ell \le 2\beta _n}\sup _{|x|\le n^2}\frac{\text {Var}_{\widetilde{{\text {P}}}_{x,\omega }}(T_{n,\ell })}{n^2}\rightarrow 0\quad \hbox {as }n\rightarrow \infty , \end{aligned}$$

where \(\text {Var}_{\widetilde{{\text {P}}}_{x,\omega }}\) denotes the variance with respect to \(\widetilde{{\text {P}}}_{x,\omega }\).

Proof

We can restrict ourselves to \(\omega \in A_2\). Define the successive sojourn times \(\tau _{n,i}= (T_{n,i}-T_{n,i-1})\). Then \(T_{n,\ell }= \tau _{n,1}+\dots +\tau _{n,\ell }\). Unlike for random walk in a homogeneous environment, the variables \(\tau _{n,i}\), \(i=1,\dots ,2\beta _n\), are in general not independent under \(\widetilde{{\text {P}}}_{0,\omega }\). However, for \(i<j\), \(\tau _{n,j}\) is conditionally independent from \(\tau _{n,i}\) given \(\hat{X}_{n,j-1}\). By the strong Markov property (with the same justification as in the proof of Lemma 6), we obtain for \(i<j\le 2\beta _n\) and \(x\in {\mathbb {Z}}^d\) with \(|x|\le n^2\),

$$\begin{aligned} \widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\tau _{n,j}\right]&=\widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,j}\,\big |\,\hat{X}_{n,j-1}, \tau _{n,i}\right] \right] \\&=\widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\widetilde{{\text {E}}}_{\hat{X}_{n,j-1},\omega }\left[ T_{n,1}\right] \right] \le \sup _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2. \end{aligned}$$

In the last step we used that the coarse grained random can bridge in \(2\beta _n\) steps a distance of at most \(4\beta _nL_n<n\) and is therefore well inside \(V_{2n^2}\) when started from \(V_{n^2}\). Similarly, we see that

$$\begin{aligned} \widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\tau _{n,j}\right] \ge \inf _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2. \end{aligned}$$

For x with \(|x|\le n^2\) and \(i,j\le 2\beta _n\), it also holds that

$$\begin{aligned} \inf _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2 \le \widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\right] \widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,j}\right] \le \sup _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2. \end{aligned}$$

Since by definition of the event \(A_2\), we have for \(\omega \in A_2\)

$$\begin{aligned} \lim _{n\rightarrow \infty }\left( \inf _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2/L_n^4\right) = \lim _{n\rightarrow \infty }\left( \sup _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2/L_n^4\right) , \end{aligned}$$

we obtain for \(i,j\le 2\beta _n\) and x with \(|x|\le n^2\),

$$\begin{aligned}&\left| \widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\tau _{n,j}\right] -\widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,i}\right] \widetilde{{\text {E}}}_{x,\omega }\left[ \tau _{n,j}\right] \right| \\&\quad \le \sup _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2 - \inf _{|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] ^2\overset{{\text {def}}}{=}\alpha (n)=o(L_n^4)\quad \hbox {for }n\rightarrow \infty . \end{aligned}$$

Using this for \(i\ne j\) and (31), (32) for the terms with \(i=j\), we conclude that for \(n\ge n(\omega )\), \(\ell \le 2\beta _n\),

$$\begin{aligned} \sup _{|x|\le n^2}\text {Var}_{\widetilde{{\text {P}}}_{x,\omega }}(T_{n,\ell }) \le C\beta _nL_n^4 +C\beta _n^2\alpha (n) = o(n^2). \end{aligned}$$

This finishes the proof. \(\square \)

We are now in position to prove a weak law of large numbers for \(k_{n,t}\).

Lemma 15

For \(\mathbb {P}\)-almost all environments, for every \(t\in [0,1]\) and every \(\epsilon >0\),

$$\begin{aligned} \sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( \Big |k_{n,t}/\beta _n -t\Big |>\epsilon \right) \rightarrow 0\quad \hbox {as }n\rightarrow \infty . \end{aligned}$$

Proof

We take \(\omega \in A_2\) as in the previous lemma. There is nothing to show for \(t=0\), so assume \(t\in (0,1]\). If the statement would not hold, then we could find \(\epsilon \), \(\epsilon '>0\) such that

$$\begin{aligned} \sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( k_{n,t}< (t-\epsilon )\beta _n\right)&>\epsilon '\quad \hbox { infinitely often, or} \end{aligned}$$
(33)
$$\begin{aligned} \sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( k_{n,t}> (t+\epsilon )\beta _n\right)&>\epsilon '\quad \hbox { infinitely often}. \end{aligned}$$
(34)

Let us first assume (33). Then, with \(i_n= \lceil (t-\epsilon )\beta _n\rceil \), by definition

$$\begin{aligned} \sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( T_{n,i_n}>tn\right) >\epsilon '\quad \hbox { infinitely often}. \end{aligned}$$

Next note that by (31), by linearity of the expectation and the fact that \(2i_nL_n<n\),

$$\begin{aligned} 0\le \frac{\sup _{|x|\le n^2}\widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] }{tn}\le \frac{i_n}{tn}\sup _{y:|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] \le 1-\epsilon /2\quad \hbox {for } n\ge n_0(\omega ). \end{aligned}$$

Chebycheff’s inequality then shows that if \(n\ge n_0(\omega )\) and x with \(|x|\le n^2\),

$$\begin{aligned} \widetilde{{\text {P}}}_{x,\omega }\left( T_{n,i_n}>tn\right)&= \widetilde{{\text {P}}}_{x,\omega }\left( T_{n,i_n}-\widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] > tn-\widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] \right) \\&\le \frac{1}{(tn)^2}\left( 1-\widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] /(tn)\right) ^{-2}\,\hbox {Var}_{\widetilde{{\text {P}}}_{x,\omega }}\left( T_{n,i_n}\right) \\&\le \frac{4}{(\epsilon tn)^2}\sup _{|y|\le n^2}\hbox {Var}_{\widetilde{{\text {P}}}_{y,\omega }}\left( T_{n,i_n}\right) . \end{aligned}$$

The right-hand side converges to zero by Lemma 14. This contradicts (33).

Now assume (34). We argue similarly. First, with \(i_n= \lfloor (t+\epsilon )\beta _n\rfloor \) by definition

$$\begin{aligned} \sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( T_{n,i_n}<tn\right) >\epsilon '\quad \hbox { infinitely often}. \end{aligned}$$

Since for \(\omega \in A_2\) and large \(n\ge n_0(\omega )\),

$$\begin{aligned} \frac{\inf _{|x|\le n^2}\widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] }{tn}\ge \frac{i_n}{tn} \inf _{y:|y|\le 2n^2}\widetilde{{\text {E}}}_{y,\omega }\left[ T_{n,1}\right] \ge 1+\epsilon /2, \end{aligned}$$

we obtain for large \(n\ge n_0(\omega )\) and x with \(|x|\le n^2\),

$$\begin{aligned} \widetilde{{\text {P}}}_{x,\omega }\left( T_{n,i_n}<tn\right)&= \widetilde{{\text {P}}}_{x,\omega }\left( \widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] -T_{n,i_n}> \widetilde{{\text {E}}}_{x,\omega }\left[ T_{n,i_n}\right] -tn\right) \\&\le \frac{4}{(\epsilon tn)^2}\sup _{|y|\le n^2}\hbox {Var}_{\widetilde{{\text {P}}}_{y,\omega }}\left( T_{n,i_n}\right) /n^2\rightarrow 0\quad \hbox {as }n\rightarrow \infty . \end{aligned}$$

Therefore, neither (33) nor (34) can hold, and the proof is complete. \(\square \)

Proof of Theorem 2

We turn to the proof of Theorem 2. Recall our notation introduced above. Since the subscript n already appears in both \(k_{n,t}\) and \(\beta _n\), we may safely write

$$\begin{aligned} \hat{X}_{k_{n,t}}\hbox { instead of }\hat{X}_{n,k_{n,t}},\quad \hat{X}_{\lfloor t\beta _n\rfloor }\hbox { instead of }\hat{X}_{n, \lfloor t\beta _n\rfloor }. \end{aligned}$$

Since both \(A_1\) and \(A_2\) have full \(\mathbb {P}\)-measure, we can restrict ourselves to \(\omega \in A_1\cap A_2\). We first prove one-dimensional convergence, uniformly in the starting point x with \(|x|\le n^2\). This will easily imply multidimensional convergence and tightness.

One-dimensional convergence

Proposition 5

For \(\mathbb {P}\)-almost all environments, for each \(t\in [0,1]\) and \(u\in {\mathbb {R}}\),

$$\begin{aligned} \sup _{|x|\le n^2}\left| {\text {P}}_{x,\omega }\left( \left( X_t^{n}-x\right) /\sqrt{n}>u\right) -\text {P}\left( {\mathcal {N}}(0,tD^{-1}\varvec{\Lambda })>u\right) \right| \rightarrow 0\quad \hbox {as }n\rightarrow \infty , \end{aligned}$$

where \({\mathcal {N}}(0,A)\) denotes a d-dimensional centered Gaussian with covariance matrix A.

Proof

Let \(t\in [0,1]\). We write

$$\begin{aligned} X_t^n= \hat{X}_{\lfloor t\beta _n\rfloor }+(X_t^n-\hat{X}_{k_{n,t}}) + (\hat{X}_{k_{n,t}}- \hat{X}_{\lfloor t\beta _n\rfloor }). \end{aligned}$$

Since by definition of the random sequence \(k_{n,t}\), one has

$$\begin{aligned} \big |X_t^n -\hat{X}_{k_{n,t}}\big |\le 1 + \big |X_{\lfloor tn\rfloor } -\hat{X}_{k_{n,t}}\big |\le 3L_n=o\left( \sqrt{n}\right) , \end{aligned}$$

our claim follows from the following two convergences when \(n\rightarrow \infty \).

  1. (i)

    For each \(u\in {\mathbb {R}}\),

    $$\begin{aligned}\sup _{|x|\le 2n^2}\left| \widetilde{{\text {P}}}_{x,\omega }\left( \left( \hat{X}_{\lfloor t\beta _n\rfloor }-x\right) /\sqrt{n}>u\right) -\text {P}\left( {\mathcal {N}}(0,tD^{-1}\varvec{\Lambda })>u\right) \right| \rightarrow 0. \end{aligned}$$
  2. (ii)

    For each \(\epsilon >0\), \(\sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( \big |\hat{X}_{k_{n,t}}-\hat{X}_{\lfloor t\beta _n\rfloor }\big |/\sqrt{n}> \epsilon \right) \rightarrow 0\).

We first prove (i). For notational simplicity, we restrict ourselves to the case \(t=1\); the general case \(t\in [0,1]\) follows exactly the same lines, with \(\beta _n\) replaced everywhere by \(\lfloor t\beta _n\rfloor \). For later use, it will be helpful to consider here the supremum over x bounded by \(2n^2\) instead of \(n^2\). We let \((Z_{n,i})_{i=0,\dots ,n}\) be an i.i.d. sequence of random vectors distributed according to \(q_n(0,\cdot )\), independently of the RWRE. Since \(|Z_{n,i}|\le 2L_n =o(\sqrt{n})\), it suffices to show the statement for \(\hat{X}_{\beta _n}\) inside the probability replaced by \(\hat{X}_{\beta _n} + Z_{n,0}\) (tacitly assuming that \(\hat{X}_{\beta _n}\) under \(\widetilde{{\text {P}}}_{x,\omega }\) and \(Z_{n,0}\) are defined on the same probability space, whose probability measure we again denote by \(\widetilde{{\text {P}}}_{x,\omega }\)). Now let \(\hat{Y}_{i}= Z_{n,1}+\dots +Z_{n,i}\). Since \(\hat{X}_{\beta _n}+Z_{n,0}\) has law \((Q_n)^{\beta _n}q_n(x,\cdot )\) under \(\widetilde{{\text {P}}}_{x,\omega }\), and \(x + \hat{Y}_{i}\) has law \((q_n)^{i}(x,\cdot )\), we get

$$\begin{aligned}&\sup _{|x|\le 2n^2}\left| \widetilde{{\text {P}}}_{x,\omega }\left( \left( \hat{X}_{\beta _n}+Z_{n,0}-x\right) /\sqrt{n}>u\right) - \text {P}\left( (x + \hat{Y}_{\beta _n +1}-x)/\sqrt{n}>u\right) \right| \\&\quad \le \sup _{|x|\le 2n^2}\left\| \left( (Q_n)^{\beta _n}-(q_n)^{\beta _n}\right) q_n(x,\cdot )\right\| _1. \end{aligned}$$

For \(\omega \in A_1\), we obtain by iteration, uniformly in x with \(|x|\le 2n^2\),

$$\begin{aligned}&\left\| \left( (Q_n)^{\beta _n}-(q_n)^{\beta _n}\right) q_n(x,\cdot )\right\| _1\\&\quad \le \left\| (Q_n)^{\beta _n-1}\left( Q_n-q_n\right) q_n(x,\cdot )\right\| _1 +\left\| \left( (Q_n)^{\beta _n-1}-(q_n)^{\beta _n-1}\right) q_n^2(x,\cdot )\right\| _1\\&\quad \le \sup _{|x|\le 3n^2}\left\| \left( Q_n-q_n\right) q_n(x,\cdot )\right\| _1 + \sup _{|x|\le 2n^2}\left\| \left( (Q_n)^{\beta _n-1}-(q_n)^{\beta _n-1}\right) q_n(x,\cdot )\right\| _1\\&\quad \le \beta _n(\log L_n)^{-9}\rightarrow 0\quad \hbox {as }n\rightarrow \infty . \end{aligned}$$

It remains to show that \(\hat{Y}_{\beta _n}/\sqrt{n}\) converges in distribution to a d-dimensional centered Gaussian vector with covariance matrix \(D^{-1}\varvec{\Lambda }\). This will be a consequence of the following multidimensional version of the Lindeberg-Feller theorem.

Proposition 6

Let \(W_{m,\ell }\), \(1\le \ell \le m\), be centered and independent \({\mathbb {R}}^d\)-valued random vectors. Put \(\varvec{\Sigma }_{m,\ell } = (\sigma _{m,\ell }^{(ij)})_{i,j=1,\dots , d}\), where \({\sigma }_{m,\ell }^{(ij)}= \text {E}\left[ W_{m,\ell }^{(i)}W_{m,\ell }^{(j)}\right] \) and \(W_{m,\ell }^{(i)}\) is the ith component of \(W_{m,\ell }\). If for \(m\rightarrow \infty \),

  1. (a)

    \(\sum _{\ell =1}^m\varvec{\Sigma }_{m,\ell }\rightarrow \varvec{\Sigma }\),

  2. (b)

    for each \(\mathbf{v}\in {\mathbb {R}}^d\) and each \(\epsilon > 0\), \(\sum _{\ell =1}^m\text {E}\left[ |\mathbf{v}\cdot W_{m,\ell }|^2;\, \left| \mathbf{v}\cdot W_{m,\ell }\right| > \epsilon \right] \rightarrow 0\),

then \(W_{m,1}+\dots + W_{m,m}\) converges in distribution as \(m\rightarrow \infty \) to a d-dimensional Gaussian random vector with mean zero and covariance matrix \(\varvec{\Sigma }\).

Proof (of Proposition 6)

By the Cramér–Wold device, it suffices to show that for fixed \(\mathbf{v}\in {\mathbb {R}}^d\), \(\mathbf{v}\cdot (W_{m,1}+\dots +W_{m,m})\) converges in distribution to a Gaussian random variable with mean zero and variance \(\mathbf{v}^T\varvec{\Sigma }\mathbf{v}\). Under (a) and (b), this follows immediately from the classical one-dimensional Lindeberg-Feller theorem. \(\square \)

We now finish the proof of (i). Recall that \(\hat{Y}_{\beta _n}= Z_{n,1}+\dots +Z_{n,\beta _n}\), where the \(Z_{n,\ell }\) are independent random vectors with law \(q_n(0,\cdot )\). Since the underlying one-step transition kernel \(p_{L_n}\) is symmetric, the \(Z_{n,\ell }\) are centered. Moreover, denoting by \(Z_{n,\ell }^{(i)}\) the ith component of \(Z_{n,\ell }\),

$$\begin{aligned} \text {E}\left[ Z_{n,\ell }^{(i)}Z_{n,\ell }^{(j)}\right] =0\quad \hbox {for }i\ne j,\quad i,j=1,\dots ,d. \end{aligned}$$

For \(i=j\), we obtain by definition

$$\begin{aligned}&\frac{1}{n}\left( \text {E}\left[ \left( Z_{n,1}^{(i)}\right) ^2\right] +\dots +\text {E}\left[ \left( Z_{n,\beta _n}^{(i)}\right) ^2\right] \right) \nonumber \\&\quad = \frac{\beta _n}{n}\sum _{y\in {\mathbb {Z}}^d}q_n(0,y)y_i^{2} =\frac{\beta _n}{n}\int _1^2\varphi (s)\sum _{y\in {\mathbb {Z}}^d}\pi _{V_{sL_n}}^{(p_{L_n})}(0,y)y_i^{2}\text {d}s\nonumber \\&\quad =\frac{\beta _n}{n}\int _1^2(L_ns)^2\varphi (s)\sum _{y\in {\mathbb {Z}}^d}\pi _{V_{sL_n}}^{(p_{L_n})}(0,y)(y_i/sL_n)^{2}\text {d}s. \end{aligned}$$
(35)

We next recall that [2, Lemma 3.1] shows how to recover the kernel \(p_{L_n}\) out of the exit measure \(\pi _{V_{sL_n}}^{(p_{L_n})}\), namely

$$\begin{aligned} 2p_{L_n}=\sum _{y\in {\mathbb {Z}}^d}\pi _{V_{sL_n}}^{(p_{L_n})}(0,y)(y_i/sL_n)^{2} + O(L_n^{-1}). \end{aligned}$$

Replacing \(\beta _n\) by its value, we therefore deduce from (35) that

$$\begin{aligned}&\frac{1}{n}\left( \text {E}\left[ \left( Z_{n,1}^{(i)}\right) ^2\right] +\dots +\text {E}\left[ \left( Z_{n,\beta _n}^{(i)}\right) ^2\right] \right) \\&\quad =\frac{L_n^2\beta _nc_\varphi }{n}2p_{L_n}(e_i) + O\left( L_n^{-1}\right) =\frac{2p_{L_n}(e_i)}{D} + O\left( (\log n)^{-2}\right) . \end{aligned}$$

Since \(p_{L_n}(e_i)\rightarrow p_\infty (e_i)\) as \(n\rightarrow \infty \), see (12), we obtain with \( W_{\beta _n,\ell } = Z_{n,\ell }/\sqrt{n}\), \(\ell =1,\dots ,\beta _n\), in the notation of Proposition 6,

$$\begin{aligned} \sum _{\ell =1}^{\beta _n}\varvec{\Sigma }_{\beta _n,\ell }\rightarrow D^{-1}\left( 2p_\infty (e_i)\delta _i(j)\right) _{i,j=1}^d=D^{-1} \varvec{\Lambda }\quad \hbox {as }n\rightarrow \infty . \end{aligned}$$

Since \(W_{\beta _n,\ell }\le 2L_n/\sqrt{n}\le 2(\log n)^{-1}\), point (b) of Proposition 6 is trivially fulfilled. Applying this proposition finally shows that \( \hat{Y}_{\beta _n}/\sqrt{n}=W_{ \beta _n,1}+\dots +W_{\beta _n,\beta _n}\) converges in distribution to a d-dimensional centered Gaussian random vector with covariance matrix \(D^{-1}\varvec{\Lambda }\). This finishes the proof of (i).

It remains to prove (ii). In view of Lemma 15, it suffices to show that for each \(\epsilon >0\),

$$\begin{aligned} \lim _{\theta \downarrow 0}\lim _{n\rightarrow \infty }\sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( \big |\hat{X}_{k_{n,t}}-\hat{X}_{\lfloor t\beta _n\rfloor }\big |>\epsilon \sqrt{n};\,\big |k_{n,t}-\lfloor t\beta _n\rfloor \big |\, < \theta \beta _n\right) = 0. \end{aligned}$$

Fix \(\epsilon >0\), \(\theta >0\). Define the set of integers

$$\begin{aligned} A_{n}=\{\lfloor t\beta _n\rfloor - \lceil \theta \beta _n\rceil ,\dots ,\lfloor t\beta _n\rfloor + \lceil \theta \beta _n\rceil \}, \end{aligned}$$

and let \(\underline{\ell _n}=\lfloor t\beta _n\rfloor - \lceil \theta \beta _n\rceil \). Then

$$\begin{aligned}&\widetilde{{\text {P}}}_{x,\omega }\left( \big |\hat{X}_{k_{n,t}}-\hat{X}_{\lfloor t\beta _n\rfloor }\big |>\epsilon \sqrt{n};\,\big |k_{n,t}-\lfloor t\beta _n\rfloor \big |\, < \theta \beta _n\right) \\&\quad \le \widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \in A_n}\big |\hat{X}_{n,\ell }-\hat{X}_{\lfloor t\beta _n\rfloor }\big |>\epsilon \sqrt{n}\right) \\&\quad \le \widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \in A_n}\big |\hat{X}_{n,\ell }-\hat{X}_{n,\underline{\ell _n}}\big |>(\epsilon /2)\sqrt{n}\right) \\&\qquad + \widetilde{{\text {P}}}_{x,\omega }\left( \big |\hat{X}_{n,\underline{\ell _n}}-\hat{X}_{\lfloor t\beta _n\rfloor }\big |>(\epsilon /2)\sqrt{n}\right) . \end{aligned}$$

We only consider the first probability behind the last inequality; the second one is treated in a similar (but simpler) way. We first remark that after \(\underline{\ell _n}\) steps, the coarse grained RWRE with transition kernel \(Q_n\) starting in \(V_{n^2}\) is still within \(V_{2n^2}\). Therefore, by the Markov property, for x with \(|x|\le n^2\),

$$\begin{aligned}&\widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \in A_n}\big |\hat{X}_{n,\ell }-\hat{X}_{n,\underline{\ell _n}}\big |>(\epsilon /2)\sqrt{n}\right) \nonumber \\&\quad \le \sup _{|y|\le 2n^2} \widetilde{{\text {P}}}_{y,\omega }\left( \max _{\ell \le 2\lceil \theta \beta _n\rceil }\big |\hat{X}_{n,\ell }-y\big |>(\epsilon /2)\sqrt{n}\right) . \end{aligned}$$
(36)

For estimating (36), we follow a strategy similar to Billingsley [6, Theorem 9.1]. Put

$$\begin{aligned} E_\ell =\left\{ \max _{j<\ell }\big |\hat{X}_{n,j}-\hat{X}_{n,0}\big |<(\epsilon /2)\sqrt{n}\le \big |\hat{X}_{n,\ell }-\hat{X}_{n,0}\big |\right\} . \end{aligned}$$

Then

$$\begin{aligned}&\widetilde{{\text {P}}}_{y,\omega }\left( \max _{\ell \le 2\lceil \theta \beta _n\rceil }\big |\hat{X}_{n,\ell }-y\big |> (\epsilon /2)\sqrt{n}\right) \le \widetilde{{\text {P}}}_{y,\omega }\left( \big |\hat{X}_{n,2\lceil \theta \beta _n\rceil }-y\big |\ge (\epsilon /4)\sqrt{n}\right) \\&\quad + \sum _{\ell =1}^{2\lceil \theta \beta _n\rceil -1}\widetilde{{\text {P}}}_{y,\omega }\left( \big |\hat{X}_{n, 2\lceil \theta \beta _n\rceil }-\hat{X}_{n,\ell }\big | \ge (\epsilon /4)\sqrt{n};\,E_\ell \right) . \end{aligned}$$

Concerning the first probability on the right, we already know from (i) that for \(\theta < 1/2\), as \(n\rightarrow \infty \),

$$\begin{aligned} \sup _{|y|\le 2n^2}\widetilde{{\text {P}}}_{y,\omega }\left( \big |\hat{X}_{n,2\lceil \theta \beta _n\rceil }-y\big |\ge (\epsilon /4)\sqrt{n}\right) \rightarrow \text {P}\left( \big |{\mathcal {N}}(0,2\theta D^{-1}\varvec{\Lambda })\big |\ge \epsilon /4\right) . \end{aligned}$$

For fixed \(\epsilon \), the right side converges to zero as \(\theta \downarrow 0\) by Chebycheff’s inequality. For the sum over the probabilities in the above display, we stress that the increments of the coarse grained walk \(\hat{X}_{n,\ell }\) are neither independent nor stationary under \(\widetilde{{\text {P}}}_{y,\omega }\). But we have by the Markov property at time \(\ell \), for \(|y|\le 2n^2\),

$$\begin{aligned}&\sum _{\ell =1}^{2\lceil \theta \beta _n\rceil -1}\widetilde{{\text {P}}}_{y,\omega }\left( \big |\hat{X}_{n, 2\lceil \theta \beta _n\rceil }-\hat{X}_{n,\ell }\big | \ge (\epsilon /4)\sqrt{n};\,E_\ell \right) \nonumber \\&\quad \le \sum _{\ell =1}^{2\lceil \theta \beta _n\rceil -1} \widetilde{{\text {P}}}_{y,\omega }(E_\ell )\sup _{|z|\le 3n^2}\widetilde{{\text {P}}}_{z,\omega }\left( \big |\hat{X}_{n,2\lceil \theta \beta _n\rceil -\ell }-z\big | \ge (\epsilon /4)\sqrt{n}\right) . \end{aligned}$$
(37)

Similar to the proof of (i), we estimate for \(\ell =1,\dots ,2\lceil \theta \beta _n\rceil -1\)

$$\begin{aligned}&\sup _{|z|\le 3n^2}\widetilde{{\text {P}}}_{z,\omega }\left( \big |\hat{X}_{n,\ell }-z \big | \ge (\epsilon /4)\sqrt{n}\right) \le \sup _{|z|\le 3n^2}(Q_n)^{\ell }q_n\left( z,{\mathbb {Z}}^d\backslash V_{(\epsilon /8)\sqrt{n}}(z)\right) \\&\quad \le \sup _{|z|\le 3n^2}\left\| \left( (Q_n)^{\ell }-(q_n)^{\ell }\right) q_n(z,\cdot )\right\| _1 + (q_n)^{\ell +1}\left( 0,{\mathbb {Z}}^d\backslash V_{(\epsilon /8)\sqrt{n}}\right) . \end{aligned}$$

For environments \(\omega \in A_1\), the first summand is estimated by \(\ell (\log L_n)^{-9}\) as in the proof of (i). For the expression involving \(q_n\), we use the following standard large deviation estimate (a proof is for example given in [2, Proof of Lemma A.5]): There exist constants \(C_1\), \(c_1\) depending only on the dimension such that

$$\begin{aligned} (q_n)^{\ell }\left( 0,{\mathbb {Z}}^d\backslash V_{r}\right) \le C_1\exp \left( -c_1r^2/(\ell L_n^2)\right) ,\quad r>0,\,\ell \in {\mathbb {N}}. \end{aligned}$$

In our setting, we obtain

$$\begin{aligned} (q_n)^{\ell }\left( 0,{\mathbb {Z}}^d\backslash V_{(\epsilon /8)\sqrt{n}}\right) \le C \exp \left( -c\epsilon ^2/\theta \right) \quad \hbox {uniformly in }1\le \ell \le 2\lceil \theta \beta _n\rceil . \end{aligned}$$

Back to (37), the fact that the \(E_i\)’s are disjoint leads to

$$\begin{aligned}&\sum _{\ell =1}^{2\lceil \theta \beta _n\rceil -1}\widetilde{{\text {P}}}_{y,\omega }\left( \big |\hat{X}_{n, 2\lceil \theta \beta _n\rceil }-\hat{X}_{n,\ell }\big | \ge (\epsilon /4)\sqrt{n};\,E_\ell \right) \\&\quad \le \beta _n^2(\log L_n)^{-9} + \sum _{\ell =1}^{2\lceil \theta \beta _n\rceil -1}\widetilde{{\text {P}}}_{y,\omega }\left( E_\ell \right) (q_n)^{2\lceil \theta \beta _n\rceil +1-\ell }\left( 0,{\mathbb {Z}}^d\backslash V_{(\epsilon /8)\sqrt{n}}\right) \\&\quad \le o(1) + C\exp \left( -c\epsilon ^2/\theta \right) , \end{aligned}$$

everything uniformly in \(|y|\le 2n^2\). The last expression converges to zero as \(\theta \downarrow 0\). This concludes the proof of (ii) and hence of the one-dimensional convergence. \(\square \)

Convergence of finite-dimensional distributions

In order to prove convergence of the two-dimensional distributions under \({\text {P}}_{0,\omega }\), we have to show that for \(0\le t_1<t_2\le 1\) and \(u_1,u_2\in {\mathbb {R}}\), as \(n\rightarrow \infty \),

$$\begin{aligned}&\Big |{\text {P}}_{0,\omega }\left( X_{t_1}^n/\sqrt{n} > u_1,\,(X_{ t_2}^n-X_{t_1}^n)/\sqrt{n}>u_2\right) \nonumber \\&\quad - \text {P}\left( {\mathcal {N}}(0,t_1D^{-1}\varvec{\Lambda })>u_1\right) \text {P}\left( {\mathcal {N}}(0,(t_2-t_1)D^{-1}\varvec{\Lambda })>u_2\right) \Big |\rightarrow 0. \end{aligned}$$
(38)

This follows easily from the uniform one-dimensional convergence. First, we may replace \(X_{t_1}^n\) by \(X_{\lfloor t_1n\rfloor }\) and \(X_{t_2}^n\) by \(X_{\lfloor t_2n\rfloor }\), since their difference is bounded by one. Then, by the Markov property,

$$\begin{aligned}&{\text {P}}_{0,\omega }\left( X_{\lfloor t_1n\rfloor }/\sqrt{n} > u_1,\,(X_{\lfloor t_2n\rfloor }-X_{\lfloor t_1n\rfloor })/\sqrt{n}>u_2\right) \\&\quad ={\text {E}}_{0,\omega }\left( {\text {P}}_{X_{\lfloor t_1n\rfloor },\omega }\left( \left( X_{\lfloor t_2n\rfloor -\lfloor t_1n\rfloor }-X_0\right) /\sqrt{n}>u_2\right) ;\, X_{\lfloor t_1n\rfloor }/\sqrt{n} > u_1\right) \\&\quad \le {\text {P}}_{0,\omega }\left( X_{\lfloor t_1n\rfloor }/\sqrt{n} > u_1\right) \sup _{|x|\le n}{\text {P}}_{x,\omega }\left( \left( X_{\lfloor t_2n\rfloor -\lfloor t_1n\rfloor }-x\right) /\sqrt{n}>u_2\right) . \end{aligned}$$

The product of the two probabilities converges by Proposition 5 towards

$$\begin{aligned} \text {P}\left( {\mathcal {N}}(0,t_1D^{-1}\varvec{\Lambda })>u_1\right) \text {P}\left( {\mathcal {N}}(0,(t_2-t_1)D^{-1}\varvec{\Lambda })>u_2\right) . \end{aligned}$$
(39)

For the lower bound,

$$\begin{aligned}&{\text {P}}_{0,\omega }\left( X_{\lfloor t_1n\rfloor }/\sqrt{n} > u_1,\,(X_{\lfloor t_2n\rfloor }-X_{\lfloor t_1n\rfloor })/\sqrt{n}>u_2\right) \\&\quad \ge {\text {P}}_{0,\omega }\left( X_{\lfloor t_1n\rfloor }/\sqrt{n} > u_1\right) \inf _{|x|\le n}{\text {P}}_{x,\omega }\left( \left( X_{\lfloor t_2n\rfloor -\lfloor t_1n\rfloor }-x\right) /\sqrt{n}>u_2\right) , \end{aligned}$$

and the right-hand side converges again towards the product in (39). This proves convergence of the two-dimensional distributions under \({\text {P}}_{0,\omega }\). The general case of finite-dimensional convergence is obtained similarly.

Tightness

The sequence of \({\text {P}}_{0,\omega }\)-laws of \((X_t^n/\sqrt{n} : 0\le t\le 1)\) is tight, if the following Condition T holds true.

For each \(\epsilon >0\) there exist a \(\lambda >1\) and an integer \(n_0\) such that, if \(n\ge n_0\),

$$\begin{aligned} {\text {P}}_{0,\omega }\left( \max _{\ell \le n}\big |X_{k+\ell }-X_k\big | \ge \lambda \sqrt{n}\right) \le \frac{\epsilon }{\lambda ^2}\quad \hbox {for all }k\le n\lambda ^2/\epsilon . \end{aligned}$$

See [6, Theorem 8.4] for a proof of this standard criterion.

Let us now show that Condition T is indeed satisfied in our setting. First, by the Markov property at time k,

$$\begin{aligned} {\text {P}}_{0,\omega }\left( \max _{\ell \le n}\big |X_{k+\ell }-X_k\big | \ge \lambda \sqrt{n}\right) \le \sup _{|x|\le k} {\text {P}}_{x,\omega }\left( \max _{\ell \le n}\big |X_{\ell }-x\big | \ge \lambda \sqrt{n}\right) . \end{aligned}$$

The random walk \(X_k\) under \({\text {P}}_{x,\omega }\) has the same law as the first coordinate process on \(({\mathbb {Z}}^d)^{{\mathbb {N}}}\times \varXi \) under \(\widetilde{{\text {P}}}_{x,\omega }\), which we also denote by \(X_k\) (see the beginning of Sect. 5). We shall now consider the latter under \(\widetilde{{\text {P}}}_{x,\omega }\). We recall that \(k_{n,1}=k_{n,1}(\omega )\) counts the number of steps the coarse grained walk performs up to time n. Now we have

$$\begin{aligned}&{\text {P}}_{x,\omega }\left( \max _{\ell \le n}\big |X_{\ell }-x\big |\ge \lambda \sqrt{n}\right) \\&\quad \le \widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \le n}\big |X_{\ell }-x\big |\ge \lambda \sqrt{n};\, k_{n,1}\le 2\beta _n\right) + \widetilde{{\text {P}}}_{x,\omega }\left( k_{n,1}> 2\beta _n\right) . \end{aligned}$$

The second probability on the right converges to zero as n tends to infinity by Lemma 15, uniformly in starting points x with \(|x|\le n^2\). For the first probability, we find on the event \(\{k_{n,1}\le 2\beta _n\}\) for each \(j\le n\) an \(\ell \le 2\beta _n\) such that \(|X_j-\hat{X}_{n,\ell }| \le 2L_n\). We therefore obtain for large n,

$$\begin{aligned}&\widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \le n}\big |X_{\ell }-x\big |\ge \lambda \sqrt{n};\, k_{n,1}\le 2\beta _n\right) \\&\quad \le \widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \le 2\beta _n}\big |\hat{X}_{n,\ell }-x\big |\ge (\lambda /2)\sqrt{n}\right) . \end{aligned}$$

For bounding this last probability, we can follow the same steps as for estimating (36). Leaving out the details, we arrive at

$$\begin{aligned} \sup _{|x|\le n^2}\widetilde{{\text {P}}}_{x,\omega }\left( \max _{\ell \le 2\beta _n}\big |\hat{X}_{n,\ell }-x\big |\ge (\lambda /2)\sqrt{n}\right) \le \frac{C}{\lambda ^3} + C\exp \left( -c\lambda ^2\right) \le \frac{\epsilon }{\lambda ^2}, \end{aligned}$$

provided \(\lambda =\lambda (d,\epsilon )\) is large enough. This proves that Condition T is satisfied. Therefore, the sequence of \({\text {P}}_{0,\omega }\)-laws of \((X_t^n/\sqrt{n} : 0\le t\le 1)\) is tight, which concludes also the proof of Theorem 2.