Einstein relation for random walk in a one-dimensional percolation model
Nina Gantert Matthias Meiners Sebastian Müller





Einstein relation for random walk in a one-dimensional percolation model        





Nina Gantert Fakultät für Mathematik,
		Technische Universität München,
		85748 Garching bei München, Germany.
		gantert@ma.tum.de		
		Matthias Meiners Institut für Mathematik, Universität Innsbruck, 6020 Innsbruck, Austria.
		matthias.meiners@uibk.ac.at
		Sebastian Müller Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France.
		sebastian.muller@univ-amu.fr






Einstein relation for random walk in a one-dimensional percolation model



    
Nina Gantert		Matthias Meiners	Sebastian  Müller 

    Received: December 30, 2023/ Accepted: date
===========================================================================




We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph.
In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0,
then its asymptotic linear speed  is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0.
In the paper at hand, we complement this result by proving that  is differentiable at λ = 0.
Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0
equals the diffusivity of the unbiased walk.


82B43 60K37




§ INTRODUCTION

We continue our study of regularity properties in <cit.> of a biased random walk on an infinite one-dimensional percolation cluster
introduced by Axelson-Fisk and Häggström <cit.>.
The model was introduced as a tractable model that exhibits similar phenomena as biased random walk on the supercritical percolation model in ^d.
The bias, whose strength is given by some parameter λ>0, favors the walk to move in a pre-specified direction.  

There exists a critical bias ∈ (0,∞) such that for λ∈ (0, ) the walk has positive speed while for λ≥ the speed is zero,
see  Axelson-Fisk and Häggström <cit.>.
The reason for the existence of these two different regimes is that the percolation cluster contains traps (or dead ends) and the walk faces two competing effects.
When the bias becomes larger the time spent in such traps (peninsulas stretching out in the direction of the bias) increases
while the time spent on the backbone (consisting of infinite paths in the direction of the bias) decreases.
Once the bias is sufficiently large the expected time the walk stays in a typical trap is infinite and the speed of the walk becomes zero.

Even though the model may be considered as one of the easiest
non-trivial models for a random walk on a percolation cluster, explicit calculation for the speed =(λ) could not be
carried out. The main result of our previous work <cit.>  is that the speed (for fixed
percolation parameter p) is continuous in λ on (0,∞).
The continuity  of the speed may seem obvious, but to our best knowledge,
it has not been proved for a biased random walk on a percolation cluster, and not even for biased random walk on Galton-Watson trees.
Moreover, we proved in <cit.> that the speed is differentiable in
λ on (0,/2) and we characterized the derivative as the
covariance of a suitable two-dimensional Brownian motion.

This paper studies the regularity of the speed in λ=0.
In particular, we establish the Einstein relation for the model: we prove that  is differentiable at λ = 0
and that the derivative at λ = 0 equals the variance of the scaling limit of the unbiased walk.

The Einstein relation is conjectured to be true in general for reversible motions which behave diffusively.
We refer to Einstein <cit.> for a historical reference and to Spohn <cit.> for further explanations.
A weaker form of the Einstein relation holds indeed true under such general assumptions and goes back to Lebowitz and Rost <cit.>.
However, the Einstein relation in the stronger form as described above was only established (or disproved) in examples.
For instance, Loulakis <cit.> considers a tagged particle in an exclusion process,
Komorowski and Olla <cit.> and Avena, dos Santos and Völlering <cit.> investigate other examples of space-time environments. 

Komorowski and Olla  <cit.> treat a first example of random walks with random conductances on ^d, d≥ 3,
and Gantert, Guo and Nagel <cit.>
establish the Einstein relation for random walks among i.i.d. uniform elliptic random conductances on ^d, d≥ 1.
In  dimension one the Einstein relation can be proved via explicit calculations, see Ferrari et al. <cit.>.
There are only few results for non-reversible situations, see Guo <cit.> and Komorowski and Olla <cit.>.
We want to stress that while the differentiability of the speed might appear as natural  or obvious,
there are examples where the speed is not differentiable, see Faggionato and Salvi <cit.>.

Despite this recent progress not much is known in models with hard traps, e.g. random conductances without uniform ellipticity condition or percolation clusters.
The first result in this direction is  Ben Arous et al. <cit.> that proves the Einstein relation for  certain biased random walks on Galton-Watson trees. 
An additional difficulty in our model is that the traps are not only hard but do also have an influence on the structure of the backbone.
Our paper is the first, to our knowledge, to prove the  Einstein relation for a model with hard traps and dependence of traps and backbone.
Although the structure of the traps is elementary the decoupling of traps and backbone is one of the major difficulties we encountered.

We prove a quenched (joint) functional limit theorem via the corrector method, see Section <ref>,
with additional moment bounds for the distance of the walk from the origin.
The law of the unbiased walk is compared with the law of the biased one using a Girsanov transform.
The difference between these measures is quantified using the above joint limit theorem.
Finally, we use regeneration times that depend on the bias and appropriate double limits to conclude
that the derivative of the speed equals a covariance, see Section <ref>.
It remains then to identify the covariance as the variance of the unbiased walk, see Equation (<ref>).




§ PRELIMINARIES AND MAIN RESULTS
	

In this section we introduce the percolation model.
The reader is referred to Figure <ref> for an illustration.



  
Percolation on the ladder graph.
Let ℒ = (V,E) be the infinite ladder graph
with vertex set V = ×{0,1} and edge set E = {⟨ v,w⟩: v,w ∈ V, |v-w|=1}
where ⟨ v,w⟩ is an unordered pair
and |·| the standard Euclidean norm in ^2.
We also write v ∼ w for ⟨ v,w⟩∈ E, and say that v and w are neighbors.

Axelson-Fisk and Häggström <cit.> introduced
a percolation model on ℒ
that may be called `i. i. d. bond percolation on the ladder graph conditioned on the existence of a bi-infinite path'.
We give a short review of this model.

Let Ω{0,1}^E. We call Ω the configuration space,
its elements ω∈Ω are called configurations.
A path in ℒ is a finite or infinite sequence of distinct edges connecting a finite or infinite sequence of neighboring vertices.
For a given ω∈Ω,
we call a path π in ℒ open if ω(e)=1 for each edge e from π.
If ω∈Ω and v ∈ V,
we denote by _ω(v) the connected component in ω that contains v,
i. e.,

    _ω(v)	= {w ∈ V: there is an open path in ω connecting  v  and  w}.

We write : V → and :V →{0,1} for the projections from V to  and {0,1}, respectively.
Then v = ((v),(v)) for every v ∈ V.
We call (v) and (v) the - and -coordinate of v, respectively.
For N_1, N_2 ∈, let Ω_N_1,N_2 be the set of configurations
in which there exists an open path from some v_1 ∈ V with (v_1)=-N_1 to some v_2 ∈ V with (v_2) = N_2.
Further, let Ω^* ⋂_N_1, N_2 ≥ 0Ω_N_1,N_2
be the set of configurations which have an infinite path connecting -∞ and +∞.

Denote by  the σ-field on Ω generated by the projections p_e: Ω→{0,1},
ω↦ω(e), e ∈ E.
For p ∈ (0,1), let μ_p be the probability distribution on (Ω,)
which makes (ω(e))_e ∈ E an independent family of Bernoulli variables
with μ_p(ω(e)=1)=p for all e ∈ E.
Then μ_p(Ω^*)=0 by the Borel-Cantelli lemma.
Write _p,N_1,N_2(·) μ_p(·∩Ω_N_1,N_2)/μ_p(Ω_N_1,N_2)
for the probability distribution on Ω
that arises from conditioning on the existence of an open path from -coordinate -N_1 to -coordinate N_2.
The measures _p,N_1,N_2(·) converge weakly as N_1,N_2 →∞
as was shown in <cit.>:


The distributions _p,N_1,N_2 converge weakly as N_1,N_2 →∞
to a probability measure ^* on (Ω,) with ^*(Ω^*)=1.


For any ω∈Ω^*, we write = _ω for the ^*-a. s. unique infinite open cluster.
We write 0 (0,0) und define Ω_0{ω∈Ω^*: 0∈} and

    (·) ^*(· | Ω_0).





  
Random walk on the infinite percolation cluster.

Throughout the paper,
we keep p ∈ (0,1) fixed and consider random walks in a percolation environment sampled according to .
The model to be introduced next goes back to Axelson-Fisk and Hägg­ström <cit.>,
who used a different parametrization.

We work on the space V^_0
equipped with the σ-algebra = σ(Y_n: n ∈_0)
where Y_n: V^_0→ V denotes the projection from V^_0 onto the nth coordinate.
Let P_ω,λ be the distribution on V^_0 that makes Y  (Y_n)_n ∈_0
a Markov chain on V with initial position 0
and transition probabilities p_ω,λ(v,w)=P_ω,λ(Y_n+1 = w | Y_n = v)
defined via

    p_ω,λ(v,w)
    =
    e^λ ((w)-(v))/e^λ+1+e^-λ_{ω(⟨ v,w ⟩)=1}   if  v ∼ w,	
    ∑_u ∼ ve^λ ((u)-(v))/e^λ+1+e^-λ_{ω(⟨ u,v ⟩)=0}   if  v=w,	
    
    0																						   otherwise.

We write P^0_ω,λ to emphasize the initial position 0,
and P^v_ω,λ for the distribution of the Markov chain with the same transition probabilities but initial position v ∈ V.
The joint distribution of ω and (Y_n)_n ∈_0 on (Ω×^_0,⊗)
when ω is drawn at random according to a probability measure Q on (Ω,)
is denoted by Q × P^v_ω,λ_Q,λ^v where v is the initial position of the walk.
(Notice that, in slight abuse of notation, we consider Y_n also as a mapping from Ω× V^_0
to V.)
We refer to <cit.> for a formal definition.
We write _λ^v for _,λ^v,
_λ for _λ^0
and ^*_λ for _^*,λ^0.
If the walk starts at v=, we sometimes omit the superscript .
Further, if λ=0, we sometimes omit λ as a subscript,
and write p_ω for p_ω,0, and  for _0.




  
The speed of the random walk.

Axelson-Fisk and Häggström <cit.>
showed that (Y_n)_n ∈_0 is recurrent under _0 and transient under _λ for λ≠ 0.
Moreover, there is a critical bias ∈ (0,∞) separating the ballistic from the sub-ballistic regime.
More precisely, if one denotes by X_n (Y_n) the projection of Y_n on the -coordinate, the following result holds.

	
For any λ≥ 0, there exists a deterministic constant (λ) = (p,λ) ∈ [0,1] such that

    X_n/n → (λ)	  _λ-a. s. as  n →∞.

Further, there is a critical bias = (p) > 0
(for which an explicit expression is available) such that

    (λ) > 0	 for  0 < λ <    and   (λ) = 0	 for λ = 0  and λ≥.



For λ > 0 this is Theorem 3.2 in <cit.>.
For λ = 0, the sequence of increments (X_n-X_n-1)_n ∈ is ergodic
by Lemma <ref> below.
Birkhoff's ergodic theorem implies
(0) = lim_n →∞X_n/n = [X_1] = 0 -a. s.




  
Functional central limit theorem for the unbiased walk.

In a preceding paper <cit.>,
we have shown that  is differentiable as a function of λ on the interval (0,/2),
and continuous on (0,∞).
In this paper, we show that  is also differentiable at 0 with '(0) = σ^2
where σ^2 is the limiting variance of n^-1/2 X_n under the distribution _0.
This is the Einstein relation for the model.
Clearly, a necessary prerequisite for the Einstein relation
is the central limit theorem for the unbiased walk.

Before we provide the central limit theorem for the unbiased walk,
we introduce some notation.
As usual, for t ≥ 0, we write ⌊ t ⌋ for the largest integer ≤ t.
Then, we define

    B_n (X_⌊ nt⌋/√(n))_0 ≤ t ≤ 1

for each n ∈.
The random function B_n takes values in the Skorokhod space D([0,1])
of right-continuous real-valued functions with existing left limits.
We denote by “⇒” convergence in distribution of random variables in the Skorokhod space D[0,1],
see <cit.> for details.

	
There exists a constant σ = σ(p) ∈ (0,∞) such that

    B_n	⇒σ B
      under  P_ω

for -almost all ω∈Ω where B is a standard Brownian motion.


It is worth mentioning that an annealed invariance principle for (B_n)_n ∈ follows
without much effort from <cit.>.
In principle, we do not require a quenched central limit theorem for the proof of the Einstein relation.
However, we do require a joint central limit theorem for B_n together with a certain martingale M_n,
see Theorem <ref> below.
Therefore, we cannot directly apply the results from <cit.>.
On the other hand, in the proof of the Einstein relation we use precise bounds on the corrector.
Using similar arguments as Berger and Biskup <cit.>,
these bounds almost immediately give the quenched invariance principle.




  
Einstein relation.

We are now ready to formulate the Einstein relation:
	
The speed  is differentiable at λ=0 with derivative

    '(0) = lim_λ↓ 0(λ)/λ = σ^2

where σ^2 is given by Theorem <ref>.





  
The joint functional central limit theorem.

As in <cit.>, the proof of the differentiability of the speed
is based on a joint central limit theorem for X_n
and the leading term of a suitable density.

To make this precise, we first introduce some notation.
For v ∈ V, let N_ω(v) {w ∈ V: p_ω,0(v,w) > 0}.
Thus, N_ω(v) ≠∅, even for isolated vertices.
For w ∈ N_ω(v), the function log p_ω,λ(v,w) is differentiable at λ=0.
Hence, we can write a first-order Taylor expansion of log p_ω,λ(v,w) around λ = 0 in the form

    log p_ω,λ(v,w)
    = log p_ω(v,w)
    + λν_ω(v,w)+ λ o(λ)

where ν_ω(v,w) is the derivative of  log p_ω,λ(v,w) at 0
and o(λ) converges to 0 as λ→0.
Since there is only a finite number of 1-step transition probabilities,
o(λ) → 0 as λ→ 0 uniformly (in v, w and ω).

For all v and all ω, p_ω,λ(v,·) is a probability measure
on N_ω(v) and hence

    ∑_w∈ N_ω(v)ν_ω(v,w) p_ω(v,w) = 0.

Therefore, for fixed ω,
the sequence (M_n)_n ≥ 0 where M_0=0 and,
for n ∈,

    M_n=∑_k=1^nν_ω(Y_k-1,Y_k)

is a P_ω-martingale with respect to the canonical filtration of the walk (Y_k)_k ∈_0.
Clearly, M_n is a (measurable) function of ω and (Y_k)_k ∈_0
and thus a random variable on Ω× V^_0.
The sequence (M_n)_n ≥ 0 is also a martingale under the annealed measure .

	
Let p ∈ (0,1). Then, for -almost all ω∈Ω,

    (B_n(t),n^-1/2 M_⌊ nt ⌋)		⇒	(B,M)
      under  P_ω

where (B,M) is a two-dimensional centered Brownian motion with deterministic covariance matrix
Σ = (σ_ij)_i,j=1,2.
Further, it holds that

    σ_12 = σ_21 = [B(1)M(1)] = [B(1)^2] = σ^2.



As the martingale M_n has bounded increments,
the Azuma-Hoeffding inequality <cit.> applies and gives the following
exponential integrability result, see the proof of Proposition 2.7 in <cit.> for details.

	
For every t > 0,

    sup_n ≥ 1_λ[e^t n^-1/2 M_n] < ∞.



We finish this section with an overview of the steps that lead to the proof of Theorem <ref>.

	
  * In Section <ref>, we prove the joint central limit theorem, Theorem <ref>.
		The proof is based on the corrector method, which is a decomposition technique
		in which Y_n is written as a martingale plus a corrector of smaller order.
		The martingale is constructed in Section <ref>.
		Many arguments are based on the method of the environment seen from the point of view of the walker.
	
  * In Lemma <ref>, we prove that
		
    sup_n ∈1/n[max_k=1,…,n X_k^2] < ∞.

		The proof is based on estimates for the almost sure fluctuations of the corrector
		derived in Section <ref>.
	
  * Using the joint central limit theorem and (<ref>), we show
		in Proposition <ref> that, for any α > 0,
		
    lim_λ→ 0,
    λ^2 n →α_λ[X_n]/λ n = [B(1) M(1)].

		Equation (<ref>) is a weak form of the Einstein relation going back to Lebo­witz and Rost <cit.>.
	
  * Finally, we show in Section <ref> that
		
    lim_α→∞lim_λ→ 0,
    λ^2 n →α[(λ)/λ - _λ[X_n]/λ n]
    		 =  0.


Notice that (<ref>) together with (0)=0 implies

    '(0) = lim_α→∞lim_λ→ 0,
    λ^2 n →α_λ[X_n]/λ n = [B(1) M(1)].





§ BACKGROUND ON THE PERCOLATION MODEL
	

In this section we provide some basic results on the percolation model.



  
Ergodicity of the percolation distribution.
To ease notation, we identify V with the additive group ×_2.
For instance, we write (k,1)+(n,1)= (k+n,0) for k,n ∈.
With this notation, for v ∈ V, we define the shift θ^v:V → V, w ↦ w-v.
The shift θ^v canonically extends to a mapping on the edges
and hence to a mapping on the configurations ω∈Ω.
In slight abuse of notation, we denote all these mappings by θ^v.
The mappings θ^v form a commutative group since θ^vθ^w = θ^v+w. 

The next result is contained in the proof of Lemma 5.5 in <cit.>.

	
The probability measure ^* is ergodic w. r. t. the family of shifts θ^v, v ∈ V,
that is, it is invariant under all shifts θ^v and for all shift-invariant sets A ∈, we have ^*(A) ∈{0,1}.





  
Cyclic decomposition.
We introduce a decomposition of the percolation cluster into independent cycles.
A similar decomposition for the given model was first introduced in <cit.>.
If (i,1) is isolated in ω, we call (i,0) a pre-regeneration point.
Cycles begin and end at pre-regeneration points.
These are bottlenecks in the graph which the walk has to visit in order to get past.
Let …, R^pre_-2, R^pre_-1, R^pre_0, R^pre_1, R^pre_2, …
be an enumeration of the pre-regeneration points such that
… < (R^pre_-1) < 0 ≤(R^pre_0) <  (R^pre_1) < … .


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	[densely dotted] (-10.5,1) – (-10,1);
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	(-8,1) – (-7,1);
	(-3,1) – (-2,1);
	(1,1) – (2,1);
	(6,0) – (7,0);
	(9,0) – (10,0);
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Let  be the set of all pre-regeneration points.
Let E^i,≤, E^i, ≥⊆ E
consist of those edges
with both endpoints having -coordinate ≤ i or ≥ i, respectively.
Further, let E^i,< E ∖ E^i, ≥ and E^i,> E ∖ E^i,≤.
We denote the subgraph of ω with vertex set {v ∈ V: a ≤(v) ≤ b}
and edge set {e ∈ E^a,≥∩ E^b,<: ω(e)=1}
by [a,b) and call [a,b) a block (of ω).
The pre-regeneration points split the percolation cluster into blocks


ω_n  [(R_n-1^pre), (R_n^pre)),	  	n ∈.


There are infinitely many pre-regeneration points on both sides of the origin -a. s.
The random walk (Y_n)_n ≥ 0 under 
can be viewed as a random walk among random conductances (ω(e))_e ∈ E (with additional self-loops).
For n ∈, we define C_n to be the effective conductance between R_n-1^pre
and R_n^pre.
To be more precise, consider the nth cycle ω_n as a finite network.
Then the effective resistance between R_n-1^pre
and R_n^pre is well-defined, see <cit.>.
We denote this effective resistance by 1/C_n and the effective conductance by C_n.
We further define L_n  L(ω_n) to be the length of the nth cycle,
i. e., L_n = (R_n^pre)-(R_n-1^pre).
We summarize the two definitions:

    C_n 𝒞_eff(R_n-1^pre↔ R_n^pre)
      and  
    L_n = (R_n^pre)-(R_n-1^pre).

For later use, we note the following lemma.

	
The family {(C_n,L_n): n ∈} is independent and the (C_n,L_n), n ∈∖{0},
are identically distributed. Further, there is some ϑ > 0 such that

    [exp(ϑ/C_n)] + [exp(ϑ L_n)] < ∞

for all n ∈.


By Lemma 3.3 in <cit.>, under ,
the family (θ^R_n-1^preω_n)_n ∈
is independent and all cycles except cycle 0 have the same distribution.
Hence the family ((C_n,L_n))_n ∈ is independent and the (C_n,L_n), n ∈, n ≠ 0
are identically distributed.
Lemma 3.3(b) in <cit.> gives that
L_1 = (R_1^pre)-(R_0^pre) has a finite exponential moment of some order ϑ' > 0.
By Raleigh's monotonicity law <cit.>, C_1^-1,
the effective resistance between R_1^pre and R_0^pre,
increases if open edges between these two points are closed.
So, the effective resistance between R_1^pre and R_0^pre
is bounded above by the effective resistance of the longest self-avoiding open path
connecting these two points.
This path has length at most 2L_1
and thus, by the series law, resistance of at most 2L_1.
Therefore, C_1^-1 has a finite exponential moment of order ϑ'/2.

The proof of the statements concerning the cycle θ^R_-1^preω_0
can be accomplished analogously, but requires revisiting the proof of Lemma 3.3 in <cit.>.
We omit further details.


We close this section with the definition of the backbone. We call a vertex v forwards-communicating (in ω)
if it is connected to +∞ via an infinite open path that does not visit any vertex u with (u)<(v).
Finally, we define (ω) {v ∈ V: v  is forwards-communicating (in ω)}.





§ THE ENVIRONMENT SEEN FROM THE WALKER AND INPUT FROM ERGODIC THEORY


We define the process  of the environment seen from the particle
(ω(n))_n ∈_0 by ω(n) θ^Y_nω, n ∈_0.
It can be understood as a process under P_ω as well as under _0.
For later use, we shall show that (ω(n))_n ≥ 0 is a reversible, ergodic Markov chain under _0.

	
The sequence (ω(n))_n ∈_0 is a Markov process with state space Ω
under P_ω, ^*_0 and _0, with initial distributions
δ_ω, ^* and , respectively.
In each of these cases, the transition kernel M(ω,') is given by

    M(ω,f)
    =	E_ω[f(θ^Y_1ω)]
    =	1/3∑_v ∼0(_{ω(0,v)=1} f(θ^v ω) + _{ω(0,v)=0} f(ω)),

ω∈Ω, f nonnegative and -measurable.
Moreover,
(ω(n))_n ∈_0 is reversible and ergodic under _0.


The proof of (<ref>) is a standard calculation and can be done along the lines of the corresponding one for random walk in random environment,
see e. g. <cit.>.


















To prove reversibility of (ω(n))_n ∈_0 under _0, notice that,
for every bounded and measurable f: Ω→ and all v ∈ V with v ∼,

    E_p [f(θ^v ω) _{ω(0,v)=1}]
     = 	E_p[f(ω) _{ω(0,-v)=1}].

This can be verified along the lines of the proof of (2.2) in <cit.>.


















Arguing as in the proof of Lemma 2.1 in <cit.>,
(<ref>) implies that

    E_p [f(ω) Mg(ω)]	 = 	E_p [g(ω) Mf(ω)]

for all measurable and bounded f,g: Ω→,
which is the reversibility of (ω(n))_n ∈_0 under _0.


















To prove ergodicity, we argue as in the proof of Lemma 4.3 in <cit.>.
Fix an invariant set A' ⊆Ω, i. e.,

    M(ω,A')	 = 	1	  for -almost all ω∈ A'.

By Corollary 5 on p. 97 in <cit.>,
it is enough to show that (A') ∈{0,1}.
If (A') = 0, there is nothing to show.
Thus assume that (A') > 0.
Since  is concentrated on Ω_0,
we can ignore the part of A' that is outside Ω_0
and can thus assume A' ⊆Ω_0.
From the form of M, we deduce that

    for -almost all ω∈ A'  we have that θ^v ω∈ A'  for all  v ∈(ω,0).

To avoid trouble with exceptional sets of -probability 0,
we define A {ω∈ A': θ^v ω∈ A'  for all  v ∈(ω,0)}.
Since A ⊆ A', it suffices to show (A)=1.
First notice that (A) = (A') > 0 and that

    ω∈ A  and   v ∈(ω,0)  imply θ^v ω∈ A.

Plainly,

    A ⊆ B {ω∈Ω: θ^v ω∈ A  for some  v ∈ V}.

By definition, B is invariant under shifts θ^v, v ∈ V.
Since ^*(B) ≥^*(A) > 0, the ergodicity of ^*
(see Lemma <ref>) implies ^*(B) = 1.
We shall now show that B ∩Ω_0⊆ A up to a set of ^* measure zero.
Once this is shown, we can conclude that ^*(A) = ^*(Ω_0),
in particular, (A)=1.
In order to show B ∩Ω_0⊆ A ^*-a. s.,
pick an arbitrary ω∈ B ∩Ω_0 such that ω has only one infinite connected component _∞.
(The set of ω with this property has measure 1 under ^*.)
By definition of the set B, there exists a v ∈ V such that θ^v ω∈ A.
Since A ⊆Ω_0, the origin 0 must be in the infinite connected component of θ^v ω
or, equivalently, v is in the infinite connected component of ω.
From the uniqueness of the infinite connected component together with ω∈Ω_0,
we thus infer v ∈(ω,0).
This is equivalent to -v ∈(θ^v ω, 0).
Together with θ^v ω∈ A this implies ω = θ^-vθ^v ω∈ A
by means of (<ref>).
The proof is complete.


The lemma has the following useful corollary.

	
Let f: Ω→ be integrable with respect to . Then, for -almost all ω∈Ω,
we have

    lim_n →∞1/n∑_k=0^n-1 f(ω(k)) = [f]	  	P_ω-a. s.



As (ω(n))_n ∈_0 is reversible and ergodic with respect to _0,
we infer

    lim_n →∞1/n∑_k=0^n-1 f(ω(k)) = _0[f(ω(0))]	  _0 -a. s.

from Birkhoff's ergodic theorem.
As the law of ω(0) under _0 is , we have _0[f(ω(0))] = [f].
Hence (<ref>) follows from (<ref>) and the definition of _0.


Next we see that if the walker sees the same environment at two different epochs,
then, with probability 1, the position of the walker at those two epochs is actually the same.
This allows to reconstruct the random walk from the environment seen from the walker.

	
We have ^*(θ^v ω = θ^wω) = 0
for all v, w ∈ V, v ≠ w.
The same statement holds with ^* replaced by .


By shift invariance, we may assume w=0 and v ≠0.
Call a vertex u ∈ V backwards-communicating (in ω)
if there exists an infinite open path in ω which contains u but no vertex
with strictly larger -coordinate.
Define T = ( T_i)_i ∈ by letting

    T_i	   00   if neither (i,0) nor (i,1) are backwards-communicating;
     01   if (i,0) is not backwards-communicating but (i,1) is;
     10   if (i,0) is backwards-communicating but (i,1) is not;
     11   if (i,0) and (i,1) are backwards-communicating.

Notice that θ^v ω = ω implies θ^v  T =  T where θ^v  T_i records
whether the vertices (i,0) and (i,1) are backwards-communicating in θ^v ω.
Under ^*, T is a time-homogeneous, stationary, irreducible and aperiodic Markov chain
with state space { 01,  10,  11}
by <cit.>.
From this one can deduce ^*(θ^v  T =  T) = 0 and, in particular, ^*(θ^v ω = ω) = 0.
Finally, notice that, for every event A ∈, ^*(A) = (A) whenever ^*(A) ∈{0,1}.


	
The increment sequences (X_n-X_n-1)_n ∈ and (Y_n-Y_n-1)_n ∈ are ergodic under .


Define a mapping φ: Ω×Ω→ V via

    φ(ω,ω') =
    
    v			   if ω' = θ^v ω for a unique  v ∈ V;	
    0   otherwise.

It can be checked that φ is product-measurable.
Further, -a. s.,
Y_n - Y_n-1 = φ(ω(n-1),ω(n))
for all n ∈.
Combining the ergodicity of (ω(n))_n ∈_0
under _0 (see Lemma <ref>)
with Lemma 5.6(c) in <cit.>,
we infer that (Y_n-Y_n-1)_n ∈ = (φ(ω(n-1),ω(n)))_n ∈
is ergodic.
(To formally apply the lemma, one may extend (ω(n))_n ∈_0
to a stationary ergodic sequence (ω(n))_n ∈ using reversibility.)
Then also (X_n-X_n-1)_n ∈ = ((Y_n-Y_n-1))_n ∈ is ergodic under 
again by Lemma 5.6(c) in <cit.>.





§ PRELIMINARY RESULTS
	



  
Hitting probabilities.

The next lemma provides bounds on hitting probabilities for biased random walk
that we use later on.

	
Let L,R ∈ and u,v,w ∈ be such that (v)-(u)=L⌊1/λ⌋
and (w)-(v)=R ⌊1/λ⌋. Then,
with T_u and T_w denoting the first hitting times of (Y_n)_n ∈_0 at u and w, respectively,
we have

    1-e^-R/1-e^-R+  6 (e^2L-1)≤ P_ω,λ^v(T_u< T_w) ≤1-e^-2R/1-e^-2R+ 1/5 (e^L-1)

for all sufficiently small λ∈ (0,λ_0] for some λ_0 > 0 not depending on L,R.
In particular, for these λ,

    1/6e^2L-5≤ P_ω,λ^v(T_u < ∞) ≤5/4 + e^L.

Moreover, for L=1 and R=∞, for all sufficiently small λ > 0, we have

    P_ω,λ^v(T_u < ∞) ≤4/10.



Since u,w are pre-regeneration points,
it suffices to consider the finite subgraph [u,w).
As v is also a pre-regeneration point,
standard electrical network theory gives

    P_ω,λ^v(T_u< T_w) = (v ↔ w)/(u ↔ v)+(v ↔ w)

where (u ↔ v) = ℛ_eff,λ(u ↔ v) denotes the effective resistance between u and v in [u,w),
see <cit.>.
Let us first prove the upper bound by applying Raleigh's monotonicity law <cit.>.
We add all edges left of v that were not present in the cluster ω.
This decreases the effective resistance (u ↔ v) between u and v.
On the right of v we delete all edges but a simple path from v to w.
This increases the effective resistance (v ↔ w). 

The graph obtained is denoted by G, see Figure <ref> for an example.
We conclude

    P_ω,λ^v(T_u< T_w) ≤ℛ_eff,G(v ↔ w)/ℛ_eff,G(u ↔ v)+ℛ_eff,G(v ↔ w)

where ℛ_eff,G denote the corresponding effective resistances in G.
We may assume without loss of generality that v = 0.
Then, by the series law, we can bound ℛ_eff,G(v ↔ w) from above by

    ℛ_eff,G(v ↔ w) ≤∑_k=1^2R ⌊1/λ⌋ e^-λ k
    = e^-λ/1-e^-λ (1-e^-λ 2R ⌊1/λ⌋)
    ≤1-e^-2R/1-e^-λ.

From the Nash-Williams inequality <cit.>, we infer

    ℛ_eff,G(u ↔ v) ≥∑_k=1^L ⌊1/λ⌋( 2 e^-λ (2k-1))^-1 
    = e^λ/2e^2λ L ⌊1/λ⌋-1/e^2λ-1≥1/2e^2(1-λ)L-1/e^2λ-1.

Consequently,

    P_ω,λ^v(T_u< T_w)
       ≤1-e^-2R/1-e^-2R+ 1/2 (e^2(1-λ)L-1) 1-e^-λ/e^2λ-1
       ≤1-e^-2R/1-e^-2R+ 1/5 (e^L-1)

for all sufficiently small λ > 0 independent of L,R. 
The proof of the lower bound is similar.
We add all edges right of v and keep only one simple path left of v.
For this new graph G we bound the effective resistances as follows. 
From the Nash-Williams inequality <cit.>, we infer

    ℛ_eff,G(v ↔ w) ≥∑_k=1^R ⌊1/λ⌋( 2 e^λ (2k-1))^-1
    = e^-λ/21-e^-λ 2R ⌊1/λ⌋/1-e^-2λ≥1/51-e^-R/1-e^-λ

for all sufficiently small λ > 0. Moreover,

    ℛ_eff,G(u ↔ v) ≤∑_k=1^2L ⌊1/λ⌋ e^λ k
    = e^λ/e^λ-1 (e^λ 2L ⌊1/λ⌋-1)
    ≤e^λ/e^λ-1 (e^2L-1).

The lower bound in (<ref>) now follows.
Equation (<ref>) follows from (<ref>) by letting R →∞.
Equation (<ref>) follows from (<ref>)
and the observation that the term on the right-hand side of (<ref>) with R = ∞ and L=1
tends to 4/3+e^2=0.3850… for λ→ 0.





§ HARMONIC FUNCTIONS AND THE CORRECTOR
	

We use harmonic functions to construct a martingale approximation for X_n.
As a result, X_n can be written as a martingale plus a corrector.



  
The corrector method.
The corrector method has been used in various setups,
see e.g. <cit.> and <cit.>.
In the present setup, the method works as follows.

We seek a function ψ: Ω× V → such that, for each fixed ω,
ψ(ω,·) is harmonic in the second argument with respect to
the transition kernel of P_ω, that is,
E^v_ω [ψ(ω,Y_1)] = ψ(ω,v) for all v ∈ V.
In what follows, we shall sometimes suppress the dependence of ψ on ω in the notation so that the above condition becomes

    E^v_ω [ψ(Y_1)] = ψ(v),	  	v ∈ V.

If we find such a function, then (ψ(Y_n))_n ∈_0 is a martingale under P_ω.
We can then define χ(ω,v) (v) - ψ(ω,v) and get that
X_n = ψ(Y_n) + χ(Y_n). In other words,
X_n can be written as the nth term in a martingale, ψ(Y_n),
plus a corrector, χ(Y_n).
In order to derive a central limit theorem for X_n,
it then suffices to apply the martingale central limit theorem to ψ(Y_n)
and to show that the contribution of χ(Y_n) is asymptotically negligible.



  
Construction of a harmonic function.
Let ω∈Ω_0 be such that there are infinitely many pre-regeneration points
to the left and to the right of the origin (the set of these ω has -probability 1).
Then, under P_ω, the walk (Y_n)_n ≥ 0
is the simple random walk on the unique infinite cluster _ω.
It can also be considered as a random walk among random conductances where
each edge e ∈ E has conductance ω(e).
Recall that C_n denotes the effective conductance between R_n-1^pre
and R_n^pre, see (<ref>).
We couple our model with the random conductance model on  with conductance C_n
between n-1 and n.
For the latter model, the harmonic functions are known.
In fact,

    Ψ(n) 
    -∑_k=n+1^01/C_k   for  n<0,	
    
    0									   for  n=0,	
    ∑_k=1^n1/C_k   for  n>0,

is harmonic for the random conductance model on .
We define ϕ(R_n^pre) Ψ(n).
Now fix an arbitrary n ∈.
For any vertex v ∈ω_n,
we then set ϕ(v) ϕ(R_n-1^pre) + (R_n-1^pre↔ v),
where the latter expression denotes the effective resistance between R_n-1^pre and v in the finite network ω_n.
This definition is consistent with the cases v = R_n-1^pre, R_n^pre,
and, by <cit.>, makes ϕ harmonic on ω_n.
As n ∈ was arbitrary, ϕ is harmonic on _ω.
We now define

    ψ(ω,v)	[L_1]/[C_1^-1] (ϕ(ω,v) - ϕ(ω,0)),	  	v ∈_ω,

where we remind the reader that the L_n, n ∈ were defined in (<ref>).
Notice that the expectations [L_1] and [C_1^-1] are finite by Lemma <ref>.
Since ϕ(ω,·) is harmonic under P_ω, so is ψ(ω,·)
as an affine transformation of ϕ(ω,·).
It turns out that ψ is more suitable for our purposes as ψ is additive in a certain sense.
Next, we collect some facts about ψ.

	
Consider the function ψ: Ω× V → constructed above.
The following assertions hold:

	
  * For -almost all ω∈Ω_0,
		the function v ↦ψ(ω,v) is harmonic with respect to (the transition probabilities of) P_ω.
	
  * For -almost all ω∈Ω_0 and all u,v ∈_ω,
		it holds that
		
    ψ(ω,u+v) = ψ(ω,u) + ψ(θ^u ω,v).

	
  * For -almost all ω∈Ω_0
		
    sup_v ∼ w |ψ(ω,v)-ψ(ω,w)| _{v ∈_ω}_{ω(⟨ v,w ⟩)=1}≤[L_1]/[C_1^-1].




Assertion (a) is clear from the construction of ψ.
For the proof of (b), in order to ease notation,
we drop the factor [L_1]/[C_1^-1] in the definition of ψ.
This is no problem as (<ref>) remains true after multiplication by a constant.
Now fix ω∈Ω_0 such that there are infinitely many pre-regeneration points
to the left and to the right of 0. The set of these ω has -probability 1.
Let u,v ∈_ω. 
We suppose that there are n,m ∈_0 such that u ∈ω_n and v ∈ω_n+m.
The other cases can be treated similarly.
We further assume that (u)=0.
Define T inf{n ∈_0: Y_n ∈} to be the first hitting time of the set of pre-regeneration points.
From Proposition 9.1 in <cit.>, we infer

    ψ(ω,u)
       =
    ∑_k=1^n-11C_k(ω) + P_ω^u(Y_T = R_n^pre) 1C_n(ω) - ϕ(ω,0)	
    and ψ(θ^uω,v)
       =
    ∑_k=n+1^n+m-11C_k(ω) + P_θ^u ω^v(Y_T = R_m^pre) 1C_n+m(ω) - ϕ(θ^u ω,0)

where we have used that (u)=0 which implies that the pre-regeneration points in θ^u ω
are the pre-regeneration points in ω but shifted by index n as u ∈ω_n.
Here,

    P_θ^u ω^v(Y_T = R_m^pre)
    =	
    P_ω^u+v(Y_T = R_n+m^pre)	
    and  	
    - ϕ(θ^u ω,0) =	
    P_θ^u ω^0(Y_T=R_-1^pre) 1/C_n = P_ω^u (Y_T=R_n-1^pre) 1/C_n.

Using the last two equations in (<ref>)
and summing over (<ref>) and (<ref>) gives:

    ψ(ω,u) + ψ(θ^u ω,v)
       =	∑_k=1^n+m-11/C_k + P_ω^u+v(Y_T = R_n+m^pre) 1/C_n+m - ϕ(ω,0) 
       =	ψ(ω,u+v).

The proof in the case (u)=1 is similar but requires more cumbersome calculations as the pre-regenerations change when considering θ^u ω
instead of ω due to the flip of the cluster. However, the pre-regeneration points ω remain pivotal edges in θ^u ω
and by the series law the corresponding resistances add. We refrain from providing further details.

We now turn to the proof of assertion (c).
According to the definition of ψ, the statement is equivalent to

    sup_v ∼ w |ϕ(ω,v)-ϕ(ω,w)| _{v ∈_ω}_{ω(⟨ v,w ⟩)=1}≤ 1
       for -almost all ω.

For the proof of (<ref>), pick ω∈Ω_0
such that there are infinitely many pre-regeneration points to the left and to the right of the origin.
Now pick v,w ∈_ω with ω(⟨ v,w ⟩) = 1.
Then there is an n ∈ such that v,w are vertices of ω_n and ⟨ v,w ⟩ is an edge of ω_n.
In this case, by the definition of ϕ,
ϕ(v) ϕ(a) + (a ↔ v)
and ϕ(w) ϕ(a) + (a ↔ w)
for a=R_n-1^pre
where (u ↔ u') denotes the effective resistance between u and u' in the finite network ω_n.
To unburden notation, we assume without loss of generality that ϕ(a)=0.
Then (·↔·) is a metric on the vertex set of ω_n, see <cit.>.
In particular, (·↔·) satisfies the triangle inequality.
This gives

    ϕ(w) = (a↔w) ≤(a↔v) + (v↔w) ≤(a↔v) + 1 = ϕ(v)+1,

where we have used that (v ↔ w) ≤ 1.
This inequality follows from Raleigh's monotonicity principle <cit.>
when closing all edges in ω_n except ⟨ v,w ⟩.
By symmetry, we also get ϕ(v) ≤ϕ(w)+1 and, hence, |ϕ(v)-ϕ(w)| ≤ 1.



For v ∈ V (and fixed ω), we define χ(v) (v) - ψ(v).
Then X_n = χ(Y_n) + ψ(Y_n).
For the proof of the Einstein relation,
we require strong bounds on the corrector χ.
These bounds are established in the following lemma.

	
For any ε∈ (0,1/2) and every sufficiently small δ > 0 there is a random variable K on Ω with
[K^2]<∞ such that

    |χ(ω,v)| ≤ K(ω) + ε |(v)|^1/2+δ   for all  v ∈_ω -almost surely.

Further, there is a random variable D ∈ L^2() such that

    |χ(Y_k)| ≤ D + k^1/4+δ  -almost surely for all  k ∈.



For k ∈, set η_k  L_k - [L_1]/[C_1^-1]1/C_k.
Then, for n ∈,

    χ(R^pre_n)
       =	(R^pre_n) - ψ(R^pre_n)
    = ∑_k=1^n (L_k - [L_1]/[C_1^-1]1/C_k) + [L_1]/[C_1^-1]ϕ(ω,0)	
       ∑_k=1^n η_k + [L_1]/[C_1^-1]ϕ(ω,0)

where η_1,…,η_n are i.i.d. centered random variables.
Here [e^ϑ |η_1|] < ∞ for some ϑ > 0 by Lemma <ref>.
From (<ref>), the fact that |ϕ(ω,0)| ≤ 1/C_0 and again
Lemma <ref>, which guarantees that [1/C_0^2]<∞,
we thus infer, for arbitrary given ε∈ (0,1/4) and δ∈ (0,1/2),

    |χ(R^pre_n)|
    ≤  |η_1+…+η_n| + [L_1]/[C_1^-1]1/C_0≤ K_2 + ε n^1/2+δ

for all n ∈ and a random variable K_2 on (Ω,) satisfying [K_2^2]<∞.
If v ∈ω_n for some n > 0,
then

    |χ(R^pre_n) - χ(v)| ≤ L_n + [L_1]/[C_1^-1]1/C_nξ_n.

The ξ_n, n ∈ are nonnegative and i.i.d. under .
Hence, (<ref>) gives

    ξ_n ≤ K_1 + ε n^1/2+δ

for all n ∈, where K_1 is a nonnegative random variable on (Ω,)
with [K_1^2] < ∞.
Analogous arguments apply when v ∈ω_n with n ≤ 0.
Combining (<ref>) and (<ref>),
we infer that for sufficiently small δ > 0
there is a random variable K ≥ 1 on (Ω,) with [K^2]<∞
such that

    |χ(ω,v)| ≤ K(ω) + 2 ε |(v)|^1/2+δ

for all v ∈_ω, i.e., (<ref>) holds.

Finally, we turn to the proof of (<ref>).
We use (<ref>) with ε = 1/2 twice
and |(Y_k)| ≤ k to infer
for some sufficiently small δ∈ (0,1/2),

    |χ(Y_k)|
       ≤ K(ω) + 12 |X_k|^1/2+δ = K(ω) + 12 |ψ(Y_k) + χ(Y_k)|^1/2+δ
       ≤ K(ω) + 12 |ψ(Y_k)|^1/2 + δ + 12 |χ(Y_k)|^1/2+δ
       ≤ K(ω) + 12 |ψ(Y_k)|^1/2 + δ + 12 K(ω)^1/2+δ + 12 k^(1/2+δ)^2

P_ω-a. s. for all k ∈_0.
Now (ψ(ω,Y_k))_k ∈_0 is a martingale with respect to P_ω
and also a -martingale with respect to its canonical filtration.
As it has bounded increments, we may apply Lemma <ref>(d)
to infer the existence of a variable K_2 ∈ L^2() such that
|ψ(ω,Y_k)| ≤ K_2 + k^1/2 + δ for all k ∈_0 -a. s.
Using this in (<ref>), we conclude

    |χ(Y_k)|
       ≤ K(ω) + 12 K_2^1/2 + δ + 12 k^(1/2 + δ)^2 + 12 K(ω)^1/2+δ + 12 k^(1/2+δ)^2
       ≤ K(ω) + 12 K(ω)^1/2+δ + K_2^1/2 + δ + k^(1/2 + δ)^2

-a. s. for all k ∈_0.
The assertion now follows with
D(ω,(v_k)_k ∈_0)  K(ω) + 12 K(ω)^1/2+δ + K_2(ω,(v_k)_k ∈_0)^1/2 + δ
and with the observation that (1/2 + δ)^2 →1/4 as δ↓ 0.






§ QUENCHED JOINT FUNCTIONAL LIMIT THEOREM VIA THE CORRECTOR METHOD
	

From Proposition <ref> and the martingale central limit theorem,
we infer the following result.

	
For -almost all ω∈Ω,

    1/√(n) (ψ(Y_⌊ nt⌋),M_⌊ nt ⌋)		⇒	(B,M)
      under  P_ω

where (B,M) is a two-dimensional centered Brownian motion with covariance matrix
Σ = (σ_ij)_i,j=1,2 of the form

    [ σ_11 σ_12; σ_21 σ_22 ]
    =
    [          [ψ(Y_1)^2] [ψ(Y_1) ν_ω(0,Y_1)]; [ψ(Y_1) ν_ω(0,Y_1)]      [ν_ω(0,Y_1)^2] ].



Let α,β∈.
We prove that n^-1/2(αψ(Y_⌊ nt⌋) + β M_⌊ nt ⌋)
converges to a centered Brownian motion in the Skorokhod space D([0,1]).
To this end, we invoke the martingale functional central limit theorem <cit.>.
Let

    ξ_n,k(α,β) α/√(n) (ψ(Y_k)-ψ(Y_k-1)) + β/√(n)ν_ω(Y_k-1,Y_k),

for n ∈ and k=1,…,n.
In order to apply the result, it suffices to check the following two conditions:

    ∑_k=1^⌊ nt ⌋ E_ω[ξ_n,k(α,β)^2 |_k-1]
    → c(α,β) t	  in  P_ω-probability
    ∑_k=1^⌊ nt ⌋ E_ω[ξ_n,k(α,β)^2 _{|ξ_n,k(α,β)| > ε}|_k-1]
    → 0	  in  P_ω-probability

for every ε > 0
where c(α,β) is a suitable constant depending on α,β.
Here, _k σ(Y_0,…,Y_k) ⊂ where Y_0,Y_1,… are considered
as functions V^_0→ V.
To check (<ref>), we first define the functions f,g,h: Ω→,

    f(ω)	    E_ω[ψ(ω,Y_1)^2],	
    
    g(ω)	    E_ω[ψ(ω,Y_1) ν_ω(0,Y_1)],	
    
    h(ω)     E_ω[ν_ω(0,Y_1)^2].

These three functions are finite
and integrable with respect to .
This follows from Proposition <ref>(c) for ψ and from the boundedness of ν_ω(v,w)
as a function of ω,v,w.
From Proposition <ref>(b) (with u=Y_k-1 and v= Y_k-Y_k-1), we infer for every k ∈:

    f(ω(k-1))
       =
    E_ω[(ψ(ω,Y_k)-ψ(ω,Y_k-1))^2 |_k-1],	
    
    g(ω(k-1))
       = 
    E_ω[(ψ(Y_k)-ψ(Y_k-1)) ν_ω(Y_k-1,Y_k) |_k-1],	
    
    h(ω(k-1))
       = 
    E_ω[ν_ω(Y_k-1,Y_k)^2 |_k-1].

Lemma <ref> thus gives

    ∑_k=1^⌊ nt ⌋ E_ω[ξ_n,k(α,β)^2 |_k-1]		
        =		1/n∑_k=1^⌊ nt ⌋(α^2 f(ω(k-1)) + 2αβ g(ω(k-1)) + β^2 h(ω(k-1)))	
        →(α^2 [f] + 2αβ[g] + β^2 [h])t.

This gives (<ref>) with P_ω-almost sure convergence instead
of the weaker convergence in P_ω-probability.
Equation (<ref>) follows by an argument in the same spirit.
We therefore conclude that

    α/√(n)ψ(Y_⌊ nt⌋) + β/√(n) M_⌊ nt ⌋⇒α B(t) + β M(t)

in the Skorokhod space D[0,1] as n →∞.
This implies convergence of the finite-dimensional distributions and, hence,
by the Cramér-Wold device,
we conclude that the finite-dimensional distributions of
n^-1/2(ψ(Y_⌊ nt⌋),M_⌊ nt ⌋)
converge to the finite-dimensional distributions of (B,M).
As the sequences n^-1/2ψ(Y_⌊ nt⌋) und n^-1/2 M_⌊ nt ⌋
are tight in the Skorokhod space D[0,1], so is n^-1/2 (ψ(Y_⌊ nt⌋),M_⌊ nt ⌋),
cf. <cit.>.
This implies (<ref>).
The formula for the covariances follows from (<ref>).


We can now give the proof of Theorem <ref>.


In view of (<ref>) and Theorem 4.1 in <cit.>,
it suffices to check that, for -almost all ω,

    max_k=0,…,n|χ(ω,Y_k)|/√(n)→	0	  in  P_ω-probability.

For a sufficiently small δ∈ (0,1/2),
we conclude from (<ref>),
with K slightly increased if necessary (it is sufficient to replace the original K by 2K + 1),
that

    |χ(ω,Y_k)| ≤ K(ω) + 2ε |ψ(ω,Y_k)|^1/2+δ a. s.

for all k ∈_0.
By Proposition <ref>,
max_k=0,…,n |ψ(Y_k)|/√(n) converges in distribution
to the maximum max_0 ≤ t ≤ 1 |B(t)| of the absolute value of a Brownian motion on the unit interval.
Hence

    max_k=0,…,n |χ(ω,Y_k)|/√(n)≤K(ω)/√(n) + 2ε(max_k=0,…,n |ψ(ω,Y_k)|/√(n))^1/2+δ1/n^1/4-δ/2

which converges to 0 in distribution by Slutsky's theorem, and hence in probability.

It remains to prove (<ref>),
that is, [B(1)M(1)] = [B(1)^2] = σ^2.
Uniform integrability, see (<ref>) and (<ref>) below, implies

    [B(1)M(1)] = lim_n →∞1/n[X_n M_n]
      and  [B(1)^2] = lim_n →∞1/n[X_n^2].

It follows from (<ref>) and (<ref>)
that 1/n[χ(Y_n)^2] → 0.
Thus, from X_n = ψ(Y_n)+χ(Y_n), we deduce

    lim_n →∞1/n[X_n M_n] = lim_n →∞1/n[ψ(Y_n) M_n]
     and lim_n →∞1/n[X_n^2] = lim_n →∞1/n[ψ(Y_n)^2].

It thus remains to show

    lim_n →∞1/n[ψ(Y_n) M_n]	= lim_n →∞1/n[ψ(Y_n)^2].

Since (ψ(Y_n))_n ∈_0 and (M_n)_n ∈_0 are martingales with respect to the same filtration,
their increments at different times are uncorrelated. Consequently,

    1/n[ψ(Y_n) M_n] = 1/n∑_k=1^n [(ψ(Y_k)-ψ(Y_k-1))ν_ω(Y_k-1,Y_k)].

For ω∈Ω and w ∈ N_ω(v) = {w ∈ V: p_ω,0(v,w) > 0},
an elementary calculation yields

    ν_ω(v,w)
    = (w)-(v)											   if 	w ≠ v,	
    ∑_u ∼ v: ω(⟨ u,v ⟩)=1 ((u)-(v))		   if 	w = v.

In particular, ν_ω(Y_k-1,Y_k) = X_k-X_k-1 if Y_k ≠ Y_k-1.
Therefore, we can replace ν_ω(Y_k-1,Y_k) by X_k-X_k-1 in (<ref>)
and infer

    1/n   [ψ(Y_n) M_n]
    = 1/n∑_k=1^n [(ψ(Y_k)-ψ(Y_k-1))(X_k-X_k-1)]	
       = 1/n∑_k=1^n [(ψ(Y_k)-ψ(Y_k-1))(ψ(Y_k)-ψ(Y_k-1)+χ(Y_k)-χ(Y_k-1))]	
       = 1/n[ψ(Y_n)^2] + 1/n∑_k=1^n [(ψ(Y_k)-ψ(Y_k-1))(χ(Y_k)-χ(Y_k-1))].

The second sum vanishes as n →∞ since (ψ(Y_n))_n ∈_0 has bounded increments
and 1/n[|χ(Y_n)|] → 0 as n →∞.





§ THE PROOF OF THE EINSTEIN RELATION




 §.§ Proof of Equation (<ref>).


For the proof of the Einstein relation, we use a combination of the approaches from <cit.>.

	
It holds that

    lim sup_n →∞1/n[max_k=1,…,n X_k^2] < ∞.



By Lemma <ref>,
(ω(n))_n ≥ 0 is a stationary and ergodic sequence under .
This sequence can be extended canonically to a two-sided stationary and ergodic sequence (ω(n))_n ∈
on the underlying space Ω_0^.
The increment sequences (X_n-X_n-1) and (Y_n - Y_n-1)_n ∈
also form stationary and ergodic sequences by Lemma <ref>.
Therefore, we can invoke Theorem 1 in <cit.> and conclude that

    [max_k=1,…,n X_k^2] ≤ 4 n (1+80 δ_n,2)^2

where δ_n,2 = ∑_j=1^n j^-3/2([E_ω[X_j]^2])^1/2.
Here, we have

    E_ω[X_j]^2
    = E_ω[ψ(Y_j) + χ(Y_j)]^2
    = E_ω[χ(Y_j)]^2
    ≤ E_ω[|χ(Y_j)|^2].

There exists a random variable D ∈ L^2()
such that (<ref>) holds for some δ > 0 sufficiently small,
i.e., |χ(Y_j)| ≤ D + j^1/4+δ/2 for all j ∈_0 -a. s.
Hence,

    E_ω[|χ(Y_j)|^2]
       ≤ E_ω[|D + j^1/4+δ/2|^2]
    ≤ E_ω[2 D^2] + 2j^1/2+δ.

Consequently,

    sup_n ≥ 1δ_n,2≤∑_j≥1 j^-3/2(2 [D^2] + 2j^1/2+δ])^1/2
    < ∞

if we pick δ > 0 sufficiently small.
Consequently, (<ref>) follows from (<ref>).





 §.§ Proof of Equation (<ref>).


The first two steps of the proof of Theorem <ref> are completed.
We continue with Step 3, i.e., the proof of (<ref>).
It is based on a second order Taylor expansion for
∑_j=1^n log p_ω,λ(Y_j-1,Y_j) at λ = 0:

    ∑_j=1^n log p_ω,λ(Y_j-1,Y_j)	
        =		λ M_n	 + λ^2/2∑_j=1^n (
    p_ω,0”(Y_j-1,Y_j)/p_ω,0(Y_j-1,Y_j) - ν_ω(Y_j-1,Y_j)^2 )
    + λ^2 ∑_j=1^n o_ω,λ(Y_j-1,Y_j)

where o_ω,λ(v,w) tends to 0 uniformly in ω∈Ω and v,w ∈ V as λ→ 0.
Set

    A_n
    1/2∑_j=1^n(ν_ω(Y_j-1,Y_j)^2 - p_ω”(Y_j-1,Y_j)/p_ω(Y_j-1,Y_j))

where we write p_ω for p_ω,0,
and 

    R_λ,nλ^2 ∑_j=1^n o_ω,λ(Y_j-1,Y_j).

Both, A_n and R_λ,n are random variables on (Ω× V^_0,⊗).

	
Let λ→ 0 and n→∞
such that lim_n →∞λ^2n = α≥ 0.
Then

    λ^2 A_n →α/2[M(1)^2]
      -a. s. and in  L^1

and R_λ,n→ 0 -a. s.


The convergence R_λ,n→ 0 follows from
the fact that o_ω,λ(v,w) tends to 0 uniformly in ω∈Ω and v,w ∈ V as λ→ 0.

For the proof of (<ref>),
notice that A_n-A_n-1 is a function of (ω(n-1),ω(n)).
To make this more transparent, we write

    A_n   -A_n-1 = 1/2(ν_ω(Y_n-1,Y_n)^2 - p_ω”(Y_n-1,Y_n)/p_ω(Y_n-1,Y_n))	
       = 
    1/2(ν_ω(n-1)(0,φ(ω(n-1),ω(n)))^2
    - p_ω(n-1)”(0,φ(ω(n-1),ω(n)))/p_ω(n-1)(0,φ(ω(n-1),ω(n))))

with the function φ from the proof of Lemma <ref>.
Since (ω(n))_n ∈_0 is ergodic under ,
so is (A_n-A_n-1)_n ≥ 1, see e.g. Lemma 5.6(c) in <cit.>.
Birkhoff's ergodic theorem gives

    lim_n →∞1/n A_n
    = 1/2[∑_w∈ N_ω(0)(ν_ω(0,w)^2 - p_ω”(0,w)/p_ω(0,w)) p_ω(0,w)]
      -a. s.
 
Further, for all v and all ω, p_ω(v,·)
is a probability measure on N_ω(v), hence
∑_w∈ N_ω(v) p_ω”(v,w) = 0.
Consequently,

    lim_n →∞1/n A_n
    =
    1/2[∑_w∈ N_ω(0)ν_ω(0,w)^2 p_ω(0,w)]
      -a. s.

On the other hand, by Theorem <ref>, we have

    [M(1)^2]
       =
    lim_n →∞1/n[M_n^2]
    = lim_n →∞1/n∑_k=1^n [ν_ω(Y_k-1,Y_k)^2]	
       =
    [ν_ω(Y_0,Y_1)^2]
    = [∑_w∈ N_ω(0)ν_ω(0,w)^2 p_ω, 0(0,w)],

where the second equality follows from the fact that the increments of square-integrable martingales are uncorrelated
and the third equality follows from the fact that (ν_ω(Y_k-1,Y_k))_k ∈ is an ergodic sequence
under .


	
For any α > 0, it holds that

    <ref>lim_λ→ 0,
    λ^2 n →α_λ[X_n]/λ n
    = [B(1) M(1)].



We follow Lebowitz and Rost <cit.>
and use the (discrete) Girsanov transform introduced in Section <ref>.
Indeed, using (<ref>), we get

    _λ[X_n]
       =
    [X_n
    exp(∑_j=1^nlogp_ω,λ(Y_j-1,Y_j)/p_ω(Y_j-1,Y_j))]	
       =
    [X_n exp(λ M_n - λ^2 A_n + R_λ,n)].

Now divide by λ n ∼√(α n) and use Theorem <ref>, Lemma <ref>,
Slutsky's theorem and the continuous mapping theorem to conclude that

    X_n/λ n
    e^λ M_n - λ^2 A_n + R_λ,n1/√(α) B(1) e^√(α) M(1) - α/2[M(1)^2].

Suppose that along with convergence in distribution, convergence of the first moment holds.
Then we infer

    lim_n →∞_λ[X_n]/λ n   = 1/√(α)[B(1) exp(√(α) M(1) - α/2[M(1)^2])]
    = [B(1) M(1)]

where the last step follows from the integration by parts formula for two-dimensional Gaussian vectors.
It remains to show that the family on the left-hand side of (<ref>) is uniformly integrable.
To this end, use Hölder's inequality to obtain

    sup_λ,n    [|X_n/λ n
    e^λ M_n - λ^2 A_n + R_λ,n|^6/5]	
       ≤sup_λ,n[|X_n/λ n|^2]^3/5sup_λ,n[e^3λ M_n - 3λ^2 A_n
    + 3 R_λ,n]^2/5.

By Lemma <ref>, the first supremum in the last line is finite.
Concerning the finiteness of the second,
notice that λ^2 A_n and R_λ,n are bounded
when λ^2n stays bounded
(see the proof of Lemma <ref> for details),
whereas (<ref>) gives sup_λ,n [e^3λ M_n] < ∞.





 §.§ Regeneration points and times
	

Given λ∈ (0,1],
define λ-dependent pre-regeneration points by:

    R^pre,λ_n= R^pre_n ⌊ 1/λ⌋,	  	n ∈.

The set of λ-pre-regeneration points is denoted by .
The cluster is decomposed into independent pieces ω_n^λ [R^pre,λ_n-1,R^pre,λ_n), n ∈.
The λ-regeneration times are defined as
τ^λ_0ρ^λ_0 0
and, inductively,

    τ_n^λ   inf{k > τ^λ_n-1: Y_k∈,
    Y_j≠ Y_k for all  j < k  and
       
    Y_j ∉ for all  j ≥ k  with  X_j < X_k },	
    
    R^λ_n   
    Y_τ^λ_n

for n ∈.
We further set ρ_n^λ X_τ^λ_n = (R^λ_n).
In words, a λ-regeneration point is a λ-pre-regeneration point R^pre,λ_n
such that the walk after the first visit to R^pre,λ_n never returns to R^pre,λ_n-1,
the λ-pre-regeneration point to the left.
In the context of regeneration-time arguments
it will be useful at some points to work with a different percolation law than  or ^*,
namely, the cycle-stationary percolation law ^∘,
which is defined below.

	
The cycle-stationary percolation law ^∘
is defined to be the unique probability measure on (Ω,) such that the
cycles ω_n, n ∈ are i.i.d. under ^∘
and such that each ω_n has the same law under ^∘
as ω_1 under ^*.
We write ^∘_λ for _^∘,λ^0.



We define

    _̋n^λ  σ(τ_1^λ,…,τ_n^λ, Y_0, Y_1, …, Y_τ_n^λ,
    ω_k^λ: (R^pre,λ_k) ≤ρ_n^λ),

the σ-algebra of the walk up to time τ_n^λ and of the environment up to ρ_n^λ.
The distances between λ-regeneration times are not i.i.d.,
but 1-dependent.

	
For any n ∈ and all measurable sets F ∈_≥ = σ(p_⟨ v,w ⟩: (v) ∧(w) ≥ 0) and G ∈,

    _λ((θ^R^λ_n    Y_τ_n^λ+k)_k ≥ 0∈ G, θ^R_n-1^λω∈ F |_̋n-1^λ)	
         = _λ^∘((Y_k)_k ≥ 0∈ G, ω∈ F | X_k > (R^pre,λ_-1)  for all  k ∈).

In particular, ((τ_n+1^λ-τ_n^λ, ρ_n+1^λ-ρ_n^λ))_n ∈
is a 1-dependent sequence of random variables under _λ.


Since Lemma <ref> is a natural observation and its proof is a rather straightforward
but tedious adaption of the proof of Lemma 4.1 in <cit.>, we omit the details of the proof.

The subsequent lemma provides the key estimate for the distances between λ-regeneration points.

	
There exist finite constants C,ε>0 depending only on p such that, for every sufficiently small λ > 0,

    _λ(ρ_2^λ-ρ_1^λ≥ n)	 ≤  C e^- λε n  for all  n ∈_0.

In particular,

    lim sup_λ→ 0λ^2 _λ[(ρ_1^λ)^2] < ∞,
        lim sup_λ→ 0λ^2 _λ[(ρ_2^λ-ρ_1^λ)^2] < ∞,

and

    lim sup_λ→ 0λ^2 ∑_n ≥ 1 n _λ(ρ_2^λ-ρ_1^λ≥ n)^1/2 < ∞.



For the proof, we require the following lemma.

	
There exist finite constants ε=ε(p)>0, c_*=c_*(p) > 0 such that, for all x ≥ 0,

    ^∘((R^pre_⌊ε x⌋) > x)
    ≤ e^-c_* x.



It follows from Lemma 3.3(b) of <cit.> that there exists a constant c(p) ∈ (0,1)
depending only on p such that

    ^∘((R_1^pre) > m) ≤ c(p)^m

for all m ∈_0. Hence, the moment generating function ϑ↦^∘[e^ϑ(R_1^pre)]
is finite in some open interval containing the origin,
in particular, (R_1^pre) has positive finite mean μ(p) (depending only on p).
Let ε∈ (0,μ(p)^-1).
Then, for some sufficiently small u>0,

    1 > ^∘[e^u (R_1^pre)] e^-u ε^-1 e^-c_* ε^-1.

Fix x ≥ 0.
Since the ω_n, n ∈ are i.i.d. under ^∘,
(R^pre_⌊ε x ⌋) is the sum of ⌊ε x ⌋
i.i.d. random variables each having the same law as (R_1^pre) under ^∘.
Consequently, Markov's inequality gives

    ^∘((R^pre_⌊ε x⌋) > x)
       ≤^∘[e^u (R_1^pre)]^⌊ε x ⌋ e^-ux≤^∘[e^u (R_1^pre)]^ε x e^-ux≤ e^-c_* x.




We first derive (<ref>) and (<ref>)
from (<ref>).
We only prove the second relation of (<ref>).
For λ>0, summation by parts and (<ref>) give

    λ^2 _λ[(ρ_2^λ-ρ_1^λ)^2]
       = λ^2 ∑_n ≥ 0 (2n+1) _λ(ρ_2^λ-ρ_1^λ > n)
    ≤ C λ^2 ∑_n ≥ 0 (2n+1) e^- λε n,

which remains bounded as λ↓ 0. Analogously,

    λ^2    ∑_n ≥ 1 n _λ(ρ_2^λ-ρ_1^λ≥ n)^1/2≤ C^1/2λ^2 ∑_n ≥ 0 n e^-λε n/2,

which again remains bounded as λ→ 0 and thus gives (<ref>).

We now turn to the proof of (<ref>).
By Lemma <ref>
the law of ρ_2^λ-ρ_1^λ under _λ
is the same as the law of ρ_1^λ under _λ^∘
given that (Y_n)_n ≥ 0 never visits R^pre,λ_-1:

    _λ(ρ_2^λ-ρ_1^λ∈·)
    = _λ^∘(ρ_1^λ∈·  |   X_k > R^pre,λ_-1 for all  k ∈).

Let C {X_k > (R^pre,λ_-1)  for all  k ∈}.
In order for C^ to occur, the walk (Y_n)_n ∈_0 must travel at least 1/λ steps to the left on the backbone
as the distance of 0 = R^pre,λ_0 and R^pre,λ_-1
is at least 1/λ.
From Lemma 6.3 in <cit.>, we thus conclude that

    _λ^∘(C^) ≤2(e^2λ-1)/e^λ-11/e^2-1.

As λ→ 0, the bound on the right-hand side tends to 4/(e^2-1) =  0.626….
Hence, for all sufficiently small λ > 0, we have _λ^∘(C^) ≤ 2/3,
and therefore, _λ^∘(C) ≥ 1/3.
Fix such a λ. Then

    _λ^∘(ρ_1^λ∈· | C)
    = _λ^∘(C)^-1_λ^∘(ρ_1^λ∈·, C)
    ≤ 3 _λ^∘(ρ_1^λ∈·, C)
    ≤ 3 _λ^∘(ρ_1^λ∈·)

and it thus remains to bound _λ^∘(ρ_1^λ≥ n) for n ∈_0.

The basic idea is that ρ_1^λ≥ n if
either there are unusually few λ-pre-regeneration points in [0,n]
or the walk (Y_k)_k ∈_0 has to make too many excursions of length at least ⌊1/λ⌋ to the left.
To turn this idea into a rigorous proof, we first observe that for ε = ε(p)>0 from Lemma <ref>,
we have

    _λ^∘(ρ_1^λ≥ n)
    ≤_λ^∘((R^pre,λ_⌊ελ n ⌋) > n)
    + _λ^∘(ρ_1^λ≥(R^pre,λ_⌊ελ n ⌋)).

The first probability on the right-hand side of (<ref>) is bounded by

    _λ^∘((R^pre,λ_⌊ελ n ⌋) > n)
       = ^∘((R^pre,λ_⌊ελ n ⌋) > n)
    = ^∘((R^pre_⌊ελ n⌋·⌊1/λ⌋) > n)		
       ≤^∘((R^pre_⌊ε n⌋) > n) ≤ e^-c_* n

where we have used the elementary inequality ⌊ a ⌋·⌊ b ⌋≤⌊ ab ⌋ for all a,b > 0
and then Lemma <ref>.

We now turn to the second probability on the right-hand side of (<ref>).
Observe that a λ-pre-regeneration point
R^pre,λ_i is a λ-regeneration point
iff after the first visit to it, the random walk (Y_k)_k ≥ 0 never returns to R^pre,λ_i-1.
We define Z_0',Z_1',Z_2',… to be the sequence of indices of the λ-pre-regeneration points visited by (Y_k)_k ≥ 0
in chronological order, i.e., Z_j'=i if the jth visit of (Y_k)_k ≥ 0 to 
is at the point R^pre,λ_i.
We then define Z_0,Z_1,Z_2,… to be the corresponding agile sequence,
that is, each multiple consecutive occurrence of a number in the string
is reduced to a single occurrence.  
For instance, if

    (Z_0',Z_1',Z_2',Z_3',Z_4',Z_5',Z_6',…) = (0,-1,0,0,0,1,2,…),

then

    (Z_0,Z_1,Z_2,Z_3,Z_4,…) = (0,-1,0,1,2,…).

Then R^pre,λ_i is a λ-regeneration point if for the first j ∈ with Z_j = i,
we have Z_k≥ i for all k ≥ j.
Let

    ϱ^* inf{Z_k: k ∈ with  Z_j < Z_k ≤ Z_l  for all  0 ≤ j < k ≤ l}.

Then

    _λ^∘(ρ_1^λ≥(R^pre,λ_⌊ελ n ⌋))
    = _λ^∘(ϱ^* ≥⌊ελ n⌋).

We compare the latter probability with the corresponding probability for a biased nearest-neighbor random walk on 
which at any vertex is more likely to move left than the walk (Z_k)_k ∈_0.
More precisely, we may assume without loss of generality
that on the underlying probability space there exists a  biased nearest-neighbor random walk on 
which we denote by (S_k)_k ∈_0
such that _λ^∘(S_0 = 0)=1 and

    610_λ^∘(S_k+1 = j+1 | S_k = j) = 1 - _λ^∘(S_k+1 = j-1 | S_k = j).

According to (<ref>), we have

    _λ^∘(Z_k+1 = j-1 | Z_k = j) ≤410

for λ > 0 sufficiently small.
This means that we may couple the walks (S_k)_k ∈_0 and (Z_k)_k ∈_0
such that {Z_k-Z_k-1 = -1}⊆{S_k-S_k-1 = -1} for all k ∈.
Define

    ϱinf{S_k: k ∈ with  S_i < S_k ≤ S_j  for all  0 ≤ i < k ≤ j}.

A moment's thought reveals that ϱ≥ϱ^* and hence, for every n ∈_0, by Lemma <ref>,

    _λ^∘(ϱ^* ≥⌊ελ n⌋) ≤_λ^∘(ϱ≥⌊ελ n⌋)
    ≤ C^* e^-c^* ⌊ελ n⌋≤ C^* e^c^* e^-c^* ελ n.

This completes the proof of (<ref>).


	
We have

    lim sup_λ→ 0λ^4_λ[(τ_1^λ)^2]<∞lim sup_λ→ 0λ^4_λ[(τ_2^λ-τ_1^λ)^2]<∞,

and

    lim inf_λ→ 0λ^2_λ[τ_1^λ]>0 lim inf_λ→ 0λ^2_λ[τ_2^λ-τ_1^λ]>0.

As a consequence,

    lim sup_λ→ 0λ^2_λ[τ_1^λ]<∞lim sup_λ→ 0λ^2_λ[τ_2^λ-τ_1^λ]<∞,

and 

    lim sup_λ→ 0_λ[(τ_1^λ)^2]/_λ[τ_1^λ]^2<∞lim sup_λ→ 0_λ[(τ_2^λ-τ_1^λ)^2]/_λ[τ_2^λ-τ_1^λ]^2<∞.



The uniform bounds in (<ref>) follow from (<ref>) and Jensen's inequality.
The bounds (<ref>) follow from (<ref>)  and (<ref>).
Let us first prove (<ref>).
The time spent until the first λ-regeneration is bounded below by the number of visits to the pre-regeneration points
R^pre_k with 0 ≤ k < ⌊ 1/λ⌋^-1,
the pre-regeneration points between 0 and R^pre,λ_1.
Fix such k ∈{0,…,⌊1/λ⌋^-1-1} and write N_k for the number of returns of (Y_n)_n ≥ 0 to R^pre_k.
We shall give a lower bound for E_ω,λ[N_k].
We may assume without loss of generality that R^pre_k=.
Under P_ω,λ, the number of returns of the walk (Y_n)_n ≥ 0 to  is geometric with success probability being the escape probability

    P_ω,λ(Y_n ≠ for all  n ≥ 1) = (,∞)/e^λ+1+e^-λ,

where the identity is standard in electrical network theory, see for instance <cit.>.
Consequently,

    E_ω,λ[N_k] = (1-(,∞)/e^λ+1+e^-λ) / ((,∞)/e^λ+1+e^-λ)
    ≥(,∞)

for all sufficiently small λ > 0. From the Nash-Williams inequality <cit.>,
we infer

    (,∞) ≥∑_k=1^∞(2 e^λ(2k-1))^-1 = e^λ/2·1/1-e^-2λ,

a bound which is independent of ω. Since there are ⌊ 1/λ⌋ such pre-regeneration points to the left of R^pre,λ_1,
we conclude that

    lim inf_λ→ 0λ^2_λ[τ_1^λ]
    ≥lim inf_λ→ 0λ^2⌊1λ⌋e^λ/2·1/1-e^-2λ > 0.

This proves the first part in (<ref>).
The second part is analogous or follows using Lemma <ref>.

Let us turn to (<ref>).
We shall prove the unconditioned case for τ_1^λ;
the conditioned case involving τ_2^λ-τ_1^λ follows similarly.
We again use the decomposition:

    τ_1^2 = (τ_1^λ, + τ_1^λ, traps)^2.

First we treat _λ[(τ_1^traps)^2]. 

In order to control the time spent in traps we first bound the time spent in a fixed trap of finite length.
Unfortunately the upper bound given in Lemma 6.1(b) in <cit.> is too rough. However, we follow the arguments there but only consider κ=4.
Let us  consider a discrete line segment {0,…,m}, m≥ 2, and a nearest-neighbor random walk (S_n)_n ≥ 0
on this set starting at i ∈{0,…,m} with transition probabilities

    (S_k+1=j+1 | S_k=j) =	1-(S_k+1=j-1 | S_k=j) = e^λ/e^-λ+e^λ

for j=1,…,m-1 and

    (S_k+1=1 | S_k=0) = (S_k+1=m-1 | S_k=m) = 1.

For i=0, we are interested in τ_m inf{k ∈: S_k = 0}, the time until the first return of the walk to the origin.
Let (Z_n)_n ≥ 0 be the agile version of (Y_n)_n ≥ 0, i.e., the walk one infers after deleting all entries Y_n for which Y_n = Y_n-1
from the sequence (Y_0,Y_1,…).
The stopping times τ_m will be used to estimate the time the agile walk (Z_n)_n ≥ 0 spends
in a trap of length m given that it steps into it.

Let V_i∑_k=1^τ_m-1_{S_k=i} be the number of visits to the point i before the random walk returns to 0, i=1,…,m.
Then τ_m = 1+∑_i=1^m V_i and, by Jensen's inequality,

    _0 [τ_m^4]
     = _0 [(1+∑_i=1^m V_i)^4]
     ≤ 	(m+1)^3(1 + _0 [∑_i=1^m V_i^4] ).

For i=0,…,m, let

    σ_i inf{k ∈: S_k = i}  and  
    r_i (σ_i < σ_0).

Given S_0=i, when S_1 = i+1, then σ_i < σ_0.
When the walk moves to i-1 in its first step,
it starts afresh there and hits i before 0 with probability P_𝑖-1(σ_i < σ_0).
Determining P_𝑖-1(σ_i < σ_0) is the classical ruin problem, hence

    r_i
     = e^λ/e^-λ+e^λ + e^-λ/e^-λ+e^λ(1- e^2λ-1/1-e^-2λ i e^-2λ i)
       for  i=1,…,m-1;	
    
    1- e^2λ-1/1-e^-2λ m e^-2λ m   for  i=m.

In particular, for i=1,…,m-1, r_i does not depend on m.
Moreover, we have r_1 ≤ r_2 ≤…≤ r_m-1 and r_1 ≤ r_m ≤ r_m-1.
By the strong Markov property, for k ∈,
(V_i = k)
= (σ_i < σ_0)   r_i^k-1 (1-r_i)
and hence

    _0 [V_i^4]
    = ∑_k ≥ 1 k^4(V_i = k)
    ≤ (1-r_m-1) ∑_k ≥ 1 k^4 r_m-1^k-1≤ p_3(r_m-1) ( 1/1-r_m-1)^4,





for some polynomial p_3(x) of degree 3 in x independent of λ.
Letting λ→ 0, we find

    lim_λ→ 0 r_m-1 = 2m-3/2m-2

and hence

    lim sup_λ→ 0_0 [V_i^4] ≤  p_3(2m-3/2m-2) (2m-2)^4.

Hence, using equation (<ref>),

    lim sup_λ→ 0_0 [τ_m^4] ≤lim_λ→ 0( (m+1)^3(1 + _0 [∑_i=1^m V_i^4] )) ≤p̃(m),

for some polynomial p̃.
Let ℓ_1 denote the length of the trap with the trap entrance having the smallest nonnegative -coordinate.
Let ℓ_0 and ℓ_2 be the lengths of the next trap to the left and right, respectively, etc.
The law of ℓ_0 differs from the law of the other ℓ_n but this difference is not significant for our estimates, see Lemma 5.1 in  <cit.>.
We proceed as in  the proof of Lemma 6.2 in <cit.>.
For any ω∈Ω^* and any v on the backbone,
by the same argument that leads to (24) in <cit.>,

    P_ω,λ^v(Y_n ≠ v  for all  n ∈)
    ≥(∑_k=0^∞ e^-λ k)^-1/e^λ+1+e^-λ
    = 1-e^-λ/e^λ+1+e^-λ.

This bound is uniform in the environment ω∈Ω^* but depends on λ. Denote by v_i the entrance of the ith trap.  
By the strong Markov property,
T_i, the time spent in the ith trap, can be decomposed into M i.i.d. excursions into the trap:
T_i = T_i,1+ … + T_i,M. Since v_i is forwards-communicating, (<ref>) implies
that  P_ω,λ(M ≥ n) ≤ (1-)^n, n ∈.
In particular, M is stochastically bounded by a geometric random variable M̃ with success parameter .
Moreover, T_i,1, …, T_i,j are i.i.d. conditional on {M ≥ j}.
We now derive an upper bound for E_ω,λ[T_i,j^4 | M ≥ j]. To this end, we have to take into account the times the walk stays put.
Each time, the agile walk (Z_n)_n ≥ 0 makes a step in the trap,
this step is preceded by an independent geometric number of times the lazy walk stays put.
The success parameter of this geometric random variable depends on the position inside the trap.
However, it is stochastically dominated by a geometric random variable G with
P_0(G ≥ k) = γ_λ^k for γ_λ = (1+e^λ)/(e^λ+1+e^-λ).
Plainly, γ_λ→2/3 as λ→ 0. Consequently, estimate (<ref>) and Jensen's inequality give

    lim sup_λ→ 0E_ω,λ[T_i,j^4 | M ≥ j] ≤lim sup_λ→ 0_0[τ_ℓ_i^4] _0[G^4] ≤p̂(ℓ_i),

where ℓ_i is the length of the ith trap (which is treated as a constant under the expectation _0) and p̂ = _0[G] ·p̃ is again a polynomial.
Moreover, by Jensen's inequality and the strong Markov property,

    E_ω,λ[T_i^4]
       =  ∑_m=1^∞  
    E_ω,λ[(∑_j=1^m T_i,j)^4| M = m ] P_ω,λ(M=m) 
       ≤∑_m=1^∞ m^4  E_ω,λ[T_i,1^4 | M≥ 1] P_ω,λ(M = m) 
       ≤ [M̃^4]  E_ω,λ[T_i,1^4 | M≥ 1]	
       ≤c̃( 1/)^4 E_ω,λ[T_i,1^4 | M≥ 1],

for some constant c̃ independent of ω and λ.
We have

    lim sup_λ→ 0λ( 1/)<∞.

Hence

    lim sup_λ→ 0λ^4_λ[T_i^4 | ℓ_i=m] ≤ p^*(m),

where p^* is a polynomial with coefficients independent of λ. Using Lemma 3.5 in <cit.>, we find

    _λ[T_i^4]
       =	∑_m≥ 1_λ(ℓ_i=m) _λ[T_i^4 | ℓ_i=m]    ≤	c(p) ∑_m≥ 1 m e^-2 m_λ[T_i^4 | ℓ_i=m],

where c(p) is a constant only depending on p.
Due to (<ref>), the dominated convergence theorem applies and gives the following bound:

    lim sup_λ→ 0λ^4_λ[T_i^4] < ∞.

Now let L -min{X_k, k∈} be the absolute value of the leftmost visited -coordinate of the walk.
Since

    _λ[ (τ_1^λ, traps)^2]
    ≤_λ[ (∑_i=-L^-1 T_i + T_0 + ∑_i=1^ρ_1^λ T_i)^ 2]

we first consider

    _λ[ (∑_i=1^ρ_1^λ T_i)^ 2]
       =	_λ[∑_i,j=1^∞  T_i T_j _{ρ_1^λ≥ i ∨ j}]	
       ≤∑_j=1^∞_λ[T_j^2 _{ρ_1^λ≥ j}]
    + 2 ∑_j=1^∞∑_i=1^j-1_λ[T_i T_j_{ρ_1^λ≥ j}].

One application of the Cauchy-Schwarz inequality for the first sum and two applications for the second give

    _λ[ (∑_i=1^ρ_1^λ T_i)^ 2]
       ≤∑_j=1^∞ (_λ[T_j^4])^1/2_λ(ρ_1^λ≥ j)^1/2
        + 2 ∑_j=1^∞∑_i=1^j-1 (_λ[T_i^4] _λ[T_j^4])^1/4_λ(ρ_1^λ≥ j)^1/2
       ≤ (_λ[T_1^4])^1/2( ∑_j=1^∞  C^1/2 e^- λε j/2 + 2 ∑_j=1^∞ j _λ(ρ_1^λ≥ j)^1/2).

With the estimates (<ref>) and (<ref>) we obtain

    lim sup_λ→ 0λ^4_λ[ (∑_i=1^ρ_1^λ T_i)^2]<∞.

The first term in the upper bound in (<ref>) is treated in the same way.
Next, we show that _λ(L ≥ m) decays exponentially fast in m.
Indeed, L ≥ 2m implies that there is an excursion on the backbone to the left of length at least m
or the origin is in a trap that covers the piece [-m,0) and thus has length at least m.
The probability that there is an excursion on the backbone of length at least m is bounded by a constant (independent of λ)
times e^-2λ m by Lemma 6.3 in <cit.>.
The probability that a trap that covers the piece [-m,0) is bounded by a constant (again independent of λ) times e^-2 m
by <cit.> or <cit.>.
We may thus argue as above to conclude that

    lim sup_λ→ 0λ^4_λ[ (∑_i=-L^-1 T_i)^2]<∞.

Regarding the term _λ[T_0^2], we can apply (<ref>).
Controlling the mixed terms in 
(<ref>) using the Cauchy-Schwarz inequality we obtain

    lim sup_λ→ 0λ^4_λ[(τ_1^λ, traps)^2] <∞.

Next we treat the time on the backbone. Since the strategy of proof is the same as for the traps we try to be as brief as possible.
Write N(v) ∑_n ≥ 0_{Y_n=v} for the number of visits of the walk (Y_n)_n ≥ 0 to v ∈ V.
We have

    _λ[ (τ_1^λ, )^2]
       ≤_λ[  (∑_-L ≤(v) ≤ρ_1^λ N(v) _{v ∈})^2]	
       = _λ[  (∑_-L ≤(v) < 0 N(v) _{v ∈} + ∑_0 ≤(v) ≤ρ_1^λ N(v) _{v ∈})^2]

We treat the second moment of the second sum first. Using the Cauchy-Schwarz inequality twice, we infer

    _λ[(∑_0 ≤(v) ≤ρ_1^λ N(v) _{v ∈})^2]	
       =	_λ[∑_(v),(w) ≥ 0 N(v) N(w) _{v,w ∈}_{ρ_1^λ≥(v) ∨(w)}]	
       ≤ 2 _λ[∑_j=0^∞∑_i=0^j ∑_(v)=i, 
    (w)=j N(v) N(w) _{v,w ∈}_{ρ_1^λ≥ j}]
       ≤ 2 ∑_j=0^∞∑_i=0^j ∑_(v)=i, 
    (w)=j(_λ [N(v)^4 _{v ∈}])^1/4(_λ[N(w)^4 _{w ∈}])^1/4_λ(ρ_1^λ≥ j)^1/2.

The number of visits to v ∈ is stochastically dominated by a geometric random variable with success probability , see (<ref>).
Hence

    [N(v)^4 _{v ∈}] ≤c̃( 1/)^4.

Using (<ref>) and (<ref>), we infer

    lim sup_λ→ 0   λ^4 _λ[(∑_0 ≤(v) ≤ρ_1^λ N(v) _{v ∈})^2]	
       ≤ 8 c̃^1/2lim sup_λ→ 0λ^4 ( 1/)^2∑_j=0^∞ (j+1) _λ(ρ_1^λ≥ j)^1/2 < ∞.

We may argue similarly to infer the analogous statement for the first sum in (<ref>).
Hence, using again the Cauchy-Schwarz inequality for the mixed terms in (<ref>), we conclude that

    lim sup_λ→ 0λ^4_λ[ (τ_1^λ, )^2] <∞.

Using the Cauchy-Schwarz inequality for the mixed terms in decomposition (<ref>)
together with (<ref>) and (<ref>), we finally obtain the first statement in (<ref>).
The second statement in  (<ref>) then follows from Lemma <ref>.


The existence of a regeneration structure allows to express the linear speed in terms of regeneration points and times. 
	
Let λ>0. Then

    (λ) =  _λ[ρ_2^λ - ρ_1^λ]/_λ[τ_2^λ-τ_1^λ ].


We omit the proof as it is standard and can be derived as <cit.>,
with references to classical renewal theory replaced by references to renewal theory for 1-dependent variables
as presented in <cit.>.
As a consequence of Lemmas <ref>, <ref> and <ref>, we obtain

    lim sup_λ→ 0(λ)/λ < ∞.





 §.§ Proof of Equation (<ref>).
	

It remains to prove (<ref>), i.e.,

    <ref>lim_α→∞lim_λ→ 0,
    λ^2 n →α[(λ)/λ - _λ[X_n]/λ n]
     =  0.

The proof follows along the lines of Section 5.3 in <cit.>.
In order to keep this paper self-contained, we repeat the corresponding arguments from <cit.> in the present context.

For λ>0, we set

    k(n)	⌊n/_λ[τ^λ_2-τ^λ_1]⌋,	  	n ∈.

Notice that k(n) is deterministic but depends on λ even though this dependence does not figure in the notation.
Analogously, we shall sometimes write τ_n for τ_n^λ and, thereby, suppress the dependence on λ.
For the proof of (<ref>), it is sufficient to show that

    lim_α→∞lim sup_λ→ 0,
    λ^2 n ∼α| _λ[X_τ_k(n)]/λ n - (λ)/λ|
        = 	0	
    and    lim_α→∞lim sup_λ→ 0,
    λ^2 n ∼α| 1/λ n_λ[X_n-X_τ_k(n)]|	    = 	0.

For the proof of (<ref>), notice that

    | _λ[X_τ_k(n)]/λ n - (λ)/λ|
       =
    | 1/λ n_λ[∑_j=1^k(n) (X_τ_j-X_τ_j-1) ] - (λ)/λ|	
       ≤1/λ n_λ[X_τ_1]
    + | 1/λ n (k(n)-1) _λ[X_τ_2-X_τ_1] - (λ)/λ|.

Here, using that λ^2 n ∼α,
we have that 1/λ n_λ[X_τ_1] ∼α^-1λ_λ[X_τ_1].
Thus, the first term in (<ref>) vanishes as first λ→ 0 and n →∞ simultaneously
and then α→∞ by (<ref>).
Turning to the second summand in (<ref>), we first notice that by Lemma <ref>,
we have (λ) = _λ[X_τ_2-X_τ_1] / _λ[τ_2-τ_1] and hence

    |    1/λ n (k(n)-1) _λ[X_τ_2-X_τ_1] - (λ)/λ|	
       = (λ)/λ| _λ[τ_2-τ_1]/n(⌊n/_λ[τ_2-τ_1]⌋-1) - 1|.

This expression vanishes as λ→ 0 since lim sup_λ→ 0(λ)/λ < ∞ by (<ref>).

It finally remains to prove (<ref>).
We begin by proving the analogue of Lemma 5.13 in <cit.>.
While the proof is essentially the same, we have to replace the independence property of the times between two regenerations by the 1-dependence property.
	
For all ε >0,

    _λ(| τ_k- k _λ[τ_2-τ_1]| ≥ε  k _λ[τ_2-τ_1]) → 0  for  k→∞

uniformly in λ∈ (0,λ^*] for some sufficiently small λ^* > 0.


An application of Markov's inequality yields

    _λ   (| τ_k- k _λ[τ_2-τ_1]| ≥ε  k _λ[τ_2-τ_1])	
       ≤1/ε^2 k^2 (_λ[τ_2-τ_1])^2_λ[| τ_k-k_λ[τ_2-τ_1]|^2]	
       = 1/ε^2 k^2_λ[τ_2-τ_1]^2_λ[( τ_1 -_λ[τ_2-τ_1]
    +∑_j=2^k (τ_j-τ_j-1-_λ[τ_2-τ_1]) )^2].

Denote the summands under the square by A_1,…,A_k.
Expanding the square and using the 1-dependence of the A_j and the fact that all A_j but A_1 are centered, we infer

    _λ   (| τ_k- k _λ[τ_2-τ_1]| ≥ε  k _λ[τ_2-τ_1])	
       ≤1/ε^2 k^2_λ[τ_2-τ_1]^2(∑_j=1^k _λ[A_j^2] + ∑_|i-j|=1_λ[A_i A_j] )	
       ≤1/ε^2 k^2_λ[τ_2-τ_1]^2(∑_j=1^k _λ[A_j^2] + 2(_λ[A_1^2] _λ[A_2^2])^1/2
        + 2(k-1) (_λ[A_2^2] _λ[A_3^2 ])^1/2).

The assertion now follows from Lemma <ref>.

Now fix ε > 0 and write

    1/λ n|_λ[X_n-X_τ_k(n)]|
       ≤1/λ n|_λ[(X_n-X_τ_k(n)) _{|τ_k(n)-n| < ε n}]|	
        +1/λ n|_λ[(X_n-X_τ_k(n))_{|τ_k(n)-n| ≥ε n}]|.

Using the Cauchy-Schwarz inequality, the first term on the right-hand side of (<ref>) can be bounded as follows.

    1/λ n|_λ   [(X_n-X_τ_k(n)) _{|τ_k(n)-n| < ε n}]|
    ≤2/λ n_λ[max_|j-n| < ε n |X_j-X_⌊(1-ε)n⌋|]	
       = 2/λ n_0[max_|j-n| < ε n |X_j-X_⌊(1-ε)n⌋| e^λ M_n - λ^2 A_n + R_λ,n]	
       ≤2/λ n_0[max_|j-n| < ε n |X_j-X_⌊(1-ε)n⌋|^2 ]^1/2_0[e^2 λ M_n - 2 λ^2 A_n + 2 R_λ,n]^1/2.

We infer lim sup_λ^2n →α_0[e^2 λ M_n - 2 λ^2 A_n + 2 R_λ,n] < ∞ as in the proof of Proposition <ref>.
Regarding the first factor, we find

    1/λ^2 n^2_0[max_|j-n| < ε n |X_j-X_⌊(1-ε)n⌋|^2 ]
       = 1/λ^2n1/n[ E_ω(⌊(1-ε)n⌋),0[max_0  ≤ j ≤ 2ε n X_j^2 ]]	
       = 2ε/λ^2n1/2ε n_0[max_0  ≤ j ≤ 2ε n X_j^2 ].

This term vanishes as first λ^2 n →α (by Lemma <ref>) and then α→∞.
The second term on the right-hand side of (<ref>) can be bounded using the Cauchy-Schwarz inequality, namely,

    1/λ n   |_λ[(X_n-X_τ_k(n))_{|τ_k(n)-n| ≥ε n}]|	
       ≤1/λ n_λ[(X_n-X_τ_k(n))^2]^1/2·_λ(|τ_k(n)-n| ≥ε n)^1/2
       ≤√(2)/λ^2 n( λ^2 _λ[X_n^2]
    + λ^2 _λ[X_τ_k(n)^2])^1/2·_λ(|τ_k(n)-n| ≥ε n)^1/2.

The first factor stays bounded as λ^2n →α
whereas the second factor tends to 0 as n →∞ and λ^2n →α by Lemma <ref>.
Altogether, this finishes the proof of (<ref>).



The research of M. Meiners was supported by DFG SFB 878 “Geometry, Groups and Actions”
and by short visit grant 5329 from the European Science Foundation (ESF)
for the activity entitled `Random Geometry of Large Interacting Systems and Statistical Physics'.
The research was partly carried out during mutual visits of the authors
at Aix-Marseille Université, Tech­nische Universität Graz, Technische Universität Darmstadt, Universität Innsbruck,
and Tech­nische Universität München.
Grateful acknowledgement is made for hospitality from all five universities.





§ AUXILIARY RESULTS FROM RANDOM WALK THEORY


We use the following result, which may be of interest in its own right.

	
Let ξ_1, ξ_2,… be random variables
on some probability space with underlying probability measure  (and expectation ),
and let S_n ξ_1+…+ξ_n, n ∈_0.

	
  * Let α > 0. If K is an _0-valued random variable
		with [K^α]<∞ and if ξ_1,ξ_2,… are i.i.d. with
		[|ξ_1|^α+1]<∞, then [S_K^α]<∞.
	
  * Assume that ξ_1,ξ_2,… are nonnegative and i.i.d. under 
		with [e^ϑξ_1] < ∞ for some ϑ > 0.
		Then, for every ε > 0 and δ∈ (0,1/2) 
		there is a random variable K_1 with [K_1^2] < ∞ such that, for all n ∈,
		
    ξ_n ≤ K_1 + ε n^1/2+δ a. s.

	
  * Assume that ξ_1,ξ_2,… are i.i.d., centered random variables under 
		with [e^ϑ |ξ_1|] < ∞ for some ϑ > 0.
		Then, for every ε > 0 and δ∈ (0,1/2) 
		there exists a random variable K_2 with [K_2^2] < ∞ such that, for all n ∈,
		
    |S_n| ≤ K_2 + ε n^1/2+δ a. s.

	
  * Assume that (S_n)_n ∈_0 is a martingale and that there is a constant C > 0
		with (|ξ_n| ≤ C) = 1 for all n ∈.
		Then, for every ε > 0 and δ∈ (0,1/4)
		there exists a random variable K_2 with [K_2^2] < ∞
		such that (<ref>) holds.



Assertion (a) follows from <cit.>.

For the proof of (b), fix ε > 0 and δ∈ (0,1/2). Then define

    L max{n ∈_0: n=0  or  n ≥ 1  and ξ_n > ε n^1/2+δ}.

For n ≥ 1, the union bound and Markov's inequality give

    (L ≥ n)
       = (ξ_k > ε k^1/2+δ for some  k ≥ n)	
       ≤∑_k ≥ n(ξ_k ≥ε k^1/2+δ)
    ≤[e^ϑξ_1] ∑_k ≥ n e^-ϑε k^1/2+δ.

Hence, (L ≥ n) decays faster than any negative power of n as n →∞.
With K_1  S_L, we have [K_1^2]<∞ from (a) and, for all n ∈, 

    ξ_n ≤ K_1 + ε n^1/2+δ.


For the proof of assertion (c), we use 
moderate deviation estimates (see e.g. <cit.>).
The cited theorem gives

    lim_n →∞ n^-2δlog(|S_n| ≥ε n^1/2+δ) = -ε^22 [ξ_1].

Hence, for any c ∈ (0,ε^2/2 [ξ_1]), we have

    (|S_n| ≥ε n^1/2+δ) ≤ e^- c n^2δ

for all sufficiently large n. With
L max{n ∈_0: |S_n| ≥ε n^1/2+δ},
we infer for sufficiently large n

    (L ≥ n)
       = (|S_k| ≥ε k^1/2+δ for some  k ≥ n)	
       ≤∑_k ≥ n(|S_k| ≥ε k^1/2+δ)
    ≤∑_k ≥ n e^- c k^2δ.

In particular, L has finite power moments of all orders.
Now define K_2  L ∨ (∑_j=1^L |ξ_j|).
Then [K_2^2] < ∞ by assertion (a) and, for all n ∈,

    |S_n| ≤ K_2 + ε n^1/2+δ.


Assertion (d) follows from an application of the Azuma-Hoeffding inequality <cit.>.
The cited inequality gives for L_±max{n ∈_0: ± S_n ≥ε n^1/2+δ}

    (L_±≥ n)
    ≤∑_k ≥ n(± S_k ≥ε k^1/2+δ)
    ≤∑_k ≥ n e^-ε^2 k^2δ/2 C^2.

As above, we conclude that L  L_+ ∨ L_- has finite power moments of all orders and,
for all n ∈, (<ref>) holds with K_2  CL.


Finally, we use the following lemma for biased nearest-neighbor random walk on .
It is possible that the result is available in the literature. However, we have not been able to locate it.

	
Let (S_n)_n ∈_0 be a biased nearest-neighbor random walk on 
with respect to some probability measure , i.e.,
(S_0=0)=1 and

    12 < r (S_n+1 = k+1 | S_n = k) = 1-(S_n+1 = k-1 | S_n = k)

for all k ∈ and n ∈_0. Further, let

    ϱinf{j ∈: S_i < S_j ≤ S_k  for all  0 ≤ i < j ≤ k}

be the first positive point the walk visits from which it never steps to the left.
Then there exist finite constants C^*,c^* = c^*(r) > 0 such that
(τ≥ k) ≤ C^* e^-c^* k for all k ∈_0.


The proof is standard and relies on the usual recursive construction of regeneration times,
see e.g. <cit.>, and the Gambler's ruin formula.
We omit the details. 











spmpsci


