Fluctuations of the quenched mean of a planar random walk in an i.i.d. random environment with forbidden direction

We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove a functional CLT for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.


Introduction
One of several models in the study of random media is random walks in random environment (RWRE).An overview of this topic can be found in the lecture notes by Sznitman [9] and Zeitouni [10].While one dimensional RWRE is fairly well understood, there are still many simple questions (transience/recurrence, law of large numbers, central limit theorems) about multidimensional RWRE which have not been resolved.In recent years, much progress has been made in the study of multidimensional RWRE but it is still far from complete.Let us now describe the model.
Let Ω = {ω = (ω x• ) x∈Z d : ω x• = (ω x,y ) y∈Z d ∈ [0, 1] Z d , y ω x,y = 1} be the set of transition probabilities for different sites x ∈ Z d .Let T z denote the natural shifts on Ω so that (T z ω) x• = ω x+z,• .T z can be viewed as shifting the origin to z.Let S be the product σ-field on the set Ω. A probability measure P over Ω is chosen so that (Ω, S, P, (T z ) z∈Z d ) is stationary and ergodic.The set Ω is the environment space and P gives a probability measure on the set of environments.Hence the name "random environment".For each ω ∈ Ω and x ∈ Z d , define a Markov chain (X n ) n≥0 on Z d and a probability measure P ω x on the sequence space such that P ω x (X 0 = x) = 1, P ω x (X n+1 = z|X n = y) = ω y,z−y for y, z ∈ Z d .
There are thus two steps involved.First the environment ω is chosen at random according to the probability measure P and then we have the "random walk" with transition probabilities P ω x (assume x is fixed beforehand).P ω x (•) gives a probability measure on the space (Z d ) N and is called the quenched measure.The averaged measure is Other than the behaviour of the walks itself, a natural quantity of interest is the quenched mean E ω x (X n ) = z zP ω x (X n = z), i.e. the average position of the walk in n steps given the environment ω.Notice that as a function of ω, this is a random variable.A question of interest would be a CLT for the quenched mean.A handful of papers have dealt with this subject.Bernabei [2] and Boldrighini, Pellegrinotti [3] deal with this question in the case where P is assumed to be i.i.d. and there is a direction in which the walk moves deterministically one step upwards (time direction).Bernabei [2] showed that the centered quenched mean, normalised by its standard deviation, converges to a normal random variable and he also showed that the standard deviation is of order n 1 4 .In [3], the authors prove a central limit theorem for the correction caused by the random environment on the mean of a test function.Both these papers however assume that there are only finitely many transition probabilities.Balázs, Rassoul-Agha and Seppäläinen [1] replace the assumption of finitely many transition probabilites with a "finite range" assumption for d = 2 to prove an invariance principle for the quenched mean.In this paper we restrict to d = 2 and look at the case where the walk is allowed to make larger steps upward.We prove a functional CLT for the position of the walk on crossing level n; there is a nice martingale structure in the background as will be evident in the proof.
Another reason for looking at the quenched mean is that recently a number of papers by Rassoul-Agha and Seppäläinen (see [6], [7]) prove quenched CLT's for X n using subdiffusivity of the quenched mean.Let us now describe our model.

Model:
We restrict ourselves to dimension 2. The environment is assumed to be i.i.d.over the different sites.The walk is forced to move at least one step in the e 2 = (0, 1) direction; this is a special case of walks with forbidden direction ( [7]).We assume a finite range on the steps of the walk and also an ellipticity condition .
(iii)There exists some δ > 0 such that Remark 1.2.Conditon (iii) is quite a strong condition.The only place where we have used it are in Lemma 3.5 and in showing the irreducibility of the random walk q in Section 3.2 and the Markov chain Z k in the proof of Lemma 3.10.Condition (iii) can certainly be made weaker.
Before we proceed, a few words on the notation.For x ∈ R, [x] will denote the largest integer less than or equal to x.For x, y ∈ Z 2 , P ω x,y (•) will denote the probabilities for two independent walks in the same environment ω and P x,y = EP ω x,y .E denotes P-expectation.E ω x , E ω x,y , E x , E x,y are the expectations under P ω x , P ω x,y , P x , P x,y respectively.For r, s ∈ R, P ω r,s , P r,s , E ω r , E ω r,s , E r,s will be shorthands for and E ([r],0),([s],0) respectively.C will denote constants whose value may change from line to line.Elements of R 2 are regarded as column vectors.For two vectors x and y in R 2 , x • y will denote the dot product between the vectors.

Statement of Result
Let for positive constants β, σ, c 1 defined further below in (14), (13) and (11) respectively.Let e 1 = (1, 0) T .Define for s, r ∈ R Notice that for any fixed r ∈ R, ξ n (s, r) is the centered quenched mean of the position on crossing level [ns] of a random walk starting at (r √ n, 0).The theorem below gives a functional CLT for ξ n (s, r).
Theorem 2.1.Fix a positive integer N .For any N distinct real numbers where the above convergence is the weak convergence of processes in D[0, ∞) with the Skorohod topology.
) is a mean zero Gaussian process with covariances where the above convergence is the weak convergence of processes in D[0, ∞) with the Skorohod topology.Here 14), ( 13) and (11) respectively.

Proof of Theorem 2.1
We begin with a variation of the well known Martingale Functional Central Limit Theorem whose proof is deferred to the Appendix.Lemma 3.1.Let {X n,m , F n,m , 1 ≤ m ≤ n} be an R d -valued square integrable martingale difference array on a probability space (Ω, F, P ).Let Γ be a symmetric, non-negative definite d × d matrix.Let h(s) be an increasing α-Hölder continuous function on [0, 1] with h(0) = 0 and h(1 for each 0 ≤ s ≤ 1, and for each > 0. Then S n (•) converges weakly to the process Ξ(•) on the space D[0, 1] with the Skorohod topology.Here Ξ(s) = B(h(s)) where B(•) is a Brownian motion with diffusion matrix Γ.
Let F 0 = {∅, Ω} and F k = σ ωj : j ≤ k − 1 where ωj = {ω x• : x • e 2 = j}.F k thus denotes the part of the environment strictly below level k (level k here denotes all points {x : x • e 2 = k}).Notice that for all x and for each i ∈ {1, 2, The above computation tells us The main work in the paper is checking the condition (4) in the above lemma.First note that Using the fact that E D(x, ω) − ED = 0 and that D(x, ω) − ED is independent of F k−1 , we get Here Γ = E(DD T )−E(D)E(D) T and X, X are independent walks in the same environment starting at ([r i √ n], 0) and ([r j √ n], 0) respectively.We will later show that the above quantity converges in P-probability as n → ∞ to h(s)Γ where h(s) is the function in (2).We will also show that h is increasing and Hölder continuous.Lemma 3.1 thus gives us From ( 8) we can complete the proof of Theorem 2.1 as follows.Recalling the definiton in (3), is the number of levels hit by the walk up to level n.Since from equation ( 6), So (recall the definiton of ŵ in (1)), Returning to equation (7), the proof of Theorem 2.1 will be complete if we show as n → ∞.We will first show the averaged statement (with ω integrated away) and then show that the difference of the two vanishes: We would like to get rid of the inequality L k−1 ≤ n − 1 in the expression above so that we have something resembling a Green's function.Denote by ∆L j = L j+1 − L j .Call L k a meeting level (m.l.) if the two walks meet at that level, that is be the consecutive ∆L j where j ∈ I and R 1 , R 2 , • • • be the consecutive ∆L j where j / ∈ I.We start with a simple observation.
Lemma 3.4.Fix x, y ∈ Z.Under P x,y , Proof.We prove the second statement.The first statement can be proved similarly.Call a level a non meeting common level (n.m.c.l) if the two walks hit the level but do not meet at that level.For positive integers The fourth equality is because P ω (l,i),(l+j,i) (∆L 0 = k n ) depends on ω i whereas the previous term depends on the part of the environment strictly below level i.The proof is complete by induction.Lemma 3.5.There exists some a > 0 such that E (0,0),(0,0) (e aL 1 ) < ∞ and E (0,0),(1,0) (e aL 1 ) < ∞.
Proof.By the ellipticity assumption (iii), we have . Since L 1 is stochastically dominated by a geometric random variable, we are done.
Let us denote by X [0,n] the set of points visited by the walk upto time n.It has been proved in Proposition 5.1 of [7] that for any starting points x, y ∈ Z 2 This inequality is obtained by control on a Green's function.The above lemma and the inequality that follows tell us that common levels occur very frequently but the walks meet rarely.Let We will need the following lemma.Lemma 3.6.For each > 0, there exist constants C > 0, b( ) > 0, d( ) > 0 such that Thus Ln n → c 1 P 0,0 a.s.Proof.We prove the first inequality.From Lemma 3.5, we can find a > 0 and some ν > 0 such that for each n, E n (i,j) exp(aL n ) ≤ ν n .We thus have Let γ = 4c 0 .Denote by I n = {j : 0 ≤ j < n, L j is a meeting level } and recall ∆L j = L j+1 − L j .We have Let T 1 , T 2 , . . .be the increments of the successive meeting levels.By an argument like the one given in Lemma 3.4, {T i } i≥1 are i.i.d.Also from (10), it follows that E(T 1 ) = ∞.Hence we can find for some b 2 > 0. The last inequality follows from standard large deviation theory.Also Let {M j } j≥1 be i.i.d. with the distribution of L 1 under P 0,e 1 and {N j } j≥1 be i.i.d. with the distribution of L 1 under P 0,0 .We thus have that the above expression is less than for some b 3 ( ), b 4 ( ) > 0. Recall that 2γ > c 0 by our choice of γ.Combining all the inequalities, we have for some b( ) > 0. The proof of the second inequality is similar.
Returning to (9), let us separate the sum into two parts as Now the second term above is which goes to 0 as n tends to infinity by Lemma 3.6.Similarly ] ≥ n also goes to 0 as n tends to infinity.Thus where a n ( ) → 0 as n → ∞.Now we will show the second term in in the right hand side of the above equation is negligible.
Using the Markov property for the second line below, we get In the expression after the last equality, we have by Lemma 3.6.This gives us where b n ( ) → 0 as n → ∞.By the Markov property again and arguments similar to above where d n ( ) → 0 as n → ∞.

3.2.
Control on the Green's function.We follow the approach used in [1] to find the limit as n → ∞ of in the right hand side of (12).Since > 0 is arbitrary, this in turn will give us the limit of the left hand side of (12).
In the averaged sense Y k is a random walk on Z perturbed at 0 with transition kernel q given by q(0, y) = P (0,0),(0,0) Denote the transition kernel of the corresponding unperturbed walk by q.
The q walk is easily seen to be aperiodic(from assumption 1.1 (iii)), irreducible and symmetric and these properties can be transferred to the q walk.The q can be shown to have finite first moment (because L 1 has exponential moments) with mean 0. Green functions for the q and q walks are given by The potential kernel a of the q walk is It is a well known result from Spitzer [8](sections 28 and 29) that lim x→±∞ a(x) |x| = 1 σ 2 where σ 2 = variance of the q walk.(13) Furthermore we can show that (see [1] )} under the measure P 0,0 .We will use the martingale central limit theorem ([4] page 414) to show that Yn √ n converges to a centered Gaussian.We first show We already have from Lemma 3.5 that for some a > 0 Note that where u 0 = E 0,0 (Y 2 1 ) and E (1,0),(0,0) ((Y 1 − 1) 2 ) = σ 2 (as defined in (13)), the variance of the unperturbed walk q.So To complete, by choosing b large enough we get Hence → 0 in P 0,0 -probability.We have checked both the conditions of the martingale central limit theorem and so we have . This will follow if we can show that n − 1 2 Y n uniformly integrable.But that is clear since we have It is easily shown that We already know by the local limit theorem ([4] section 2.6) that lim which is equal to lim βG n (0, 0) by the above computations.The arguments in ( [1] page 518 (4.9)) allow us to conclude that sup Now returning back to (12), the local limit theorem ([4] section 2.6) and a Riemann sum argument gives us Hence the right hand side of (12) tends to This completes the proof of Proposition 3.2 .
3.3.Proof of Proposition 3.3.This section is based on the proof of Theorem 4.1 in Section of [5] and (5.20) in [1].Recall F 0 = {∅, Ω}, Proof.Notice that R l is 0 unless one of the walks hit level l because otherwise the event in question does not need ω l .We then have and it follows that 1 n n l=0 ER 2 l,2 → 0. The remaining parts of the proposition can be similarly proved.This completes the proof of Proposition 3.8 and hence Proposition 3.3 We have thus shown where h(s) is as in (2).From the left hand side of the above expression, we can conclude that h(s) is nondecreasing.For if h(s) > h(t) for some s < t, we have The left hand side is a nonpositive definite matrix whereas the right hand side is nonnegative definite.We show that h is Hölder continuous.
For 0 < s < t, we have for some s ≤ u ≤ t.Since the right hand side of the above equation is bounded for 0 < u ≤ 1, we are done .
Theorem 2.1 is now proved except for Lemma 3.10.sup We first prove the following Lemma 3.11.
where Z is a standard Normal random variable.
denote the first time when the walk reaches level n or above.Denote the drift D(x, ω) at the point x by D(x, ω) •) be a Brownian motion in R 2 with diffusion matrix Γ = E(DD T ) − E(D)E(D) T .For a fixed positive integer N and real numbers r 1 , r 2 , • • • , r N and θ 1 , θ 2 , • • • , θ N , define the function h : R → R by

1 √ n n− 1 l=0
k and both walks hit level k F l .Call R l = n k=l+1 • • • .It is enough to show that E( R l ) 2 → 0.By orthogonality of martingale increments, ER l R m = 0 for l = m.Letφ n = |{k : k ≤ n, k is a meeting level }|,the number of levels at which the two walks meet up to level n.

1 and 2 .
The sum 1 is over all 0 = x 0 , x 1 , • • • , x n and u = x0 , x1 , • • • , xn such that the first level where x i = xj occurs is before the first level where x i = xj + e 1 occurs.Similarly 2 is over all 0 = x 0 , x 1 , • • • , x n and u = x0 , x1 , • • • , xn such that x i = xj occurs after x i = xj + e 1 .The above expression now becomes