The quenched central limit theorem for a model of random walk in random environment

A short proof of the quenched central limit theorem for the random walk in random environment introduced by Boldrighini, Minlos, and Pellegrinotti is given.


Introduction
In this article we give a short proof of the quenched central limit theorem for a model of the random walk in random environment. At each site the transition probability kernel is affected by the current state of the environment at this site. The model was introduced by Boldrighini, Minlos, and Pellegrinotti, see in particular [BMP94,BMP97,BMP07] and more recent papers Boldrighini et al. [BMPZ15] and Di Persio [DP10]. The model is described in Section 2. Boldrighini et al. [BMPZ15] contains a nice overview of the literature on the subject. For a survey on the recent progress on this and similar models see Zeitouni [Zei06] or Biskup [Bis11]. A related model is considered by Barraquand and Corwin [BC16] and Thiery and Le Doussal [TLD17].
The proof makes use of the multidimensional martingale CLT by Küchler and Sørensen [KS99]. The paper is organized as follows. In Section 2 we describe the model and give the statement. In Section 3 we give the proofs and some further comments.

Model, conditions and results
Consider a particle moving in a n-dimensional infinite lattice and denote by X t is position at time t. On the lattice, a dynamical random environment is considered. It is described by the random field Note that the time is discrete. We assume that ξ is the result of independent copies of the same random variable taking values in some finite space S. The space of configurations is given byΩ = S Z n ×Z + . The values of the field for each (t, are i.i.d random variables, distributed according to a given probability measure denoted by π.
The one step transition probability from position x at same time t to position y at the subsequent time step is given by where P 0 is the transition probability of a free random walk and c is the function which provides the influence of the environment on the particle's dynamic.
In order for the probability P to be well-defined, the following conditions must be fulfilled: Moreover we assume that the random environment has the following property: which means that P 0 is the mean transition probability.
Additionally, let P 0 and c be of bounded range. We denote by P ξ be the conditional probability with respect to the environment.
Let us define the 'average' transition probabilitȳ We further assume that for some b c ∈ R, Theorem 2.1. For almost every realisation ξ of the random environment we have P ξ -a.s., where U is a standard normal vector and η 2 is the positive semidefinite matrix with entries Lemma 3.2. We have The above sum by y is taken over the countable set (Note that P {Y t ∈ Y for all t ∈ N} = 1).
Proof. By definition of Y and P ξ , P ξ -a.s. for a.a. ξ.
Proof. Note that for 1 ≤ i, j ≤ n, Under P ξ a.s. on {Y t = y} the distribution of Y t+1 − Y t is P 0 (u) + c(u, ξ t (y)). Since under P ξ the random variables Y t+1 − Y t are independent of each other for different t, the statement of the lemma follows from the law of large numbers.
(ii) We also have Proof (i) We start by noting that for i, j ∈ {1, ..., n}, Indeed, By (10), Conditioning on Y t , we get (ii) (10) holds for E ξ too, since The proof continues as in (i).
Proof. Let us only prove the second convergence in (11). By Lemma 3.5, The statement of the lemma would follow once we show that for every s ∈ S a.s. #{(r, y) : r ≤ t, Y r = y, ξ r (y) = s} t = π(s).
Since the events {Y r = y} and {ξ r (y) = s} are independent, so by the law of large numbers (12) holds P-a.s. Hence (12) also holds P ξ -a.s for Π-a.a. ξ, otherwise, denoting the event of the left hand side of (12) by A, we would have P(A) = P ξ (A)π(dξ) < 1.
where U a standard n-dimensional Gaussian vector.