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I am very grateful to the anonymous referee for his valuable comments (incorporated into the current presentation as Re- marks 1–3), providing the proof of Lemma 5 and
In this section, we prove results concerning local (i.e., restricted to proper subspaces) vs global linear dependence of operators that will be used in the proof of Theorem 1.5, and
It is very similar to the proof of Lemma 4 in Kesten (1987), which deals with the probability of four disjoint paths to ∆S(n), two occupied ones and two vacant ones, with the
Our proof of Theorem 1.1 is based on estimates for the structure function, i.e., the spatial Fourier transform of the correlation function with respect to h·i , the
In section 5, we study the integrability of the Green function of the walk which ensures the existence of the original (non killed) Kalikow’s auxiliary walk and finish the proof of
The proof of Theorem 3 is based on Burkholder’s technique, which reduces the problem of proving a martingale inequality to finding a certain spe- cial function.. The description of
We now show how the uniform law of large numbers in Theorem 1.1 can be used to prove the hydrodynamic limits for the system of RWRE as stated in Theorem 1.4.. Proof of Theorem 1.4:
The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins [2] to prove uniqueness of the martingale problem in the framework
