Models of random walks in uniformly elliptic i.i.d. random environment (RWRE) \(X_n\), \(n=1,2,\dots\), on \(\mathbb{Z}^d\), \(d \geq 4\), are considered. The RWRE is assumed to be ballistic, i.e., the limiting velocity \(v=\lim_{n \to \infty} n^{-1}X_n\) is a non-zero almost sure constant. The ballisticity is assured by a condition slightly weaker than Sznitman’s condition \((T')\). Generally RWREs may be plain nestling, marginally nestling and non-nestling. Since the two latter cases have been studied previously, in the article, plain nestling RWREs are considered.
It is known that, for every \(\epsilon >0\) and \(n\) large enough, the annealed probability \(\operatorname{P}(\|n^{-1}X_n-a\|_{\infty}<\epsilon)\) decays exponentially with \(n\) for every \(a \notin A\), with \(A\) being a convex hull of \(0\) and \(v\). For every \(a \in A\), the annealed probability \(\operatorname{P}(\|n^{-1}X_n-a\|_{\infty}<\epsilon)\) decays more slowly than exponentially. This property is called linear slowdown. In the article, the annealed probability of a linear slowdown is bounded from above by \(\exp(-(\log n)^{d-\epsilon})\). This bound almost matches the known lower bound \(\exp(-C(\log n)^d)\), and significantly improves previously known lower bounds. As a corollary, almost sharp estimates for the quenched probability of slowdown are provided. As a tool for obtaining the main result, an almost local version of the quenched central limit theorem under the assumption of the same condition is obtained.