Aging properties of Sinai's model of random walk in random environment

We study in this short note aging properties of Sinai's (nearest neighbour) random walk in random environment. With $\PP^o$ denoting the annealed law of the RWRE $X_n$, our main result is a full proof of the following statement due to P. Le Doussal, C. Monthus and D. S. Fisher: $$\lim_{\eta\to0} \lim_{n\to\infty} \PP^o (\frac{|X_{n^h} - X_n|}{(\log n)^2} < \eta) = \frac{1}{h^2} [ {5/3} - {2/3} e^{-(h-1)} ]. $$
Comment: This note will be part of the forthcoming lecture notes of O. Zeitouni on RWRE, to appear as proceedings of the St Flour summer school 2001