On fluctuations of the drift in hyperbolic groups

Let $\mathrm{\Gamma }$ refer to a convex-cocompact group of isometries of a CAT($-1$) space $X$ and $Y\to X\mathrm{/}\mathrm{\Gamma }$ to a Galois cover with a word-hyperbolic group of deck transformations. We show that, for almost every geodesic $\xi$ with respect to the Bowen-Margulis-Sullivan measure on the lift of $X\mathrm{/}\mathrm{\Gamma }$, there exist $\mathfrak{m},\sigma >0$ and a standard Brownian motion $B_s$ such that, for any $\lambda >1\mathrm{/}4$, $$d(p(g_s(\xi )),\mathbf{o})=\mathfrak{m}s+\sigma B_s+o(s^{\lambda }),$$ with $g_s$ referring to the geodesic flow acting on the geodesics of $Y$ and $p(g_s)$ to the canonical projection to $Y$. The result is a consequence of an almost sure invariance principle for random walks on hyperbolic groups with dependent increments, whose proof makes use of a new Ruelle operator theorem for skew products and Martin boundary techniques for random walks with dependent increments.