Martin boundary of brownian motion on Gromov hyperbolic metric graphs

Let \begin{document}$ \widetilde{X} $\end{document} be a locally finite Gromov hyperbolic graph whose boundary consists of infinitely many points and with cocompact isometric action discrete group \Gamma . We show the uniform Ancona inequality for Brownian motion which implies that \lambda -Martin coincides any \in [0, \lambda_0], in particular at bottom spectrum \lambda_0