We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range (2,3). In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is n(logn)2, the maximum hitting time is nlogn
Nearby in the stack
, and the average hitting time is
n
. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in
n
, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.