A group endowed with a probability measure comes naturally with a random walk. If moreover this group acts on some space, then one can push the random walk to the space under consideration, up to the choice of a basepoint. The resulting random walk is called the image random walk.
In this talk, motivated by numerous examples (hyperbolic groups acting on one of their Cayley graphs, mapping class groups acting on the corresponding curve complex...), the (discrete) groups will act on Gromov hyperbolic spaces. We will discuss the escape rate for such image random walks, and more precisely the associated large deviations problem.