Many classical theorems in probability concern sums of iid real random variables. For a group G, one can instead consider a product of iid G-valued random variables. A setting of interest is when G acts on a hyperbolic space, such as the hyperbolic plane or a d-regular tree. This includes mapping class groups acting on their curve complexes, Gromov-hyperbolic groups acting on their Cayley graphs, etc.

The corresponding random walk exhibits many phenomena which differ from the abelian case. On the other hand, one can still obtain laws of large numbers and central limit theorems. In 2020, Boulanger, Mathieu, Sert, and Sisto proved that a random walk on a hyperbolic group satisfies a large deviation principle. Recently, Gouëzel showed with a clever geometric argument that a non-elementary random walk on a hyperbolic space satisfies linear growth with exponential decay.

In this talk I will describe the geometry of random walks on hyperbolic spaces, and sketch Gouëzel’s proof.