Toronto Probability

Given a countable group $G$ equipped with a probability measure $\mu$, one can define a random walk on $G$ as the Markov process whose increments are iid sampled by $\mu$. The large-scale properties of this random walk can exhibit a plethora of exotic behaviours, relating to the algebraic and geometric structure of $G$. One way of capturing the large-scale behaviour is via the Poisson boundary of $(G,\mu)$, a canonical (abstract) measure space associated with the Markov chain. This object has been studied intensely over the decades. In particular, mathematicians have found concrete ‘realizations’ of this measure space as topological boundaries for the group (i.e. the Gromov boundary of a hyperbolic group). In 1983, Kaimanovich and Vershik asked whether the Poisson boundary can always be realized in this way. I will describe the complete resolution of this problem in the negative. This is joint work with Joshua Frisch.