  Two discrete dynamical systems are discussed and analyzed whose trajectories
encode significant explicit information about a number of problems in
combinatorial probability, including graphical enumeration on Riemann surfaces
and random walks in random environments. The two models are integrable and our
analysis uncovers the geometric sources of this integrability and uses this to
conceptually explain the rigorous existence and structure of elegant closed
form expressions for the associated probability distributions. Connections to
asymptotic results are also described. The work here brings together ideas from
a variety of fields including dynamical systems theory, probability theory,
classical analogues of quantum spin systems, addtion laws on elliptic curves,
and links between randomness and symmetry.
