Integrable mappings from a unified perspective

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, addition laws on elliptic curves, and links between randomness and symmetry.