# Integrable Mappings from a Unified Perspective

## TL;DR

This paper explores two integrable discrete dynamical systems that reveal geometric origins of their integrability, providing insights into combinatorial probability problems and connecting diverse mathematical fields.

## Contribution

It offers a unified geometric framework explaining the integrability and closed-form solutions of models in combinatorial probability and dynamical systems.

## Key findings

- Identification of geometric sources of integrability
- Explicit closed-form expressions for probability distributions
- Connections to asymptotic behavior and diverse mathematical fields

## Abstract

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.