The purpose of this talk is to advertise a program that brings some new approaches to bear on combinatorial problems in mathematical physics and probability theory. The origins of these approaches lie in older ideas of dynamical systems theory. Examples of the problems are calculating geodesic distance on random surfaces, random walks in random environments, and continuum limits of map enumeration. Sources of the new approaches include Moser's classical approach to billiard dynamics and Arnold's notion of algebraic entropy, generalizing the classical concepts of Kolmogorov-Sinai and topological entropy. But the focus of the talk will be on some explicit examples. This program is an outgrowth of the thesis of recent Arizona PhD Tova Brown.