How could one define a “uniform random continuous path” in d dimensions? A natural answer is given by Brownian motion, a stochastic process that arises as a limit of discrete random walks in R^d, appropriately rescaled in space and time. This will be our first example of a “scaling limit”.
Planar maps are planar graphs “drawn” on the surface of the sphere. What does a large random planar map look like? A first step will be to study plane trees, maps with a single face: what does a large plane tree look like? The procedure of taking scaling limits is an amazingly fruitful tool in this context; it highlights a connection to Brownian motion and sparks a beautiful theory of “continuous” random geometries, with far-reaching implications in several fields of Mathematics and Physics.