Asymptotics and scaling properties of two-dimensional lattice paths and polygons

We consider the generating functions of different models of two-dimensional lattice polygons, weighted with respect to their area and perimeter. Many of them are expected to show similar asymptotic features which do not depend on the exact details of the model, such as the existence of a scaling limit, which is described by the Airy function. For rooted self-avoiding polygons, this scaling behaviour has been conjectured by Guttmann, Richard and Jensen. However, a proof of it seems to be currently out of reach. One can make progress by considering models which are analytically more tractable, such as area-weighted Dyck paths. In this talk I will show how a uniform asymptotic expression for their area-perimeter generating function can be obtained by using the method of steepest descents, generalised to the case of two coalescing saddle points. Afterwards I will explain how a new type of asymptotic behaviour can be observed by adding a perturbation to the area-perimeter generating function of Dyck paths, leading to a model which we call deformed Dyck paths.