The Tricritical Scaling Function of Partially Directed Vesicles Polyg\^ones convexes en dimension 3

In my second talk I will discuss the asymptotic analysis of the generating function for partially directed vesicle models. In a semi-continuous version of the model this leads to a simple application of the method of dominated balance on a non-linear ODE. However, for the original discrete model the equivalent of the ODE is a $q$-difference equation for which there is no known equivalent of this method. Here, the asymptotics can be worked out with the help of a new contour integral representation. This leads to an explicit calculation of the scaling behaviour around the critical point in terms of the Airy function. In particular, one gets a uniform asymptotic expansion of the involved $q$-series as $q$ approaches $1$ from below.