Embedded trees and the support of the ISE

Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by ±1. Recently it has been proved (see Chassaing and Schaeffer (2004) [8] and Janson and Marckert (2005) [11]) that the distribution of the maximum and minimum labels are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is to make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier et al. (2003) [6] that is based on properties of Jacobi's θ-functions. In particular, we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass ℘-function. Furthermore we re-derive the limiting radius distribution in random quadrangulations (by Chassaing and Schaeffer (2004) [8]) with the help of exact counting generating functions.