The density of the ISE and local limit laws for embedded trees

Abstract

It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local version of this result: ISE has a (random) H\"{o}lder continuous density, and the vertical profile of embedded trees converges to this density, at least for some such trees. As a consequence, we derive a formula for the distribution of the density of ISE at a given point. This follows from earlier results by Bousquet-M\'{e}lou on convergence of the vertical profile at a fixed point. We also provide a recurrence relation defining the moments of the (random) moments of ISE.