The density of the ISE and local limit laws for embedded trees
Probability
2008-05-05 v2 Combinatorics
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.
Keywords
Cite
@article{arxiv.math/0509322,
title = {The density of the ISE and local limit laws for embedded trees},
author = {Mireille Bousquet-Mélou and Svante Janson},
journal= {arXiv preprint arXiv:math/0509322},
year = {2008}
}