Gilles Schaeffer: Random Quadrangulations and Aldous' ISE

This is a joint work with Philippe Chassaing.

A surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius $r_n$ of a random quadrangulation with $n$ faces is shown to converge, up to scaling, to the width $r=R-L$ of the support of the one-dimensional ISE, or precisely:


More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero.


Philippe Chassaing and Gilles Schaeffer.
Random planar lattices and Integrated SuperBrownian Excursion.
Preprint available as arXiv:math.CO/0205226.

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