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 of a random quadrangulation with
faces is shown to
converge, up to scaling, to the width
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.