4 research outputs found
Random Planar Lattices and Integrated SuperBrownian Excursion
In this paper, 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. 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.
The first combinatorial ingredient is an encoding of quadrangulations by
trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's
well labelled trees. The second step relates these trees to embedded (discrete)
trees in the sense of Aldous, via the conjugation of tree principle, an
analogue for trees of Vervaat's construction of the Brownian excursion from the
bridge.
From probability theory, we need a new result of independent interest: the
weak convergence of the encoding of a random embedded plane tree by two contour
walks to the Brownian snake description of ISE.
Our results suggest the existence of a Continuum Random Map describing in
term of ISE the scaled limit of the dynamical triangulations considered in
two-dimensional pure quantum gravity.Comment: 44 pages, 22 figures. Slides and extended abstract version are
available at http://www.loria.fr/~schaeffe/Pub/Diameter/ and
http://www.iecn.u-nancy.fr/~chassain