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Random Planar Lattices and Integrated SuperBrownian Excursion

Philippe Chassaing 1 Gilles Schaeffer 2
2 ADAGE - Applying discrete algorithms to genomics
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 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 (e^{(n)}, W^{(n)}) to the Brownian snake description (e,\hat W) 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.
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Submitted on : Tuesday, September 26, 2006 - 2:55:12 PM
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Philippe Chassaing, Gilles Schaeffer. Random Planar Lattices and Integrated SuperBrownian Excursion. [Intern report] A02-R-215 || chassaing02a, 2002, 44 p. ⟨inria-00101065⟩

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