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Random planar lattices and integrated superBrownian excursion
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  • Published: 26 November 2003

Random planar lattices and integrated superBrownian excursion

  • Philippe Chassaing1 &
  • Gilles Schaeffer2 

Probability Theory and Related Fields volume 128, pages 161–212 (2004)Cite this article

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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, 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. 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.

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Author information

Authors and Affiliations

  1. Institut Élie C artan, Université Henri Poincaré, B.P. 239, 54506, Vandoeuvrelès-Nancy, France

    Philippe Chassaing

  2. CNRS – Laboratoire d’Informatique, École Polytechnique, 91128, Palaiseau, France

    Gilles Schaeffer

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  1. Philippe Chassaing
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  2. Gilles Schaeffer
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Correspondence to Philippe Chassaing.

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Chassaing, P., Schaeffer, G. Random planar lattices and integrated superBrownian excursion. Probab. Theory Relat. Fields 128, 161–212 (2004). https://doi.org/10.1007/s00440-003-0297-8

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  • Received: 22 May 2002

  • Revised: 17 July 2003

  • Published: 26 November 2003

  • Issue Date: February 2004

  • DOI: https://doi.org/10.1007/s00440-003-0297-8

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Keywords

  •  Random planar lattices
  • Planar maps
  • Quadrangulations
  • Dynamical triangulations
  • Fluid lattices
  • Internal Hausdorff dimension
  • Profile
  • Radius
  • Well labelled trees
  • ISE
  • Brownian snake
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