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Random Planar Lattices and Integrated SuperBrownian Excursion

  • Philippe Chassaing
  • Gilles Schaeffer
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this extended abstract, 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 rn 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:
$$ n^{ - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} r_n \xrightarrow{{law}} (8/9)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} r. $$

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.

Keywords

Root Vertex Dyck Path Label Distribution Incipient Infinite Cluster Maximal Label 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Philippe Chassaing
    • 1
  • Gilles Schaeffer
    • 2
  1. 1.Université Henri PoincaréVandoeuvreFrance
  2. 2.CNRS-LORIAVandoeuvreFrance

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