Random Planar Lattices and Integrated SuperBrownian Excursion

Autor: Philippe Chassaing, Gilles Schaeffer
Přispěvatelé: Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Applying discrete algorithms to genomics (ADAGE), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Gardy, D. and Mokkadem, A., Chauvin, B. and Flajolet, P., and Gardy, D. and Mokkadem, A.
Jazyk: angličtina
Rok vydání: 2004
Předmět:
Statistics and Probability
distances
[INFO.INFO-OH]Computer Science [cs]/Other [cs.OH]
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
FOS: Physical sciences
MSC: 05C80 (Primary) 60C05 (Secondary)
01 natural sciences
Combinatorics
010104 statistics & probability
symbols.namesake
serpent brownien
Planar
Probability theory
Mathematics::Probability
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
FOS: Mathematics
Mathematics - Combinatorics
0101 mathematics
Connection (algebraic framework)
distance
diameter
Scaling
Mathematical Physics
Mathematics
brownian snake
random maps
Probability (math.PR)
010102 general mathematics
Excursion
triangulations
Mathematical Physics (math-ph)
05C80 (Primary) 60C05 (Secondary)
arbres
trees
Radius
Planar graph
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
cartes aléatoires
Hausdorff dimension
symbols
[MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA]
Combinatorics (math.CO)
Statistics
Probability and Uncertainty
Mathematics - Probability
Analysis
diamètre
Zdroj: Probability Theory and Related Fields
Probability Theory and Related Fields, Springer Verlag, 2004, 128(2), pp.161-212. ⟨10.1007/s00440-003-0297-8⟩
[Intern report] A02-R-215 || chassaing02a, 2002, 44 p
Mathematics and Computer Science II ISBN: 9783034894753
Probability Theory and Related Fields, 2004, 128(2), pp.161-212. ⟨10.1007/s00440-003-0297-8⟩
Colloquium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities
Colloquium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, Gardy, D. and Mokkadem, A., Sep 2002, Versailles, France, pp.123--141
ISSN: 0178-8051
1432-2064
Popis: 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.
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/
Databáze: OpenAIRE
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