Limits of the boundary of random planar maps View Full Text


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Article Info

DATE

2018-12

AUTHORS

Loïc Richier

ABSTRACT

We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter α∈(1,2). First, in the dense phase corresponding to α∈(1,3/2), we prove that the scaling limit of the boundary is the random stable looptree with parameter 1/(α-1/2). Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to α∈(3/2,2), it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid O(n) loop model on quadrangulations, proving thereby a conjecture of Curien and Kortchemski. More... »

PAGES

1-39

References to SciGraph publications

  • 2018-12. Martingales in self-similar growth-fragmentations and their connections with random planar maps in PROBABILITY THEORY AND RELATED FIELDS
  • 2005-04. Probabilistic and fractal aspects of Lévy trees in PROBABILITY THEORY AND RELATED FIELDS
  • 1978-02. Planar diagrams in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2011-01. Condensation in Nongeneric Trees in JOURNAL OF STATISTICAL PHYSICS
  • 2010-12. Geodesics in large planar maps and in the Brownian map in ACTA MATHEMATICA
  • 2017-10. Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2007-09. The topological structure of scaling limits of large planar maps in INVENTIONES MATHEMATICAE
  • 2018-03. Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps in JOURNAL OF THEORETICAL PROBABILITY
  • 2015-10. Percolation on random triangulations and stable looptrees in PROBABILITY THEORY AND RELATED FIELDS
  • 2008-09. Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere in GEOMETRIC AND FUNCTIONAL ANALYSIS
  • 2013. Séminaire de Probabilités XLV in NONE
  • 2015-03. Random Walk on Random Infinite Looptrees in JOURNAL OF STATISTICAL PHYSICS
  • 2013-06. The Brownian map is the scaling limit of uniform random plane quadrangulations in ACTA MATHEMATICA
  • Identifiers

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    http://scigraph.springernature.com/pub.10.1007/s00440-017-0820-y

    DOI

    http://dx.doi.org/10.1007/s00440-017-0820-y

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    47 schema:description We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter α∈(1,2). First, in the dense phase corresponding to α∈(1,3/2), we prove that the scaling limit of the boundary is the random stable looptree with parameter 1/(α-1/2). Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to α∈(3/2,2), it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid O(n) loop model on quadrangulations, proving thereby a conjecture of Curien and Kortchemski.
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