Access the full text.
Sign up today, get DeepDyve free for 14 days.
E. Bender, E. Canfield, Robert Robins (1986)
The asymptotic number of rooted maps on a surfaceJ. Comb. Theory, Ser. A, 43
D. Aldous (1991)
The Continuum Random Tree IIIAnnals of Probability, 19
J. Marckert, A. Mokkadem (2004)
Limit of normalized quadrangulations: The Brownian mapAnnals of Probability, 34
R. Cori, B. Vauquelin (1981)
Planar Maps are Well Labeled TreesCanadian Journal of Mathematics, 33
P. Chassaing, B. Durhuus (2003)
Local limit of labeled trees and expected volume growth in a random quadrangulationAnnals of Probability, 34
J. Gall, M. Weill (2005)
Conditioned Brownian treesAnnales De L Institut Henri Poincare-probabilites Et Statistiques, 42
W. Tutte (1963)
A Census of Planar MapsCanadian Journal of Mathematics, 15
B. Duplantier, S. Sheffield (2008)
Liouville quantum gravity and KPZInventiones mathematicae, 185
T. Duquesne, J. Gall (2005)
Probabilistic and fractal aspects of Lévy treesProbability Theory and Related Fields, 131
J. Mickelsson (1989)
The Kp Hierarchy
G. Miermont (2007)
On the sphericity of scaling limits of random planar quadrangulationsElectronic Communications in Probability, 13
Omer Angel, O. Schramm (2002)
Uniform Infinite Planar TriangulationsCommunications in Mathematical Physics, 241
B. Mohar, C. Thomassen (2001)
Graphs on Surfaces
M. Weill (2007)
Asymptotics for Rooted Bipartite Planar Maps and Scaling Limits of Two-Type Spatial TreesElectronic Journal of Probability, 12
P. Mattila (1995)
Geometry of sets and measures in Euclidean spaces
G. Hooft (1974)
A Planar Diagram Theory for Strong InteractionsNuclear Physics, 72
(2006)
Rayleigh processes
D. Aldous (1991)
Stochastic Analysis: The Continuum random tree II: an overview
I. Goulden, D. Jackson (2008)
The KP hierarchy, branched covers, and triangulationsAdvances in Mathematics, 219
R. Durrett, D. Iglehart (1977)
Functionals of Brownian meander and Brownian excursionAnnals of Probability, 5
G. Miermont (2006)
Invariance principles for spatial multitype Galton–Watson treesAnnales De L Institut Henri Poincare-probabilites Et Statistiques, 44
Daniel Revuz, M. Yor (1990)
Continuous martingales and Brownian motion
G. Halsted (1900)
The International Congress of MathematiciansNature, 62
G. Miermont (2006)
An invariance principle for random planar mapsDiscrete Mathematics & Theoretical Computer Science
J. Doob (1953)
Stochastic processes
W. Kaigh (1976)
An Invariance Principle for Random Walk Conditioned by a Late Return to ZeroAnnals of Probability, 4
E. Brézin, C. Itzykson, G. Parisi, J. Zuber (1978)
Planar diagramsCommunications in Mathematical Physics, 59
J. Bouttier, E. Guitter (2007)
Statistics of geodesics in large quadrangulationsJournal of Physics A: Mathematical and Theoretical, 41
M. Weill (2006)
Asymptotics for rooted planar maps and scaling limits of two-type spatial treesarXiv: Probability
(2004)
A uniformly distributed infinite planar triangulation and a related branching process
Gilles Schaeffer (1998)
Conjugaison d'arbres et cartes combinatoires aléatoires
J. Bouttier, P. Francesco, E. Guitter (2003)
Geodesic distance in planar graphsNuclear Physics, 663
R. Bass (2011)
Convergence of probability measures
J. Ambjorn, B. Durhuus, T. Jonsson (1997)
Quantum Geometry: Monte Carlo simulations of random geometry
M. Gromov (1999)
Metric Structures for Riemannian and Non-Riemannian Spaces
G. Miermont (2007)
Tessellations of random maps of arbitrary genusarXiv: Probability
J. Bouttier, E. Guitter (2008)
The three-point function of planar quadrangulationsJournal of Statistical Mechanics: Theory and Experiment, 2008
J. Gall (1999)
Spatial Branching Processes, Random Snakes, and Partial Differential Equations
J. Gall, F. Paulin (2006)
Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-SphereGeometric and Functional Analysis, 18
Jean-François Delmas (2003)
Computation of Moments for the Length of the OneDimensional ISE SupportElectronic Journal of Probability, 8
D. Arquès (1986)
Les hypercartes planaires sont des arbres tres bien etiquetesDiscret. Math., 58
S. Evans, J. Pitman, A. Winter (2004)
Rayleigh processes, real trees, and root growth with re-graftingProbability Theory and Related Fields, 134
Omer Angel (2002)
Growth and percolation on the uniform infinite planar triangulationGeometric and Functional Analysis, 13
J. Gall (2006)
A conditional limit theorem for tree-indexed random walkStochastic Processes and their Applications, 116
J. Ambjorn, Y. Watabiki (1995)
Scaling in quantum gravityNuclear Physics, 445
D. Burago, Yu. Burago, S. Ivanov (2001)
A Course in Metric Geometry
S. Lando, A. Zvonkin (2003)
Graphs on Surfaces and Their Applications
J. Ambjorn, B. Durhuus, T. Jonsson (1997)
Quantum Geometry: Notation
P. White (2007)
REGULAR CONVERGENCE
T. Duquesne, J. Gall (2005)
The Hausdor measure of stable treesarXiv: Probability
O. Schramm (2006)
Conformally invariant scaling limits: an overview and a collection of problems
J. Marckert, G. Miermont (2005)
Invariance principles for random bipartite planar mapsAnnals of Probability, 35
P. Chassaing, G. Schaeffer (2002)
Random planar lattices and integrated superBrownian excursionProbability Theory and Related Fields, 128
J. Gall (2008)
Geodesics in large planar maps and in the Brownian mapActa Mathematica, 205
J. Bouttier, P. Francesco, E. Guitter (2004)
Planar Maps as Labeled MobilesElectron. J. Comb., 11
G. Miermont, M. Weill (2007)
Radius and profile of random planar maps with faces of arbitrary degreesElectronic Journal of Probability, 13
G. Chapuy, Michel Marcus, G. Schaeffer (2007)
A Bijection for Rooted Maps on Orientable SurfacesSIAM J. Discret. Math., 23
D. Arquès (1986)
Rooted planar maps are well labeled treesDiscrete Mathematics, 58
G. Chapuy (2008)
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled treesProbability Theory and Related Fields, 147
S. Evans (2000)
Snakes and spiders: Brownian motion on ℝ-treesProbability Theory and Related Fields, 117
(2008)
The structure of dominant unicellular maps
Jean-François Gall (2007)
The topological structure of scaling limits of large planar mapsInventiones mathematicae, 169
[We review some aspects of scaling limits of random planar maps, which can be considered as a model of a continuous random surface, and have driven much interest in the recent years. As a start, we will treat in a relatively detailed fashion the well-known convergence of uniform plane trees to the Brownian Continuum Random Tree. We will put a special emphasis on the fractal properties of the random metric spaces that are involved, by giving a detailed proof of the calculation of the Hausdorff dimension of the scaling limits.]
Published: Dec 30, 2009
Keywords: 60C05; 60F17; Random maps; random trees; scaling limits; Brownian CRT; random metric spaces; Hausdorff dimension
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.