Scaling limits of random trees and planar maps

Abstract

These are the notes for a series of lectures given at the Clay Mathematical Institute Summer School in Buzios, July 11 - August 7, 2010. We review some of the recent aspects of scaling limits of random trees and planar maps, in particular via their relations with bijective enumeration and Gromov-Hausdorff convergence.

References in corpus (3)

• The topological structure of scaling limits of large planar maps
• The three-point function of planar quadrangulations
• The topology of scaling limits of positive genus random quadrangulations

Cited by in corpus (26)

• Uniqueness and universality of the Brownian map
• The Tensor Track, III
• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
• Scaling limits of random planar maps with a unique large face
• More on the O(n) model on random maps via nested loops: loops with bending energy
• Recurrence of bipartite planar maps
• Gaussian multiplicative chaos and applications: a review
• The range of tree-indexed random walk in low dimensions
• Quadrangulations with no pendant vertices
• Quantum gravity and the KPZ formula
• Probabilistic Models for the (sub)Tree(s) of Life
• Stability of geodesics in the Brownian map
• Renormalizable Models in Rank $d\ge 2$ Tensorial Group Field Theory
• Bounds on the critical line via transfer matrix methods for an Ising model coupled to causal dynamical triangulations
• A note on weak convergence results for uniform infinite causal triangulations
• The Brownian map is the scaling limit of uniform random plane quadrangulations
• On tessellations of random maps and the $t_g$-recurrence
• Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants
• Spheres are rare
• Growth of uniform infinite causal triangulations
• Discrete analytic functions on non-uniform lattices without global geometric control
• Invariance principles for random walks in random environment on trees
• Uniform multifractal structure of stable trees
• New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications
• Local limit of sparse random planar graphs