Uniqueness and universality of the Brownian map

By Jean-François Le Gall. 2011

423 citations recorded by the source corpus.

We consider a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mn be the vertex set of Mn, which is equipped with the graph distance dgr. Both when q≥4 is an even integer and when q=3, there exists a positive constant cq such that the rescaled metric spaces (mn,cqn−1/4dgr) converge in distribution in the Gromov–Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4552b818cd93bf97e4112d12e614c1b20c9c0d91

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Random geometry on the sphere (2014, background): “A key result [31], which solves a conjecture of Oded Schramm [44], states that”
Tropical Combinatorics and Whittaker functions (2011, background): “e cases still unresolved) connections with random matrix theory. Some of these connections were probed by Okounkov [76] in terms of a mutual connection to combinatorics of maps (or fat ribbon graphs) [58, 59]. Such partition enumeration problems correspond to counting important algebraic geometric i”
Planar stochastic hyperbolic triangulations (2016, methodology): “The UIPT and its quadrangular analog the UIPQ have received a lot of attention in recent years [4, 12, 22, 27, 31] partially motivated by the connections with the physics theory of 2dimensional quantum gravity, the Gaussian free field [19] and the Brownian map [20, 30, 34].”
The Brownian Plane (2012, background): “The existence of a scaling limit for (uniformly distributed) random quadrangulations was obtained recently in the papers [15, 21], leading to a limiting random compact metric space called the Brownian map.”
Scaling limits of random planar maps with a unique large face (2012, background): “Le Gall proved independently that the same holds in the case of uniform 2p-angulations and uniform triangulations [40].”
Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps (2014, background): “There has been great progress in their probabilistic study ever since the work of Schaeffer [28], which has amongst other things led to finding the scaling limit of many large random maps (we mention [25] and [18]).”
Gaussian multiplicative chaos and applications: A review (2013, background): “Interestingly, they have also understood that such metrics can be discretized by randomly triangulated surfaces (see [6] for instance), a recent field of mathematical research that culminated with the construction of the so-called Brownian map (see [106, 107, 108, 117]) in the special case of pure g”
Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points (2012, background): “In particular, Miermont and Le Gall recently proved that a large class of random planar maps properly rescaled converge towards a universal continuous random surface called the Brownian Map [31, 36].”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, background): “This last step, however, is not mandatory in order to identify the topology of the limit: Le Gall and Paulin [LGP08], and later Miermont [Mie08], showed that any possible limit is homeomorphic to the two-dimensional sphere.”
Liouville Quantum Gravity on the Riemann Sphere (2014, background): “If this is the case, then the space (S2, d, ν) should be related to the Brownian map equipped with its volume measure (see [44,47]): more precisely, for all fixed A > 0, both metric spaces should be isometric (up to some global constant) once conditioned to have same volume A.”
Rescaled bipartite planar maps converge to the Brownian map (2013, background): “I am deeply indebted to Jean-François Le Gall for suggesting me to study this problem, for stimulating discussions and for carefully reading the manuscript and making many useful suggestions.”
The hull process of the Brownian plane (2014, background): “A fundamental object is the random compact metric space known as the Brownian map, which has been proved to be the universal scaling limit of several important classes of random planar maps conditioned to have a large size (see in particular [1, 3, 6, 23, 30]).”
Compact Brownian surfaces I: Brownian disks (2015, background): “hic to the sphere [28], this means that the boundariesoftheapproximatingrandommapsaretoosmalltobeseeninthelimit. This particularcasegeneralizestheconvergenceofuniformrandomquadrangulations,obtained in[26,34],correspondingtothecasewherel n= 2 foreveryn1. Thecasewherel n= p n!1isalsoofinterest,andisth”