The Brownian map is the scaling limit of uniform random plane quadrangulations

By Grégory Miermont. 2011

409 citations recorded by the source corpus.

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n−1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b8321ba499ac7a654cef0e519057898d38317f5

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In Mathematical Physics

Citations in the corpus

Rescaled bipartite planar maps converge to the Brownian map (2013, methodology): “The analogous constant for uniform general maps with n edges [3] is (9/8)1/4 and the one for uniform quadrangulations with n edges (n even) [11, 15] is (9/4)1/4.”
Gaussian multiplicative chaos and applications: A review (2013, background): “The prototype of such p-angulations is the Brownian map studied in (see [105, 106, 107, 114]) and corresponds to the pure gravity case c = 0.”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, methodology): “As we consider in this work surfaces with a boundary, the notion of 1-regularity used by Miermont in [Mie08] is no longer sufficient: we will also need here the notion of 0-regularity, which we will expose in Section 5.1.”
Scaling limits of random planar maps with a unique large face (2012, background): “of subsequence is superfluous and the scaling limit in fact equals the so-called Brownian map up to a scale factor [46].”
Random stable looptrees (2013, neutral): “We refer to [17] and [18] for details and proofs of the assertions contained in this section.”
Random geometry on the sphere (2014, methodology): “Note that the special case of quadrangulations, corresponding to p = 4, has been treated independently by Miermont [41] via a different method.”
Uniqueness and universality of the Brownian map (2011, background): “ary estimate relating the distances Dand D∗, which comes as an easy consequence of estimates for the volume of balls proved in [16] (a slightly diﬀerent approach to the result of Section 6 appears in [24]). Section 7 contains the proof of Theorem 1.1 in the bipartite case where qis even. The case of”
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”
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.”
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]).”
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].”
Recurrence of planar graph limits (2012, background): “the same limit, up to parametrization. For the case of random p-angulations this research has recentlyculminatedintheworkofLeGall[21]whoestablishedthisfor p =3and all even p and independentlyMiermont [27] for the case p =4. It remains open toprovethisforall p. Thestudyof distributionallimits,whilebe”
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].”