Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

Gall, Jean‐François Le; Paulin, Frédéric

doi:10.1007/s00039-008-0671-x

Cited by 103 publications

(155 citation statements)

References 26 publications

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“…In a work in preparation, we show that the limits S (β) g are supported on spaces which are homeomorphic to S g , which is a kind of higher-genus generalization of Le Gall and Paulin's result for uniform planar quadrangulations, see [26].…”

Section: Commentsmentioning

confidence: 66%

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Tessellations of random maps of arbitrary genus

Miermont¹

2009

Ann. Sci. École Norm. Sup.

133

187

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We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic.

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Section: Commentsmentioning

confidence: 66%

“…The random variable |D(q, (x, y))| = (d q (x, y) − 1) ∨ 0 is a continuous function of (m, l, t * ), since (26), (27) and (28) and the definition of w imply…”

Section: Proof Of Theoremmentioning

confidence: 99%

Tessellations of random maps of arbitrary genus

Miermont¹

2009

Ann. Sci. École Norm. Sup.

133

187

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“…The Brownian map was introduced in works by Marckert and Mokkadem and by Le Gall and Paulin [MM06,LGP08]. For a few years, the term "Brownian map" was often used to refer to any one of the subsequential Gromov-Hausdorff scaling limits of random planar maps.…”

Section: The Brownian Mapmentioning

confidence: 99%

Quantum Loewner evolution

Miller¹,

Sheffield⋆²

2016

Duke Math. J.

124

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What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ ∈ [0, 2].In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 , η). QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion ν t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ν t using an SPDE. For each γ ∈ (0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ν t .We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation.We propose QLE(2, 1) as a scaling limit for DLA on a random spanningtree-decorated planar map, and QLE(8/3, 0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3, 0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3, 0), up to a fixed time, as a metric ball in a random metric space.

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“…These properties are not used below. They will be derived in greater detail in the subsequent paper [23] where they play an important role. Clearly, the quotient space E n := [0, 1] /≈ n equipped with the metric δ n (s,…”

Section: Theorem 34 We Have Almost Surelymentioning

confidence: 99%

The topological structure of scaling limits of large planar maps

2007

Self Cite

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We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map M n which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of M n , equipped with the graph distance rescaled by the factor n −1/4 , converges in distribution as n → ∞ towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.

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Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

Abstract: We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk.

Cited by 103 publications

References 26 publications

Tessellations of random maps of arbitrary genus

Tessellations of random maps of arbitrary genus

Quantum Loewner evolution

The topological structure of scaling limits of large planar maps

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