The topological structure of scaling limits of large planar maps

Gall, Jean‐François Le

doi:10.1007/s00222-007-0059-9

Cited by 161 publications

(287 citation statements)

References 30 publications

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“…In this case, our arguments do support the notion the Hausdorff dimension of Liouville quantum gravity should be 4 for γ = 8/3, though we will not prove this statement in this paper. This is consistent with the dimension of the Brownian map [CS04,LG07] A radial SLE κ in D starting from 1 and targeted at 0 is described by the random family of conformal maps obtained by solving the radial Loewner ODĖ…”

Section: Free Boundary Gff and Quantum Zipper αsupporting

confidence: 76%

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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Section: Free Boundary Gff and Quantum Zipper αsupporting

confidence: 76%

Quantum Loewner evolution

Miller¹,

Sheffield⋆²

2016

Duke Math. J.

124

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“…From now on, we will always suppose that a is taken along this sequence. From this and the discussion above (25), we obtain that the laws of (a…”

Section: Proof Of Theoremmentioning

confidence: 63%

“…However, this would involve technical complications. Note that in the case g = 0, a result similar to Theorem 2 appears in Le Gall [25] in such a conditioned setting, though the convergence does not really take µ q into account (see however [25, Remark a., p.646]).…”

Section: Commentsmentioning

confidence: 99%

“…An important step has been accomplished by Le Gall [25], who showed that scaling limits of these random quadrangulations, considered in the Gromov-Hausdorff sense [21], must be homeomorphic to a metric quotient of the so-called Brownian Continuum Random Tree of Aldous [1]. This random quotient was introduced under the name of Brownian map by Marckert & Mokkadem [30], who proved convergence of random quadrangulations towards this limit in a sense yet different from convergence in distribution in the Gromov-Hausdorff topology.…”

Section: Introduction 1motivationmentioning

confidence: 99%

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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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“…This is very similar in spirit to Le Gall's construction of the Brownian map [LG07,LG10], starting from Aldous's continuous random tree [Ald91]. The scaling properties of the Brownian fan however are quite different.…”

Section: Construction Of the Brownian Fanmentioning

confidence: 53%

The Brownian Fan

Hairer

Weare

2014

Comm Pure Appl Math

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We provide a mathematical study of the modified Diffusion Monte Carlo (DMC) algorithm introduced in the companion article [HW14]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with FeynmanKac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "naïve" generalisation of the DMC algorithm which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan", is a very natural new mathematical object of independent interest.

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The topological structure of scaling limits of large planar maps

Cited by 161 publications

References 30 publications

Quantum Loewner evolution

Quantum Loewner evolution

Tessellations of random maps of arbitrary genus

The Brownian Fan

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