An axiomatic characterization of the Brownian map

Miller, Jason; Sheffield⋆, Scott

doi:10.48550/arxiv.1506.03806

Cited by 17 publications

(58 citation statements)

References 41 publications

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“…In the same direction, there are close relations between the present article and the recent work of Miller and Sheffield [32,33,34] aiming at proving the equivalence of the Brownian map and Liouville quantum gravity with parameter γ = 8/3. In particular, the paper [32] uses what we call Brownian snake excursions above the minimum to define the notion of a Brownian disk, corresponding to bubbles appearing in the exploration of the Brownian map: See the definition of µ L DISK in Proposition 4.4, and its proof, in [32].…”

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confidence: 68%

Excursion theory for Brownian motion indexed by the Brownian tree

Abraham¹,

Gall²

2015

Preprint

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We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connected component is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism ψ(u) = 8/3 u 3/2 . Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure M0 which is the analog of the Itô measure of Brownian excursions. We provide various descriptions of the excursion measure M0, and we also determine several explicit distributions, such as the joint distribution of the boundary length and the mass of an excursion under M0. We use the Brownian snake as a convenient tool for defining and analysing the excursions of our tree-indexed Brownian motion.

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confidence: 68%

Excursion theory for Brownian motion indexed by the Brownian tree

Abraham¹,

Gall²

2015

Preprint

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“…It is widely believed or proven in certain cases to provide, after a Riemann conformal map to a given planar domain, the correct conformal structure for the continuum limit of random planar maps, possibly weighted by the partition functions of various statistical models (see, e.g., the ICM reviews [98,47,103]). In the case of usual random planar maps with faces of bounded degrees, the universal metric structure is that of the Brownian map [97,102], which has been recently identified with that directly constructed from Liouville quantum gravity (LQG) [104,105,106,107]. Note also that different mathematical approaches to LQG exist [56,51,108,35], whose equivalence has been recently established [4].…”

Section: Comparison With Nesting In Cle Via Kpzmentioning

confidence: 99%

“…As of now, the geometry of large random planar maps with faces of bounded degrees (e.g., quadrangulations) is fairly well understood, thanks to recent spectacular progress. In particular, their scaling limit is the so called Brownian map [101,96,102,97,104], with its convergence in the Gromov-Hausdorff sense established by Le Gall and Miermont in Refs. [97,102].…”

Section: Introductionmentioning

confidence: 99%

Nesting statistics in the $O(n)$ loop model on random planar maps

Borot¹,

Bouttier²,

Duplantier³

2016

Preprint

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In the O(n) loop model on random planar maps, we study the depthin terms of the number of levels of nesting -of the loop configuration, by means of analytic combinatorics. We focus on the 'refined' generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O(n) loop model, we show that these generating series satisfy functional relations obtained by a modification of those satisfied by the unrefined generating series. In a more specific O(n) model where loops cross only triangles and have a bending energy, we explicitly compute the refined generating series. We analyze their non generic critical behavior in the dense and dilute phases, and obtain the large deviations function of the nesting distribution, which is expected to be universal. Using the framework of Liouville quantum gravity (LQG), we show that a rigorous functional KPZ relation can be applied to the multifractal spectrum of extreme nesting in the conformal loop ensemble (CLE κ ) in the Euclidean unit disk, as obtained by Miller, Watson and Wilson [111], or to its natural generalization to the Riemann sphere. It allows us to recover the large deviations results obtained for the critical O(n) random planar map models. This offers, at the refined level of large deviations theory, a rigorous check of the fundamental fact that the universal scaling limits of random planar map models as weighted by partition functions of critical statistical models are given by LQG random surfaces decorated by independent CLEs.

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“…In [7], it was shown that the collection of perimeters of the holes observed when slicing Boltzmann triangulations at all heights converges, when properly rescaled, towards a particular self-similar growth-fragmentation. We also mention Theorems 3 and 23 of Le Gall and Riera [27], which show that, when slicing directly the free Brownian disk, the holes' perimeters are described by the same growth-fragmentation as in [6] (see also [32] Section 4). When X starts from a single cell of size 0 that grows indefinitely, the geometrical connection corresponds this time to the holes in a sliced discrete approximation of the Brownian plane.…”

Section: Introductionmentioning

confidence: 91%

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

Ged¹

2019

Preprint

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We study the behaviour of a natural measure defined on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves in continuous time according to a positive self-similar Markov process and gives birth to children at negative jumps events. We are interested in the asymptotics of the mass of the ball centered at the root, as its radius decreases to 0. We obtain the almost sure behaviour of this mass when the Eve cell starts with a strictly positive size. This differs from the situation where the Eve cell grows indefinitely from size 0. In this case, we show that, when properly rescaled, the mass of the ball converges in distribution towards a non-degenerate random variable. We then derive bounds describing the almost sure behaviour of the rescaled mass. Those results are applied to certain random surfaces, exploiting the connection between growth-fragmentations and random planar maps obtained in [6]. This allows us to extend a result of Le Gall [24] on the volume of a free Brownian disk close to its boundary, to a larger family of stable disks. The upper bound of the mass of a typical ball in the Brownian map is refined, and we obtain a lower bound as well.

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An axiomatic characterization of the Brownian map

Cited by 17 publications

References 41 publications

Excursion theory for Brownian motion indexed by the Brownian tree

Excursion theory for Brownian motion indexed by the Brownian tree

Nesting statistics in the $O(n)$ loop model on random planar maps

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

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