Martingales in self-similar growth-fragmentations and their connections with random planar maps

Bertoin, Jean; Budd, Timothy A.; Curien, Nicolas; Kortchemski, Igor

doi:10.48550/arxiv.1605.00581

Cited by 11 publications

(51 citation statements)

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“…The latter fact may be viewed as a continuous analog of a recent result of Bertoin, Curien and Kortchemski [6] identifying the growth-fragmentation process arising as the scaling limit for the collection of lengths of cycles obtained by slicing random Boltzmann triangulations with a boundary at a given height, when the size of the boundary grows to infinity. In fact, the growth-fragmentation process of [6] is the same as in our main results, and this strongly suggests that the results of [6] could be extended to more general planar maps with a boundary (see also [5] for related results).…”

Section: Introductionsupporting

confidence: 83%

“…Let us now discuss growth-fragmentation processes, referring to [4] and [5] for additional details. The basic ingredient in the construction of a (self-similar) growth-fragmentation process is a self-similar Markov process (X t ) t≥0 with values in [0, ∞) and only negative jumps, which is stopped upon hitting 0.…”

Section: Introductionmentioning

confidence: 99%

“…Rather than writing down this statement about the free Brownian map, we give in Section 11 an analog of Theorem 3 for the Brownian plane, which is an infinite-volume version of the Brownian map that has been shown [10,14] to be the universal scaling limit of infinite random lattices such as the UIPT or the UIPQ. Theorem 23 below shows that the collection of boundary sizes of the connected components of the complement of the ball of radius r centered at the root of the Brownian plane evolves like the same growth-fragmentation process with indefinite growth starting from 0 (see [5,Section 4.2] for a thorough discussion of this process).…”

Section: Introductionmentioning

confidence: 99%

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Excursion theory for Brownian motion indexed by the Brownian tree

Abraham

Gall

2018

J. Eur. Math. Soc.

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We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion. IntroductionThe main purpose of the present work is to derive certain explicit distributions for the random process which we call Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts. As a key tool for the derivation of our main results we use the excursion theory developed in [1] for Brownian motion indexed by the Brownian tree. In many respects, this excursion theory is similar to the classical Itô theory, which applies in particular to linear Brownian motion and has proved a powerful tool for the calculation of exact distributions of Brownian functionals.Let us briefly describe the objects of interest in this work. We define the Brownian tree T ζ as the random compact R-tree coded by a Brownian excursion ζ = (ζ s ) s≥0 distributed according to the (infinite) Itô measure of positive excursions of linear Brownian motion. If σ stands for the duration of the excursion ζ, this coding means that T ζ is the quotient space of [0, σ] for the equivalence relation defined by s ∼ s ′ if and only if ζ s = ζ s ′ = m ζ (s, s ′ ), where m ζ (s, s ′ ) := min{ζ r : s ∧ s ′ ≤ r ≤ s ∨ s ′ }, and this quotient space is equipped with the metric induced by d ζ (s, s ′The volume measure vol(da) on T ζ is defined as the push forward of Lebesgue measure on [0, σ] under the canonical projection, and the root ρ of T ζ is the equivalence class of 0. We note that under the conditioning by σ = 1 (equivalently the total volume is equal to 1) the tree T ζ is Aldous' Brownian Continuum Random Tree (also called the CRT, see [2,3]), up to an unimportant scaling factor 2.Let us turn to Brownian motion indexed by T ζ . Informally, given T ζ , this is the centered Gaussian process (V a ) a∈T ζ such that V ρ = 0 and Var(V a − V b ) = d ζ (a, b) for every a, b ∈ T ζ . This definition is a bit informal since we are dealing with a random process indexed by a random set. These difficulties can be overcome easily by using the Brownian snake approach. We let (W s ) s≥0 be the Brownian snake (whose spatial motion is linear Brownian motion started at 0) driven by the Brownian excursion (ζ s ) s≥0 . Then, for every s ≥ 0, W s is a finite path started at 0 and with lifetime ζ s , and for every a ∈ T ζ we may define V a as the terminal point W s of the path W s , for any s ∈ [0, σ] such that a is the equivalence class of s in T ζ . The Brownian snake approach thus reduces the study of a tree-indexed Brownian motion to that a process indexed by the positive half-line, and we systematically use this approach in the next sections.The total occupation measure Θ(dx) of (V a ) a∈T ζ is the push forward of vol(da) under the mapping a → V a , or equivalently the push forward of Lebesgue measure on [0, σ] under s → W s . Under the special conditioning...

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Section: Introductionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Excursion theory for Brownian motion indexed by the Brownian tree

Abraham

Gall

2018

J. Eur. Math. Soc.

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“…Recall that a positive function is slowly varying (at infinity) if it satisfies (λx)/ (x) → 1 as x → ∞, for every λ > 0. We emphasize that Definition 2.2 is more general than that of [46], which implies that the slowly varying function is asymptotically constant (and is also the framework in [14,19,9,24]).…”

Section: Boltzmann Distributions 21 Boltzmann Distributions On Bipart...mentioning

confidence: 99%

“…These are supposed to be very different from the Brownian map because of large faces that remain present in the scaling limit. Their duals have been recently studied in [19,9], but their geometry remains widely unknown. The stable maps are believed to undergo a phase transition at α = 3/2.…”

Section: Introductionmentioning

confidence: 99%

Limits of the boundary of random planar maps

Richier

2017

Probab. Theory Relat. Fields

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We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter α ∈ (1, 2). First, in the dense phase corresponding to α ∈ (1, 3/2), we prove that the scaling limit of the boundary is the random stable looptree with parameter 1/(α − 1/2). Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to α ∈ (3/2, 2), it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid O(n) loop model on quadrangulations, proving thereby a conjecture of Curien & Kortchemski.

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Self-similar Growth Fragmentations as Scaling Limits of Markov Branching Processes

Dadoun

2019

J Theor Probab

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Martingales in self-similar growth-fragmentations and their connections with random planar maps

Cited by 11 publications

References 0 publications

Excursion theory for Brownian motion indexed by the Brownian tree

Excursion theory for Brownian motion indexed by the Brownian tree

Limits of the boundary of random planar maps

Self-similar Growth Fragmentations as Scaling Limits of Markov Branching Processes

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