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The snake in the Brownian sphere

Omer Angel, Emmanuel Jacob, Brett Kolesnik, Grégory Miermont

TL;DR

The paper tackles the problem of inverting the continuum CVS mapping that encodes a Brownian sphere as a labeled pair on an Aldous CRT; it constructs a measurable inverse that, from the doubly marked Brownian sphere and an orientation sign, recovers a Brownian tree $(\mathcal T)$ and label process $Z$ whose Brownian-snake encoding maps back to the sphere. Central to the construction is the cut locus and relative geodesic structures, which allow the authors to reconstruct the entire tree and label process from the sphere's metric-measure data, modulo the orientation sign $\epsilon$. The results establish that the Brownian snake and CRT are fundamental encodings of the Brownian sphere, with the orientation captured by an independent random coin flip and measurability ensured via an extended space $m\mathcal M^{2\bullet}\times\{\pm1\}$. This links the discrete CVS bijection with continuum limits and broader random-surfaces frameworks, highlighting the universality of the Brownian tree/snake as organizing objects in random geometry.

Abstract

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

The snake in the Brownian sphere

TL;DR

The paper tackles the problem of inverting the continuum CVS mapping that encodes a Brownian sphere as a labeled pair on an Aldous CRT; it constructs a measurable inverse that, from the doubly marked Brownian sphere and an orientation sign, recovers a Brownian tree and label process whose Brownian-snake encoding maps back to the sphere. Central to the construction is the cut locus and relative geodesic structures, which allow the authors to reconstruct the entire tree and label process from the sphere's metric-measure data, modulo the orientation sign . The results establish that the Brownian snake and CRT are fundamental encodings of the Brownian sphere, with the orientation captured by an independent random coin flip and measurability ensured via an extended space . This links the discrete CVS bijection with continuum limits and broader random-surfaces frameworks, highlighting the universality of the Brownian tree/snake as organizing objects in random geometry.

Abstract

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

Paper Structure

This paper contains 23 sections, 12 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The construction
  3. The marked points
  4. Choosing an orientation
  5. Other Brownian surfaces
  6. Organization of the paper
  7. Acknowledgments
  8. Random snakes and the Brownian sphere
  9. Deterministic encodings
  10. Trees coded by functions
  11. The mating of trees metric
  12. Gromov--Hausdorff-type spaces
  13. The continuum CVS mapping $\psi$
  14. The Brownian tree, snake and sphere
  15. The inverse continuum CVS mapping $\phi$
  16. ...and 8 more sections

Key Result

Theorem 1

Let $(X,d,\mu)$ be the Brownian sphere, and let $x^0,x^1$ be two independent points in $X$, drawn according to the measure $\mu$. Then, almost surely, there exists a measurable function of $(X,d,\mu,x^0,x^1)$, that outputs an ${\mathbb R}$-tree ${\mathcal{T}}$ and a label function $Z:{\mathcal{T}}\t

Figures (3)

  • Figure 1: Some branches of the tree $\Gamma_W$ of relative interiors of geodesics toward $x_W^1$ are represented in blue, and a part of the the cut locus $C_W$ is represented in red. The two distinguished points $x_W^0$ and $x_W^1$ are almost surely not in $\Gamma_W\cup C_W$, and we have represented two more points $x,y$ outside this set.
  • Figure 2: From the points $x,y\in \widetilde{X}_W$, there is a unique geodesic pointing towards $x^1_W$, which allows to identify $\Gamma_W(x,y)$, represented by the thick dark blue line. The curve $C_W(x,y)$, represented by the thick dark red line, consists of points from which we can find (at least) two geodesics pointing towards $x^1_W$ that separate $x$ from $y$. One of these points, called $z$, is highlighted together with the two relevant geodesics.
  • Figure 3: The curve $\gamma(x)$ is illustrated in thick lines, and the domain $D_x=p_W([0,p_W^{-1}(x)])$ is the gray area. If we choose to orient the curve $\gamma(x)$ by first following $C_W(x_W^0,x)$ from $x_W^0$ to $x$, then the Brownian sphere is canonically oriented in such a way that $D_x$ is circled counterclockwise by $\gamma(x)$, for any choice of $x\in \widetilde{X}_W\setminus \{x_W^0\}$.

Theorems & Definitions (19)

  • Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • Theorem 4
  • Proposition 5
  • proof
  • Corollary 6
  • Remark
  • Corollary 7
  • ...and 9 more