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Constructing the Brownian sphere from a continuum random unicycle

Mathieu Mourichoux

TL;DR

The paper develops a new continuum construction of the Brownian sphere biased by the distance between two distinguished points, obtained as the scaling limit of delayed quadrangulations via a continuous Miermont bijection. It provides an explicit quotient representation of the biased Brownian sphere as a continuum labelled unicycle, and introduces a free model with Poissonian encoding to study conditioning on Voronoï-like cells and distance constraints. A key contribution is a new construction of the Brownian sphere with two distinguished points at a fixed distance, along with a detailed analysis of the associated $\,\Delta$-delayed Voronoï cells whose volumes follow a Beta distribution with parameters $(\tfrac{1}{4},\tfrac{1}{4})$. The results yield a novel approach to the bigeodesic Brownian plane, giving a direct construction from the biased sphere and establishing local convergence properties, along with insights into Voronoï geometry and local limits in Brownian geometry.

Abstract

We give an explicit construction of the Brownian sphere biased by the distance between two distinguished points, which is based on the Miermont bijection for quadrangulations. We then describe various conditionings of this object, which are related to Voronoï cells in the Brownian sphere. In particular, we give a new construction of the Brownian sphere with two distinguished points at a fixed distance. We also use this construction to derive a new representation of the bigeodesic Brownian plane.

Constructing the Brownian sphere from a continuum random unicycle

TL;DR

The paper develops a new continuum construction of the Brownian sphere biased by the distance between two distinguished points, obtained as the scaling limit of delayed quadrangulations via a continuous Miermont bijection. It provides an explicit quotient representation of the biased Brownian sphere as a continuum labelled unicycle, and introduces a free model with Poissonian encoding to study conditioning on Voronoï-like cells and distance constraints. A key contribution is a new construction of the Brownian sphere with two distinguished points at a fixed distance, along with a detailed analysis of the associated -delayed Voronoï cells whose volumes follow a Beta distribution with parameters . The results yield a novel approach to the bigeodesic Brownian plane, giving a direct construction from the biased sphere and establishing local convergence properties, along with insights into Voronoï geometry and local limits in Brownian geometry.

Abstract

We give an explicit construction of the Brownian sphere biased by the distance between two distinguished points, which is based on the Miermont bijection for quadrangulations. We then describe various conditionings of this object, which are related to Voronoï cells in the Brownian sphere. In particular, we give a new construction of the Brownian sphere with two distinguished points at a fixed distance. We also use this construction to derive a new representation of the bigeodesic Brownian plane.

Paper Structure

This paper contains 32 sections, 39 theorems, 260 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Combinatorial tools
  4. The CVS bijection and its generalizations
  5. The Miermont bijection for $n=2$.
  6. Quadrilaterals with geodesic sides.
  7. Link between delayed quadrangulations and quadrilaterals
  8. Study of well-labelled unicycles
  9. The biased Brownian sphere
  10. Some preliminaries
  11. The Gromov-Hausdorff-Prokhorov topology
  12. The Brownian sphere
  13. Continuous quadrilaterals
  14. The scaling limit of delayed quadrangulations
  15. First approach
  16. ...and 17 more sections

Key Result

Theorem 2.1

The mapping $\mathrm{CVS}:\mathbb{U}_n\rightarrow\mathbb{Q}_n^{(b)}$ is one-to-two.

Figures (6)

  • Figure 1: Illustration of the Miermont bijection. In this case, the delay is $-1$.
  • Figure 2: A vertebrate with width 3,upper area 2, lower area 5 and tilt 0. The dashed line corresponds to $\mathbb{R}$.
  • Figure 3: The quadrangulation with geodesic boundaries associated to the vertebrate of Figure \ref{['vertebrate']}.
  • Figure 4: A unicycle $\mathbf{u}^\bullet\in U_{10}^\bullet$, and the corresponding vertebrate $\Phi(\mathbf{u}^\bullet)$.
  • Figure 5: The correspondence between $\mathrm{CVS}(\mathbf{u}^\bullet)$ and $\mathrm{CVS}(\Phi(\mathbf{u}^\bullet))$, where $\mathbf{u}^\bullet$ is as in Figure \ref{['Unicycle vertebre']}.
  • ...and 1 more figures

Theorems & Definitions (83)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6: Link between biased quadrangulations and quadrangles
  • proof
  • Remark 2.7
  • Theorem 2.8
  • ...and 73 more