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Local scaling limits of quadrangulations rooted on geodesics

Mathieu Mourichoux

Abstract

We identify the local scaling limit of the Uniform Infinite Planar Quadrangulation (UIPQ) and of critical Boltzmann quadrangulations, when one simultaneously rescales the distances and reroot them far away from the root of a distinguished geodesic. The limiting space is the bigeodesic Brownian plane, which appears as the local limit of the Brownian sphere around an interior point of a geodesic. We also show that the $\overline{\mathrm{UIPQ}}$ introduced by Dieuleveut, which is the local limit of the $\mathrm{UIPQ}$ rerooted at a far away point of its infinite geodesic, has the same scaling limit. These results can be seen as a commutation property between local limit, scaling limit and moving forward on a geodesic in random quadrangulations. The proofs are based on spinal decompositions of random trees and coupling results.

Local scaling limits of quadrangulations rooted on geodesics

Abstract

We identify the local scaling limit of the Uniform Infinite Planar Quadrangulation (UIPQ) and of critical Boltzmann quadrangulations, when one simultaneously rescales the distances and reroot them far away from the root of a distinguished geodesic. The limiting space is the bigeodesic Brownian plane, which appears as the local limit of the Brownian sphere around an interior point of a geodesic. We also show that the introduced by Dieuleveut, which is the local limit of the rerooted at a far away point of its infinite geodesic, has the same scaling limit. These results can be seen as a commutation property between local limit, scaling limit and moving forward on a geodesic in random quadrangulations. The proofs are based on spinal decompositions of random trees and coupling results.

Paper Structure

This paper contains 26 sections, 30 theorems, 176 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Planar maps
  4. Labeled trees
  5. Formalism about trees.
  6. Trees with labels.
  7. The CVS bijection
  8. For finite labeled trees.
  9. For infinite labelled trees.
  10. Random labeled trees
  11. The conditioned uniform labeled tree.
  12. The uniform infinite labeled tree.
  13. The infinite rerooted labeled tree.
  14. Boltzmann planar maps, the $\mathrm{UIPQ}$ and $\mathrm{\overline{UIPQ}}$
  15. The Brownian sphere and the bigeodesic Brownian plane
  16. ...and 11 more sections

Key Result

Theorem 1.1

We have in distribution for the local Gromov-Hausdorff topology. Moreover, for any sequence $(k_n)_{n\in\mathbb{N}}$ of non-negative real numbers such that $k_n\rightarrow\infty$ and $k_n=o(n)$, we have and in distribution for the local Gromov-Hausdorff topology.

Figures (8)

  • Figure 1: The diagram of convergences between the different surfaces. The arrows pointing down represent local limits, and those pointing right represent scaling limits.
  • Figure 2: An illustration of the CVS bijection when $\theta\in\mathbb{S}_{-}$.
  • Figure 3: An illustration of the CVS bijection when $\theta\in\mathbb{S}_{(1)}$.
  • Figure 4: Illustration of the sets $A_{\beta n}(\Theta_n)$ and $\widehat{A}_{\beta n}(\Theta_n)$, respectively in blue and red.
  • Figure 5: The random tree $\Theta_k$, obtained by gluing three subtrees
  • ...and 3 more figures

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.1: cori_vauquelin_1981schaeffer
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 47 more