Local scaling limits of quadrangulations rooted on geodesics

2025/11/20 by Mourichoux, Mathieu
Mathematics · Physics and Astronomy · #60C05 #60D05 #60F99 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · doi:10.48550/arxiv.2511.16538

openalex publication_date 2025/11/20 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

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 UIPQ introduced by Dieuleveut, which is the local limit of the 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.

Citations

• Compact Brownian surfaces II. Orientable surfaces
• Environment seen from infinite geodesics in Liouville Quantum Gravity
• Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation
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• Rescaled bipartite planar maps converge to the Brownian map
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• The Brownian plane
• Uniform infinite planar quadrangulations with a boundary
• The Brownian map is the scaling limit of uniform random plane\n quadrangulations
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• On the sphericity of scaling limits of random planar quadrangulations
• Invariance principles for spatial multitype Galton-Watson trees
• Local structure of random quadrangulations
• Random Trees, Levy Processes and Spatial Branching Processes

Related

• Classification of scaling limits of uniform quadrangulations with a\n boundary
• The Brownian map is the scaling limit of uniform random plane\n quadrangulations
• Scaling Limit of Random Planar Quadrangulations with a Boundary
• On the sphericity of scaling limits of random planar quadrangulations
• Scaling limit of random plane quadrangulations with a simple boundary, via restriction