2017/11/20 by Emmanuel Guitter
Mathematics · Physics and Astronomy · #Brownian motion #Combinatorics #Computer science #Geometry #Graph #Limit (mathematics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Perimeter #Planar #Scaling #Scaling limit #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Vertex (graph theory) #math-ph #math.CO #math.MP
paper · pdf · doi:10.1088/1742-5468/aacfbc
published as J. Stat. Mech. (2018) 073409 · 20 pages, 9 figures
arxiv created 2017/11/20 · openalex publication_date 2018/07/01 · arxiv updated 2018/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract We study the statistics of the Voronoï cell perimeter in large bi-pointed planar quadrangulations. Such maps have two marked vertices at a fixed given distance 2 s and their Voronoï cell perimeter is simply the length of the frontier which separates vertices closer to one marked vertex than to the other. We characterize the statistics of this perimeter as a function of s for maps with a large given volume N both in the scaling limit where s scales as N 1/4 , in which case the Voronoï cell perimeter scales as N 1/2 , and in the local limit where s remains finite, in which case the perimeter scales as s 2 for large s . The obtained laws are universal and are characteristics of the Brownian map and the Brownian plane respectively.