PreprintarXiv licencearXiv
A marked scheme representation for Voronoi volumes in the Brownian map
Abstract
For any fixed number k≥3 of independent volume-sampled sites in the unit-area Brownian map, we represent the negative-power transform of their ordinary Voronoi volumes as a finite sum of marked planar scheme integrals. These integrals arise from the critical asymptotics of generating functions for face-constrained labelled planar maps. To connect them to Voronoi volumes, we use Miermont's zero-delay correspondence to interpret the labelled maps as quadrangulations with distinguished sites and then pass to the Brownian map scaling limit.
Related
- Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion via Fractional Calculus Techniques
- Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion
- Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces
- A Lloyd-stabilized Voronoï particle method
- The optimal four-dimensional lattice quantizer and bounds from Voronoi facets
BibTeX
@misc{sun2026marked,
title = {{A marked scheme representation for Voronoi volumes in the Brownian map}},
author = {Zhenyao Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34255},
eprint = {2610.34255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34255},
}
RIS
TY - UNPB
TI - A marked scheme representation for Voronoi volumes in the Brownian map
AU - Zhenyao Sun
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34255
UR - https://arxiv.org/abs/2610.34255
AB - For any fixed number k≥3 of independent volume-sampled sites in the unit-area Brownian map, we represent the negative-power transform of their ordinary Voronoi volumes as a finite sum of marked planar scheme integrals. These integrals arise from the critical asymptotics of generating functions for face-constrained labelled planar maps. To connect them to Voronoi volumes, we use Miermont's zero-delay correspondence to interpret the labelled maps as quadrangulations with distinguished sites and then pass to the Brownian map scaling limit.
ER -