Distance statistics of block-weighted planar quadrangulations
Abstract
We study random planar quadrangulations in which each block -i.e., each simple (without multiple edges) component -is assigned a weight u. We derive an explicit expression for the distance-dependent two-point function, defined as the generating function of such block-weighted maps with two marked edges at a fixed graph distance. Using contour integral representations in two variables combined with a delicate saddle-point analysis, we compute the associated distance profile in the scaling limit of large quadrangulations. We recover the known phase transition at u = 9/5, characterized by distinct scaling exponents and different scaling functions, below, at, and above criticality. We also discuss the block distance profile, where the two marked edges are conditioned to lie within the same block.
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BibTeX
@misc{bouttier2026distance,
title = {{Distance statistics of block-weighted planar quadrangulations}},
author = {Jérémie Bouttier and Emmanuel Guitter and Hugo Manet},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06234},
eprint = {2609.06234},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05743939v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distance statistics of block-weighted planar quadrangulations
AU - Jérémie Bouttier
AU - Emmanuel Guitter
AU - Hugo Manet
PY - 2026
DA - 2026/09/05
PB - HAL
DO - 10.48550/arxiv.2609.06234
UR - https://hal.science/hal-05743939v1
AB - We study random planar quadrangulations in which each block -i.e., each simple (without multiple edges) component -is assigned a weight u. We derive an explicit expression for the distance-dependent two-point function, defined as the generating function of such block-weighted maps with two marked edges at a fixed graph distance. Using contour integral representations in two variables combined with a delicate saddle-point analysis, we compute the associated distance profile in the scaling limit of large quadrangulations. We recover the known phase transition at u = 9/5, characterized by distinct scaling exponents and different scaling functions, below, at, and above criticality. We also discuss the block distance profile, where the two marked edges are conditioned to lie within the same block.
ER -