Critical limits of generating functions for face-constrained labelled planar maps
Abstract
We study connected planar maps with a fixed number of numbered faces, whose vertices carry integer labels such that the minimum label along the boundary of each face is one. Each face has its own positive weight, and an edge shared by two faces receives the geometric mean of their weights. For any fixed number k≥3 of faces, we prove that the suitably rescaled generating function converges locally uniformly as the face weights approach the critical value 1/12, possibly at different positive rates. The limit is a finite, continuous sum of integrals over cubic schemes. Each scheme edge carries a Green kernel for a singular differential operator on the positive half-line, and each face contributes one insertion of the derivative of the tree scaling profile. The main difficulty is obtaining uniform estimates for small vertex labels, where the minimum-label constraints of different faces can interact. A positive corner-marking expansion reduces this problem to establishing uniform bounds on the contributions of marked configurations, both for small and large vertex labels. These estimates prove integrability of the surviving diagrams and show that coincident marks and noncubic schemes contribute only lower-order terms.
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BibTeX
@misc{sun2026critical,
title = {{Critical limits of generating functions for face-constrained labelled planar maps}},
author = {Zhenyao Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25322},
eprint = {2610.25322},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25322},
}
RIS
TY - UNPB
TI - Critical limits of generating functions for face-constrained labelled planar maps
AU - Zhenyao Sun
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25322
UR - https://arxiv.org/abs/2610.25322
AB - We study connected planar maps with a fixed number of numbered faces, whose vertices carry integer labels such that the minimum label along the boundary of each face is one. Each face has its own positive weight, and an edge shared by two faces receives the geometric mean of their weights. For any fixed number k≥3 of faces, we prove that the suitably rescaled generating function converges locally uniformly as the face weights approach the critical value 1/12, possibly at different positive rates. The limit is a finite, continuous sum of integrals over cubic schemes. Each scheme edge carries a Green kernel for a singular differential operator on the positive half-line, and each face contributes one insertion of the derivative of the tree scaling profile. The main difficulty is obtaining uniform estimates for small vertex labels, where the minimum-label constraints of different faces can interact. A positive corner-marking expansion reduces this problem to establishing uniform bounds on the contributions of marked configurations, both for small and large vertex labels. These estimates prove integrability of the surviving diagrams and show that coincident marks and noncubic schemes contribute only lower-order terms.
ER -