Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Centre National de la Recherche Scientifique, Université Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Commissariat à l'Énergie Atomique et aux Énergies Alternatives, Université Paris-Saclay, Institut de Physique Théorique, Institut de Recherche en Informatique Fondamentale
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212033332