Geodesic distance in planar graphs
By J. Bouttier, P. Francesco, E. Guitter. 2003
168 citations recorded by the source corpus.
We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy.
Subfields: Condensed Matter Physics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/12a3db70028a5eec590dc1575bb6f6609902c889
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Condensed Matter Physics
- Bergfinnur Durhuus
- Jean-François Le Gall
- Philippe Chassaing
- Igor Kortchemski
- C. Abraham
- Douglas Rizzolo
- Amaury Lambert
- Christina Goldschmidt
In Mathematical Physics
- Jean-François Le Gall
- Nicolas Curien
- Grégory Miermont
- Bergfinnur Durhuus
- Robert Sedgewick
- Philippe Flajolet
- Philippe Chassaing
- Jérémie Bettinelli
Citations in the corpus
- Multiscale Embedded Gene Co-expression Network Analysis (2015, unknown): “SPD is defined as distances of shortest geodesic paths among all nodes in a given network[64], thus can be naturally used to represent the degree of compactness for a given set of nodes by summarizing overall distances between the nodes on the network.”
- Random real trees (2006, background): “Large random quadrangulations, or more general planar maps, are used in theoretical physics as models of random surfaces (see in particular [9], [10] and [24]).”
- Analytic Combinatorics (2009, background): “Bouttier, Di Francesco, and Guitter [90, 91] found an amazing explicit f orm for theTj ; namely,”
- Local limit of labeled trees and expected volume growth in a random quadrangulation (2003, methodology): “This method was applied to triangulated planar surfaces (or pure two-dimensional quantum gravity) in [6], where it was argued that α = 4 (see also [10]).”