Geodesic distance in planar graphs

Abstract

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. 03/031

Geodesic Distance in Planar Graphs: An Integrable Approach

“…In [13], it was shown that the geodesic distance of all vertices from an origin vertex of the graph may translate into integer vertex labels in some corresponding trees, themselves realizing a discrete version of the Brownian snake, further studied and extended in [14] in the language of spatial branching processes. In [15], it was shown that the generating functions for planar graphs with two external legs obeyed non-linear recursion relations on the maximal geodesic distance between the legs, and it is the purpose of this note to clarify and extend the results of this paper.…”

Integrable Combinatorics

“…The first model may be equipped with an infinite family of commuting transfer matrices [9] governing the time-evolution of the surfaces, thus displaying quantum integrability, with an infinite number of conserved quantities. The second model actually addresses the correlations of marked points on the discrete surface at prescribed geodesic distance, and we show that such correlations viewed as evolutions in the geodesic distance variable, form a discrete classical integrable system with some conserved quantities modulo the equations of motion [8]. Remarkably, both cases can be reduced bijectively to statistical ensembles of trees.…”

3-D Face Recognition Using Geodesic-Map Representation and Statistical Shape Modelling

“…A geodesic is a generalization of an Euclidean distance and is defined as the length of the shortest path between two points along a continuous surface [22]. Bronstein et al [23] proposed a face recognition method based on transformation, ψ, mapping an original face S with the given geodesic distance d S (ξ 1 , ξ 2 ) onto another space S with the Euclidean distance d S (ψ(ξ 1 ), ψ(ξ 2 )) in such a way that corresponding distances are preserved:…”

“…For the faces from the same person, this can be achieved by using cross-correlation [22] in which geodesic-map's column of the given face is cross-correlated with the target face geodesic-map. The geodesic-map's column of the target face with the highest cross-correlation value is considered as being in correspondence with the point in questions from the given face as shown in Figure 5.…”