The six-vertex model on random planar maps revisited

The six-vertex model on random lattices

In this letter, the 6-vertex model on dynamical random lattices is defined via a matrix model and rewritten (following I. Kostov) as a deformation of the O(2) model. In the large N planar limit, an exact solution is found at criticality. The critical exponents of the model are determined; they vary continously along the critical line. The vicinity of the latter is explored, which confirms that ...

Vertex Degrees in Planar Maps

We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. We also discuss the possible extension to maps of higher genus.

Symmetric Vertex Models on Planar Random Graphs

We discuss a 4-vertex model on an ensemble of 3-valent (Φ) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The regular lattice equivalent of the model is the symmetric 8-vertex model on the honeycomb lattice, which can be mapped on to an Ising model in field, as was originally shown by Wu et.al. using generalised weak graph transformation technique...

The Order Dimension of Planar Maps Revisited

Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the Brightwell-Trotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyder-path...

Random cubic planar graphs revisited