Hamiltonian Cycles on Random Eulerian Triangulations

TL;DR

This paper investigates how the properties of Hamiltonian cycles change when defined on random Eulerian triangulations, revealing a shift in the associated matter field's central charge and entropy exponent.

Contribution

It demonstrates that random Eulerian triangulations alter the central charge and entropy exponent of Hamiltonian cycle models, providing new insights into 2D quantum gravity coupled with matter fields.

Findings

Central charge increases by one on Eulerian triangulations.

Entropy exponent shifts from -1 to approximately -0.768.

Enumeration confirms the change in the entropy exponent.

Abstract

A random Eulerian triangulation is a random triangulation where an even number of triangles meet at any given vertex. We argue that the central charge increases by one if the fully packed O(n) model is defined on a random Eulerian triangulation instead of an ordinary random triangulation. Considering the case n -> 0, this implies that the system of random Eulerian triangulations equipped with Hamiltonian cycles describes a c=-1 matter field coupled to 2D quantum gravity as opposed to the system of usual random triangulations equipped with Hamiltonian cycles which has c=-2. Hence, in this case one should see a change in the entropy exponent from the value gamma=-1 to the irrational value gamma=(-1-\sqrt{13})/6=-0.76759... when going from a usual random triangulation to an Eulerian one. A direct enumeration of configurations confirms this change in gamma.