Dresden 2014 – scientific programme

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SOE: Fachverband Physik sozio-ökonomischer Systeme

SOE 16: Networks - Statistics and Dynamics (joint with BP and DY)

SOE 16.14: Talk

Wednesday, April 2, 2014, 18:30–18:45, ZEU 118

Laplacian Spectrum of 2d Lattice Triangulations — •Ella Schmidt, Benedikt Krüger, and Klaus Mecke — Institut für Theoretische Physik, FAU Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen

Triangulations are an important tool in physics for describing curved geometries. Unimodular triangulations on 2d lattices can also be considered as connected, simple, plane graphs, which allows the appliance of methods from spectral graph theory on triangulations.

We calculate the distribution and averages of eigenvalues of the Laplacian matrix for random and highly ordered unimodular triangulations. Introducing a curvature energy of triangulations we measure microcanonical and canonical averages of the eigenvalues using Monte-Carlo-Simulations. We examine the probability distributions of the spectra of the ensembles of triangulations, the dependence of the eigenvalues on energy and temperature as well as the scaling with the lattice size and compare with random graph models.

In the microcanonical ensemble we find in agreement with our analytical predicitions a linear dependence of the algebraic connectivity and the spectral radius on the energy, in the canonical ensemble we encounter quasi-critical behaviour.