Spectral Properties of Unimodular Lattice Triangulations

Journal article


Publication Details

Author(s): Krüger B, Schmidt E, Mecke K
Journal: Journal of Statistical Physics
Publisher: SPRINGER
Publication year: 2016
Volume: 163
Journal issue: 3
Pages range: 514-543
ISSN: 0022-4715


Abstract

Random unimodular lattice triangulations have been recently used as an embedded random graph model, which exhibit a crossover behavior between an ordered, large-world and a disordered, small-world behavior. Using the ergodic Pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance, Markov chain Monte Carlo simulations can be applied to study these graphs. Here, we consider the spectra of the adjacency and the Laplacian matrix as well as the algebraic connectivity and the spectral radius. Power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states. For random triangulations we find a qualitative agreement of the spectral properties with well-known random graph models. In the microcanonical ensemble analytical approximations agree with numerical simulations. In the canonical ensemble a crossover behavior can be found for the algebraic connectivity and the spectral radius, thus combining large-world and small-world behavior in one model. The considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behavior allows a tuning of important transport quantities.


FAU Authors / FAU Editors

Krüger, Benedikt
Lehrstuhl für Theoretische Physik
Mecke, Klaus Prof. Dr.
Lehrstuhl für Theoretische Physik
Schmidt, Ella
Lehrstuhl für Kristallographie und Strukturphysik


How to cite

APA:
Krüger, B., Schmidt, E., & Mecke, K. (2016). Spectral Properties of Unimodular Lattice Triangulations. Journal of Statistical Physics, 163(3), 514-543. https://dx.doi.org/10.1007/s10955-016-1493-0

MLA:
Krüger, Benedikt, Ella Schmidt, and Klaus Mecke. "Spectral Properties of Unimodular Lattice Triangulations." Journal of Statistical Physics 163.3 (2016): 514-543.

BibTeX: 

Last updated on 2018-19-04 at 04:10