Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
Abstract
Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.
Keywords
Cite
@article{arxiv.0908.3944,
title = {Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)},
author = {Idan Oren and Amit Godel and Uzy Smilansky},
journal= {arXiv preprint arXiv:0908.3944},
year = {2015}
}
Comments
22 pages, 3 figures