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.

22 pages, 3 figures

References in corpus (5)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Universal spectral form factor for chaotic dynamics
• Discrete path integral approach to the trace formula for regular graphs
• The Tracy--Widom law for some sparse random matrices
• Quantum graphs where back-scattering is prohibited

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