Probability, Curvature and Spectrum on Graphs

Abstract:We show that, for a metric graph equipped with the Laplacian operator $\Delta=-\frac{d^2}{dx^2}$, the graph trace formula admits a new interpretation in terms of quantum probability and curvature. Our approach is based on a notion of graph curvature inspired by Wilson-lines and holonomy, together with von-Neumann's ergodic theorem.

Mathematical Physics (math-ph); Probability (math.PR); Spectral Theory (math.SP); Quantum Physics (quant-ph)

arXiv:2607.21639 [math-ph]

(or arXiv:2607.21639v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2607.21639 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.