Quantum ergodicity on graphs
Abstract
We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic expression in terms of a variant of the supersymmetric nonlinear sigma-model. Our estimate is based on a saddle-point analysis of this expression and leads to a criterion for when equidistribution emerges asymptotically in the limit of large graphs. Our theory predicts a rate of convergence that is a significant refinement of previous estimates, long-assumed to be valid for quantum chaotic systems, agreeing with them in some situations but not all. We discuss specific examples for which the theory is tested numerically.
Cite
@article{arxiv.0808.4110,
title = {Quantum ergodicity on graphs},
author = {S. Gnutzmann and J. P. Keating and F. Piotet},
journal= {arXiv preprint arXiv:0808.4110},
year = {2009}
}
Comments
4 pages, 1 figure