Quantum Ergodicity on Graphs

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.

Figure 1

Deviations ${\tilde{F}}_V$ from quantum ergodicity for sequences of complete graphs. Data points refer to numerical calculations, and lines refer to the saddle-point approximation. From top to bottom: Complete Neumann graph ($\beta =2$, red), complete Neumann graph with magnetic field ($\beta =1$, light blue), complete DFT graph ($\beta =2$, dark blue), and complete DFT graph with magnetic field ($\beta =1$, green).