Eigenfunction Statistics on Quantum Graphs

TL;DR

This paper analyzes the statistical properties of eigenfunctions on large quantum graphs, demonstrating a universal Gaussian Random Wave Model with system-dependent deviations linked to classical dynamics, and establishing conditions for quantum universality.

Contribution

It provides a microscopic proof of the Gaussian Random Wave Model for eigenfunctions on quantum graphs, including criteria for universality and detailed correlation analysis.

Findings

Universal component of autocorrelation functions identified

Deviations depend on classical dynamics of the graph

Asymptotic equidistribution of eigenfunctions predicted

Abstract

We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for which such a model was proposed by Berry in 1977. The autocorrelation functions we calculate for an individual quantum graph exhibit a universal component, which completely determines a Gaussian Random Wave Model, and a system-dependent deviation. This deviation depends on the graph only through its underlying classical dynamics. Classical criteria for quantum universality to be met asymptotically in the large graph limit (i.e. for the non-universal deviation to vanish) are then extracted. We use an exact field theoretic expression in terms of a variant of a supersymmetric sigma model. A saddle-point analysis of this expression leads to the estimates. In particular, intensity correlations are used to discuss the possible equidistribution of the energy eigenfunctions in the large graph limit. When equidistribution is asymptotically realized, our theory predicts a rate of convergence that is a significant refinement of previous estimates. The universal and system-dependent components of intensity correlation functions are recovered by means of an exact trace formula which we analyse in the diagonal approximation, drawing in this way a parallel between the field theory and semiclassics. Our results provide the first instance where an asymptotic Gaussian Random Wave Model has been established microscopically for eigenfunctions in a system with no disorder.