  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.
