Expansion of the form factor of quantum graphs with the symmetric group

Abstract: <br /> <br /> The form factor is the Fourier transform of the two point correlation function for eigenvalues of the scattering matrix of a quantum graph. Writing it as a sum over the symmetric group gives some insight into the origin of the diagrammatic expansions previously used to study the form factor. The appearance of the symmetric group also allows a comparison with averages in random matrix theory. Choosing the Fourier transform scattering matrix, elements in the expansion of the form factor are reduced to a modified form of a multidimensional Gauss sum. Such sums depend only on the rank of the related quadratic form which we hope to exploit to obtain new results in this special case.