Leading Off-Diagonal Correction to the Form Factor of Large Graphs

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, $K\left(\tau \right)$, of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution $-2\mathrm{}{\tau }^2$ to the form factor which agrees with random-matrix theory.