Two-point spectral correlations for star graphs

The eigenvalues of the Schrodinger operator on a graph G are related via an exact trace formula to periodic orbits on G. This connection is used to ca lculate two-point spectral statistics for a particular family of graphs, ca lled star graphs, in the limit as the number of edges tends to infinity. Co mbinatorial techniques are used to evaluate both the diagonal (same orbit) and off-diagonal (different orbit) contributions to the sum over pairs of o rbits involved. In this way, a general formula is derived for terms in the (short-time) expansion of the form factor K(tau) in powers of tau, and the first few are computed explicitly. The result demonstrates that K(tau) is n either Poissonian nor random matrix, but an intermediate between the two. O ff-diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients.