Two-point spectral correlations for star graphs
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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 calculate two-point spectral statistics for a particular family of graphs, called star graphs, in the limit as the number of edges tends to infinity. Combinatorial techniques are used to evaluate both the diagonal (same orbit) and off-diagonal (different orbit) contributions to the sum over pairs of orbits involved. In this way, a general formula is derived for terms in the (short-time) expansion of the form factor K (τ) in powers of τ, and the first few are computed explicitly. The result demonstrates that K (τ) is neither Poissonian nor random-matrix, but intermediate between the two. Off-diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients.