Correlations between Periodic Orbits and their Rôle in Spectral Statistics

We consider off-diagonal contributions to double sums over periodic orbits that arise in semiclassical approximations for spectral statistics of classically chaotic quantum systems. We identify pairs of periodic orbits whose actions are strongly correlated. For a class of systems with uniformly hyperbolic dynamics, we demonstrate that these pairs of orbits give rise to a τ² contribution to the spectral form factor K(τ) which agrees with random matrix theory. Most interestingly, this contribution has its origin in a next-to-leading-order behaviour of a classical distribution function for long times.