Universal spectral form factor for chaotic dynamics

We consider the semiclassical limit of the spectral form factor K (τ) of fully chaotic dynamics. Starting from the Gutzwiller-type double sum over classical periodic orbits we set out to recover the universal behaviour predicted by random-matrix theory, both for dynamics with and without time reversal invariance. For times smaller than half the Heisenberg time T H ∝ ℏ − f +1 , we extend the previously known τ-expansion to include the cubic term. Beyond confirming the random-matrix behaviour of individual spectra, the virtue of that extension is that the 'diagrammatic rules' come in sight which determine the families of orbit pairs responsible for all orders of the τ-expansion.