High-order level-spacing distributions for mixed systems

We apply some of the methods that have been successfully used to describe the nearest-neighbor-spacing distributions of levels of systems with mixed regular-chaotic dynamics to the calculation of high-order spacing distributions. The distributions for chaotic spectra are described in terms of a previously suggested generalization of Wigner's surmise, which assumes that the high-order level repulsion function is given by a product of the zero-order ones and that all of the spacing distributions are nearly Gaussian functions at large spacings. We compare the expressions obtained by the different methods for the next-nearest-neighbor spacing distribution with the outcome of a recently published numerical experiment on systems in transition between order and chaos. We show that the evolution of the shape of that distribution during the transition of the system from a chaotic to a regular regime is slower than the corresponding transition for the nearest-neighbor spacing distribution.