Wigner surmise for high-order level spacing distributions of chaotic systems

We suggest an extension of the Wigner surmise for the nearest-neighbor-spacing distribution of energy levels of chaotic systems to include the nth-order spacing distributions. The main assumption is that the conditional probability density of occurrence of a level at a given distance from a fixed level, provided that this distance contains n levels, is expressed in terms of the (n+1)th power of corresponding probability for a distance containing no levels. At large spacings, the nth-order level distributions are assumed to have a Gaussian shape as in the cases covered by the Wigner surmise. The expressions obtained are in good agreement with the results of numerical calculation by means of the random matrix theory for the three universal classes of symmetry.