Higher-order level spacings in random matrix theory based on Wigner's conjecture
Wen-Jia Rao
Abstract
The distribution of higher-order level spacings, i.e., the distribution of ${{s}_{i}^{(n)}={E}_{i+n}\ensuremath{-}{E}_{i}}$ with $n\ensuremath{\ge}1$ is derived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poisson ensemble. It is found ${s}^{(n)}$ in Gaussian ensembles follows a generalized Wigner-Dyson distribution with rescaled parameter $\ensuremath{\alpha}=\ensuremath{\nu}{C}_{n+1}^{2}+n\ensuremath{-}1$, whereas that in the Poisson ensemble follows a generalized semi-Poisson distribution with index $n$. Numerical evidences are provided through simulations of random spin systems as well as nontrivial zeros of the Riemann $\ensuremath{\zeta}$ function. The higher-order generalizations of gap ratios are also discussed.