Ordered level spacing probability densities

Research output: Contribution to journalResearch articleContributedpeer-review


  • Shashi C.L. Srivastava - , Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Max-Planck-Institute for the Physics of Complex Systems (Author)
  • Arul Lakshminarayan - , Indian Institute of Technology Madras (Author)
  • Steven Tomsovic - , Washington State University Pullman (Author)
  • Arnd Bäcker - , Chair of Computational Physics, Max-Planck-Institute for the Physics of Complex Systems (Author)


Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a 3 × 3 matrix model. The closest neighbour density is generalized to the kth closest neighbour spacing density, which allows for investigating long-range correlations. For larger k the probability density of kth closest neighbour spacings is well described by a Gaussian. Using these kth closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found.


Original languageEnglish
Article number025101
JournalJournal of Physics A: Mathematical and Theoretical
Publication statusPublished - 4 Dec 2018