# Gernot Akemann: Spacing Distribution in the Ginibre ensembles: Universality and applications

**Time: **
Tue 2019-09-10 15.15

**Location: **
F11, KTH

**Participating: **
Gernot Akemann, Bielefeld University and KAW guest professor at KTH

### Abstract

After reviewing the spectral statistics of the complex Ginibre ensemble I will point out that in the bulk of the spectrum it holds universally for all 3 Ginibre ensembles. For real matrices this was shown by Borodin and Sinclair and I will sketch a proof for quaternion valued matrices. This bulk universality implies that also gap probabilities and in particular the spacing distribution in radial distance is the same for all 3 ensembles. The latter is a popular tool (for real spectra) to see the transition from integrable to chaotic behaviour in quantum systems, where the integrable case is described by Poisson statistics. An example will be presented that this also holds for complex eigenvalues and that the intermediate regime is well described by a static Coulomb gas, fitting \(\beta in (0, 2)\).