Extremal spacings between eigenphases of random unitary matrices of size $N$ pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for $N=4$. We study ensembles of tensor product of $k$ random unitary matrices of size $n$ which describe independent evolution of a composite quantum system consisting of $k$ subsystems. In the asymptotic case, as the total dimension $N=n^k$ becomes large, the nearest neighbor distribution $P\left(s\right)$ becomes Poissonian, but statistics of extreme spacings $P\left(s_{\min }\right)$ and $P\left(s_{\max }\right)$ reveal certain deviations from the Poissonian behavior.

• Received 12 June 2013

DOI:https://doi.org/10.1103/PhysRevE.88.052902