  We report a study of disordered electron systems with spin-orbit coupling on
a cylinder using methods of random matrix ensembles. With a threading flux
turned on, the single particle levels will generally avoid, rather than cross,
each other. Our numerical study of the level-avoiding gaps in the disordered
Rashba model demonstrates that the normalized gap distribution is of a
universal form, independent of the random strength and the system size. For
small gaps it exhibits a linear behavior, while for large gaps it decays
exponentially. A framework based on matrix mechanical models is suggested, and
is verified to reproduce the universal linear behavior at small gaps. Thus we
propose to use the distribution of the spectral-flow gaps associated with flux
insertion as a new way to characterize 2d random systems with spin-orbit
coupling. The relevance and qualitative implications for spin (Hall) transport
are also addressed.
