Contour-integral method for transitions to the circular unitary ensemble

Abstract

The representation of correlation functions as a contour integral has been useful in the study of transitions to the Gaussian unitary ensemble (GUE). We develop the formalism for transitions to the circular unitary ensemble (CUE) and consider the general ℓCUE to CUE transition where ℓCUE denotes a superposition of ℓ independent CUE spectra in an arbitrary ratio. For large matrices, we derive the two-level correlation function for all ℓ including ℓ = ∞ (the Poisson case). The results are useful in the study of weakly broken partitioning symmetries and weakly coupled mesoscopic cavities.