Truncating Random Matrices of the Circular Ensembles: Crossover from Conserving to Lossy with Applications to Quantum Dots

In a quantum dot with three leads the transmission matrix t_12 between two of these leads is a truncation of a unitary scattering matrix S, which we treat as random. As the number of channels in the third lead is increased, the constraints from the symmetry of S become less stringent and t_12 becomes closer to a matrix of complex Gaussian random numbers with no constraints. We consider the distribution of the singular values of t_12, which is related to a number of physical quantities. When time reversal is broken this problem gives a realization of the Jacobi Unitary Ensemble.