  We consider the problem of the statistics of the scattering matrix S of a
chaotic cavity (quantum dot), which is coupled to the outside world by
non-ideal leads containing N scattering channels. The Hamiltonian H of the
quantum dot is assumed to be an M x N hermitian matrix with probability
distribution P(H) ~ det[lambda^2 + (H - epsilon)^2]^[-(beta M + 2- beta)/2],
where lambda and epsilon are arbitrary coefficients and beta = 1,2,4 depending
on the presence or absence of time-reversal and spin-rotation symmetry. We show
that this ``Lorentzian ensemble'' agrees with microscopic theory for an
ensemble of disordered metal particles in the limit M -> infinity, and that for
any M >= N it implies P(S) ~ |det(1 - \bar S^{\dagger} S)|^[-(beta M + 2 -
beta)], where \bar S is the ensemble average of S. This ``Poisson kernel''
generalizes Dyson's circular ensemble to the case \bar S \neq 0 and was
previously obtained from a maximum entropy approach. The present work gives a
microscopic justification for the case that the chaotic motion in the quantum
dot is due to impurity scattering.
