Variance of a standard vector Monte Carlo estimate in the theory of polarized radiative transfer

The spectral radius $\rho$ of the matrix integral operator defining the covariance matrix of a standard vector Monte Carlo estimate in the polarized radiative transfer problem is examined. The theory of positive operators is used to analytically calculate $\rho=\rho_0$ for transfer through an infinite homogeneous medium. For a bounded medium, it is shown that $\rho$ is approximately equal to $\rho_0$ times the spectral radius of the operator corresponding to radiative transfer without polarization. This is shown numerically by estimating the iterations of the corresponding resolvent and approximately analytically by using a perturbation of a special functional.