  Strongly Rayleigh distributions are natural generalizations of product and
determinantal probability distributions and satisfy strongest form of negative
dependence properties. We show that the "natural" Monte Carlo Markov Chain
(MCMC) is rapidly mixing in the support of a {\em homogeneous} strongly
Rayleigh distribution. As a byproduct, our proof implies Markov chains can be
used to efficiently generate approximate samples of a $k$-determinantal point
process. This answers an open question raised by Deshpande and Rademacher.
