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

Monte Carlo Markov Chain Algorithms for Sampling Strongly Rayleigh Distributions and Determinantal Point Processes