Exact conditions on physically realizable correlation functions of random media

Algorithms have been developed recently to construct realizations of random media with specified statistical correlation functions. There is a need for the formulation of exact conditions on the correlation functions in order to ensure that hypothetical correlation functions are physically realizable. Here we obtain positivity conditions on certain integrals of the autocorrelation function of $d$ -dimensional statistically homogeneous media and of statistically isotropic media. These integral conditions are then applied to test various classes of autocorrelation functions. Finally, we note some integral conditions on the three-point correlation function.