  Herein, we introduce and study a new class of discrete random fields designed
for quick simulation and covariance inference under inhomogeneous condition.
  Simulation of these correlated fields can be done in a single pass instead of
relying on multi-pass convergent methods like the Gibbs Sampler or other Markov
Chain Monte Carlo methods. The fields are constructed directly from specified
marginal probability mass functions and covariances between nearby sites.
  The proposition on which the construction is based establishes when and how
it is possible to simplify the conditional probabilities of each site given the
other sites in a manner that makes simulation quite feasible yet maintains
desired marginal probabilities and covariances between sites.
  Special cases of these correlated fields have been deployed successfully in
data authentication, object detection and image generation. The limitations
that must be imposed on the covariances and marginal probabilities in order for
the algorithm to work are studied. What's more, a necessary and sufficient
condition that guarantees the permutation property of correlated random fields
are investigated. In particular, Markov random fields as a subclass of
correlated random fields are derived by a general and natural condition.
Consequently, a direct and flexible single pass algorithm for simulating Markov
random fields follows.
