  Random fields are commonly used for modeling of spatially (or timely)
dependent stochastic processes. In this study, we provide a characterization of
the intrinsic complexity of a random field in terms of its second order
statistics, e.g., the covariance function, based on the Karhumen-Lo\'{e}ve
expansion. We then show scaling laws for the intrinsic complexity of a random
field in terms of the correlation length as it goes to 0. In the discrete
setting, it becomes approximate embeddings of a set of random vectors. We
provide a precise scaling law when the random vectors have independent and
identically distributed entires using random matrix theory as well as when the
random vectors has a specific covariance structure.
