Entropy and recurrence rates for stationary random fields

For a stationary random field {x(u): u is an element of Z(d)}, the recurrence time R-n(x) may be defined as the smallest positive k, such that the pattern {x(u): 0 less than or equal to u(i) < n} is seen again, in a new position in the cube {0 less than or equal to \u(i)\ < k}. In analogy with the case of d = 1, where the pioneering work was done by Wyner and Zlv, we prove here that the asymptotic growth of R-n(x) for ergodic fields is given by the entropy of the random field. The nonergodic case is also treated, as well as the recurrence times of central patterns in centered cubes. Both finite and countable state spaces are treated.