Entropy and recurrence rates for stationary random fields

For a stationary random field {x(u): u ∈ ℤ^d}, the recurrence time R_n(x) may be defined as the smallest positive k, such that the pattern {x(u): 0 ≤ u_i < n} is seen again, in a new position in the cube {0 ≤ |u_i| < k}. In analogy with the case of d = 1, where the pioneering work was done by Wyner and Ziv, 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.