  We establish characterization results for the ergodicity of stationary
symmetric $\alpha$-stable (S$\alpha$S) and $\alpha$-Frechet random fields. We
show that the result of Samorodnitsky [Ann. Probab. 33 (2005) 1782-1803]
remains valid in the multiparameter setting, that is, a stationary S$\alpha$S
($0<\alpha<2$) random field is ergodic (or, equivalently, weakly mixing) if and
only if it is generated by a null group action. Similar results are also
established for max-stable random fields. The key ingredient is the adaption of
a characterization of positive/null recurrence of group actions by Takahashi
[Kodai Math. Sem. Rep. 23 (1971) 131-143], which is dimension-free and
different from the one used by Samorodnitsky.
