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We establish characterization results for the ergodicity of stationarysymmetric $\alpha$-stable (S$\alpha$S) and $\alpha$-Frechet random fields. Weshow 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 andonly if it is generated by a null group action. Similar results are alsoestablished for max-stable random fields. The key ingredient is the adaption ofa characterization of positive/null recurrence of group actions by Takahashi[Kodai Math. Sem. Rep. 23 (1971) 131-143], which is dimension-free anddifferent from the one used by Samorodnitsky.

Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions