  We show that any stationary symmteric $\alpha$-stable ($S\alpha S$) random
field indexed by a countable amenable group $G$ is weakly mixing if and only if
it is generated by a null action, extending works of Samorodnitsky and
Wang-Roy-Stoev for abelian groups to all amenable groups. This enables us to
improve significantly the domain of a recently discovered connection to von
Neumann algebras. We also establish ergodicity of stationary $S\alpha S$ fields
associated with boundary and double boundary actions of a hyperbolic group $G$,
where the boundary is equipped with either the Patterson-Sullivan or the
hitting measure of a random walk, and the double boundary is equipped with the
Bowen-Margulis-Sullivan measure.
