No representation of Moore groups and affine groups has any rate of random mixing

A sequence $a_n\da 0$ forms a rate of random mixing for a unitary system $(G,\mu, \pi, {\cal H})$ if for any $u, v\in {\cal H}$ $$\limsup {|<\pi (g_n^\omega)u, v>|\over a_n} < \infty$$ a.e. $\omega$ in the probability space $(G^{\mathbb N}, \mu ^{\mathbb N})$ of the random walk induced by $\mu$. We study the class of locally compact groups none of whose representation has any rate of random mixing and prove that this class contains Moore groups and certain solvable groups which includes the group of affine transformations on a local field.