Minimal growth harmonic functions on lamplighter groups

We study the minimal possible growth of harmonic functions on lamplighters. We find that $(\mathbb{Z}\mathrm{/}2)\wr \mathbb{Z}$ has no sublinear harmonic functions, $(\mathbb{Z}\mathrm{/}2)\wr {\mathbb{Z}}^2$ has no sublogarithmic harmonic functions, and neither has the repeated wreath product $(\cdots (\mathbb{Z}\mathrm{/}2\wr {\mathbb{Z}}^2)\wr {\mathbb{Z}}^2)\wr \cdots \wr {\mathbb{Z}}^2$. These results have implications on attempts to quantify the Derriennic-Kaimanovich-Vershik theorem.