Groups with minimal harmonic functions as small as you like (With an   appendix by Nicolas Matte Bon)
  For any order of growth $f(n)=o(\log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function with slower growth. The construction uses permutational wreath products in which the base group is defined via its properly chosen Schreier graph. 
Gideon Amir and Gady Kozma
math.GR math.PR