Speed exponents of random walks on groups
  For every 3/4 <= beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any nice prescribed function up to a constant factor. 
Gideon Amir and Balint Virag
math.PR math.GR