I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly for nilpotent groups (transient ones), though subexponential or polycyclic could be also nice. 


Let $X\_n$ be the random variable of a simple random walk starting at the neutral element $e$ and $|\cdot|$ be the word length (for some generating set). Let $l\_n$ be the expected value of $|X\_n|$. To put the question in precise form: 


$\mathbf{Question}:$ For a nilpotent group $G$, does there exists $\alpha \in ]0,1[ $ so that for some constant $C>1$, $l\_n \leq C n^\alpha$? 


Is there any estimates on the possible values of $\alpha$?



Speed of random walks in groups
For nilpotent groups the speed exponent is 0.5. See this paper for the general picture <http://arxiv.org/abs/1203.6226> .


