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 $\alpha$?