# Speed of random walks in groups

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 $\alpha$?

-

For nilpotent groups the speed exponent is 0.5. See this paper for the general picture http://arxiv.org/abs/1203.6226 .

-
Very nice reference... is it obvious that there are amenable groups where the speed exponent is 1? – Antoine Jul 20 '13 at 13:47
You can have a look at Theorem 9.5 in these nice lecture notes math.bme.hu/~gabor/PGG.html. – Dan Sălăjan Jul 20 '13 at 14:39
Thanks again for this other great reference. Reading through these notes, I read that Kaimanovich showed there are generators for which Thompson's group has speed exponent 1 (after question 15.3). I could not find any reference to this...would you happen to know of it? – Antoine Oct 1 '13 at 8:35
You re welcome. Very interesting, I don´t know any reference or proof... I might think about it, maybe there is a simple reason for it. I let you know if I succeed. Dan – Dan Sălăjan Oct 1 '13 at 8:40