3
$\begingroup$

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 $\alpha$?

$\endgroup$

1 Answer 1

4
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ Very nice reference... is it obvious that there are amenable groups where the speed exponent is 1? $\endgroup$
    – ARG
    Jul 20, 2013 at 13:47
  • $\begingroup$ You can have a look at Theorem 9.5 in these nice lecture notes math.bme.hu/~gabor/PGG.html. $\endgroup$ Jul 20, 2013 at 14:39
  • $\begingroup$ 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? $\endgroup$
    – ARG
    Oct 1, 2013 at 8:35
  • $\begingroup$ 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 $\endgroup$ Oct 1, 2013 at 8:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.