Timeline for Probability that a random element of a group is trivial
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 14, 2022 at 18:58 | vote | accept | Xiyan | ||
| Oct 14, 2022 at 18:56 | answer | added | R W | timeline score: 11 | |
| Oct 14, 2022 at 16:41 | comment | added | YCor | but the phrase "probability of return" might help finding references (I don't have them in mind now) | |
| Oct 14, 2022 at 16:38 | comment | added | Xiyan | @YCor: Telling me that "it is known" is not so helpful without a reference. | |
| Oct 14, 2022 at 16:38 | comment | added | Xiyan | @HJRW: Thanks, that keyword gave me exactly what I want! The result I was looking for is contained in Woess, Wolfgang, Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41 (1983), no. 4, 363–370. | |
| Oct 14, 2022 at 16:33 | comment | added | Xiyan | @RW: Yes, that would be equivalent to my question. Thanks! | |
| Oct 14, 2022 at 5:45 | comment | added | HJRW | I think the keyword you need is “cogrowth” of a group. There are many papers on the subject. | |
| Oct 14, 2022 at 2:28 | comment | added | YCor | If one averages over all $k^n$ words ($k=|S\cup S^{-1}|$) of length $=n$, one gets the probability of return $p'_n$. Indeed it is known that $p'_n\to 0$ for an infinite group. Here $p_n=(\sum_{i=0}^n k^i p'_n)/(\sum_{i=0}^n k^i)$, it follows that $p_n$ also tends to zero. | |
| Oct 14, 2022 at 1:07 | comment | added | R W | Are you asking about the random walk on $G$ that consists in the (right) multiplication by a random increment uniformly sampled from $S\cup S^{-1}$? Just to make sure before answering. | |
| Oct 14, 2022 at 0:36 | comment | added | Benjamin Steinberg | The question I linked is looking at random walks on Schreier graphs of free groups. Every Cayley graph of a Schreier graph of a free group. They look at transitive actions. So it seems stronger | |
| Oct 14, 2022 at 0:15 | comment | added | Xiyan | @BenjaminSteinberg: I'm not sure I follow. Those seem to be about random walks on the Cayley graph of a free group, and the relevant probabilities seem to decay to $0$. Do you mean that we should write $G = F_n/K$ and think about whether a random walk in $F_n$ will land in $K$? I will have to think more about what exactly the answers are saying, but if they do answer my question it seems that they will give the answer "no" rather than "yes", right? | |
| Oct 14, 2022 at 0:10 | comment | added | Benjamin Steinberg | I think the answers to this question say yes mathoverflow.net/questions/91188/… | |
| S Oct 13, 2022 at 22:09 | review | First questions | |||
| Oct 13, 2022 at 22:56 | |||||
| S Oct 13, 2022 at 22:09 | history | asked | Xiyan | CC BY-SA 4.0 |