Timeline for Probabilities of a random walk exiting a set
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 16, 2023 at 15:00 | comment | added | ARG | For further readers, the answer is in Kaimanovich's 1992 paper: Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy. | |
| Apr 26, 2013 at 10:17 | comment | added | ARG | Thanks for the comment (and Vincent's answer). Without knowing this is equivalent to trivial Poisson boundary, on the free group (which is a tree), the intuition is that the walk is "ballistic", i.e. does not really come back. Since it will with relatively small probability cross the separating edge at the beginning, there is no reason that the exit probabilities will be small. With more effort, one can actually compute these exit probabilities. | |
| Apr 26, 2013 at 10:04 | vote | accept | ARG | ||
| Apr 26, 2013 at 9:47 | answer | added | Vincent Beffara | timeline score: 7 | |
| Apr 25, 2013 at 12:59 | comment | added | Mikael de la Salle | I discussed your question with Vincent Beffara. We arrived at the conclusion that your question has a positive answer (for every pair (s,e)) if and only if your graph is Liouville. In particular, the lamplighter group on $\mathbb{Z}^3$ gives a negative answer to your question. Vincent should soon write more on this as an answer. | |
| Apr 25, 2013 at 8:11 | comment | added | Mikael de la Salle | Why is it clearly false on the free group? | |
| Apr 24, 2013 at 13:31 | history | asked | ARG | CC BY-SA 3.0 |