Timeline for Reference request for a result on subsets unlikely to be hit by random walks in a group
Current License: CC BY-SA 3.0
5 events
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| Aug 2, 2011 at 16:54 | comment | added | Justin | @Ron: The statement is not about bounded subsets necessarily. The implication is: If we know that A is hit with exponentially decaying probability when $\mu$ is the uniform measure, then it is also true when $\mu$ is an arbitrary measure supported on S. | |
| Aug 2, 2011 at 4:21 | comment | added | Ron Maimon | @Anthony: That's puzzling. As I understood the problem, it is saying that the probability that an n-step walk with a given generator set lands in a bounded set of points decays exponentially. This is false in Z^2 because the Gaussian kernel decays as a power, but it is true in a free group on two generators because you have exponential growth. It is possible that the OP meant an arbitrary subset, but then you could take A to be the whole group minus a few points, and it is trivially false. I think that is overly uncharitable interpretation, but perhaps the question could be clarified. | |
| Aug 1, 2011 at 21:54 | comment | added | Anthony Quas | I don't see why it's false for $Z^2$: The left side of the implication is always false. | |
| Aug 1, 2011 at 20:10 | comment | added | Ron Maimon | What are the conditions on the group? I assume free, because your result is true in a free group with two generators, and false for $Z^2$. | |
| Aug 1, 2011 at 20:00 | history | asked | Justin | CC BY-SA 3.0 |