Timeline for Probability of return at step $n$ of a Random walk to its starting vertex
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Apr 4, 2011 at 9:48 | comment | added | Did | Nope, $C_d$ denotes a constant, which depends on $d$. (FYI, I read your comments by chance. If you want your comment to be notified to a given user, begin the comment with @user. An exception is when you comment on an answer, then the author of the answer is notified automatically. The same applies to comments on posts, hence this comment does not begin by @Chris.) | |
| Mar 30, 2011 at 15:38 | comment | added | Chris | Ben, Didier: reading Woess I did not really get what $C_d$ is to define. Is this the Champernowne constant? The first occurence of this notation is in Woess for infinite line: $p^(2n)(0,0) = \frac{1}{2^{2n}} \binom{2n}{n} \sim C_1 n^{-1/2}$ | |
| Mar 28, 2011 at 14:32 | vote | accept | Chris | ||
| Mar 27, 2011 at 12:18 | comment | added | Chris | Ben: after having a look at the book I agree that is is of great value. Especially right at the beginning it is stated that $p^{(2n)}(0,0) \sim C_d n^{-d/2}$ for graphs in $Z^d$. Having such a relation for finite graphs that takes into account the size of the graph would be great. Is such a relation known? | |
| Mar 27, 2011 at 9:48 | comment | added | Chris | Ben: I am interested in the CDF that is built out of the probabilities to be at the starting vertex at step $n$ over all vertices of the graph, vertices weighted by their steady state probability to actually start at this vertex. I can calculate this CDF in numerical form out of the random walk probability matrices. Now I am interested if for some families of graphs there exist easier methods. The below answer by Didier is for example exactly such a relation I am interested which is for finite circles. | |
| Mar 26, 2011 at 21:44 | answer | added | Steve Huntsman | timeline score: 0 | |
| Mar 26, 2011 at 19:29 | comment | added | Ben Green | Chris: a vast amount is known about this. See, for example, the book by Woess "Random Walks on Infinite Graphs and Groups", CUP. For finite graphs there's a huge amount of research. Do you have a more specific question? | |
| Mar 26, 2011 at 19:03 | comment | added | Chris | Didier: thanks, I have tried to make it more clear. I am interested in the probability to be at the starting vertex at a given time. | |
| Mar 26, 2011 at 19:02 | history | edited | Chris | CC BY-SA 2.5 |
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| Mar 26, 2011 at 16:38 | comment | added | Did | You seem to be talking simultaneously of recurrence time (i.e. the time of the first return to the starting vertex) and of the probability to be at the starting vertex at a given time. These are not the same and you could say which one concerns you. | |
| Mar 26, 2011 at 16:22 | answer | added | Did | timeline score: 1 | |
| Mar 26, 2011 at 14:16 | history | asked | Chris | CC BY-SA 2.5 |