Timeline for How far can a particle travel from its origin if it exhibits self-avoiding Brownian motion in two-dimensions?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 10, 2011 at 18:24 | comment | added | Yvan Velenik | Not quite the process you're looking for, but you might be interested in googling for (or checking on mathscinet for) the "myopic" or "true" self-avoiding walk. | |
| Aug 1, 2011 at 22:52 | comment | added | Ron Maimon | The question is wrongly worded, but the process might be meaningful: consider a random walk which is restricted by the condition that at each collision it reflects off the rest of the walk so as to stay outside the closed loop formed. This process is instantaneously measure zero as compared to an ordinary random walk (by scaling or by 0-1 laws), but does it define a universality class different than the usual self avoiding walk? Can't you take an ordinary Brownian path and discontinuously reparametrize it so that it becomes a reflecting path? | |
| Jul 31, 2011 at 16:42 | comment | added | George Lowther | ...as mentioned by Nate Eldridge, although you don't need Blumenthal's 0-1 law for the scaling argument. | |
| Jul 31, 2011 at 16:40 | comment | added | George Lowther | However you are defining the process, it can't be done in a scale invariant way. If W(t) is a BM then so is $a^{-1}W(a^2t)$. So, if T was the first time that it 'traps itself' then T has the same distribution as $a^2T$. This is only possible for $T=0$ or $T=\infty$ almost surely. | |
| Jul 30, 2011 at 15:56 | vote | accept | Rob Grey | ||
| Jul 30, 2011 at 15:01 | answer | added | Joseph O'Rourke | timeline score: 8 | |
| Jul 30, 2011 at 5:10 | comment | added | Ron Maimon | You can find the answer to the trapping questions numerically by simulating an infinite non-trapping walk, and looking at the probability cost of enforcing no-trapping at each time-step (I did this recreationally once, for the getting the asymptotic number of self-avoiding walks of length n-- it gives the wrong answer). Maybe you can get a different universality class for random paths which are allowed to collide, but not cross. It is possible that this condition makes sense, even if the original question is not the right one. | |
| Jul 29, 2011 at 10:17 | history | edited | Rob Grey | CC BY-SA 3.0 |
added 61 characters in body; added 10 characters in body
|
| Jul 29, 2011 at 10:10 | history | edited | Rob Grey | CC BY-SA 3.0 |
Please see the 'edit' section
|
| Jul 29, 2011 at 2:42 | comment | added | Nate Eldredge | I'm not sure that your question is meaningful as stated. First, it's typically hard to define what it means to "reflect" off a boundary which is rough, and Brownian sample paths are about as rough as they come. So I think your process may not be well-defined. Second, if it does have a positive probability of "trapping itself" (whatever that means), scaling and the Blumenthal 0-1 law would suggest that the trapping happens immediately, almost surely. | |
| Jul 29, 2011 at 2:27 | comment | added | Rob Grey | @Ron Maimon, can you help me understand your point? If the Brownian particle can interact with a reflecting boundary, why wouldn't it be able to generate a closed loop and trap itself inside by reflecting there? | |
| Jul 29, 2011 at 2:18 | comment | added | Ron Maimon | You should ask the question for the lattice version only--- the Brownian version can't trap itself. | |
| Jul 29, 2011 at 0:38 | history | edited | Rob Grey | CC BY-SA 3.0 |
added 21 characters in body
|
| Jul 29, 2011 at 0:30 | history | edited | Rob Grey | CC BY-SA 3.0 |
added 153 characters in body
|
| Jul 28, 2011 at 9:18 | history | asked | Rob Grey | CC BY-SA 3.0 |