Timeline for The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
Current License: CC BY-SA 3.0
11 events
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| Oct 2, 2012 at 5:59 | comment | added | Nathan Clisby |
I'll just add this as a comment because I don't have time to craft a careful answer. A trapped walk can also be described as a walk with atmosphere' zero, which is discussed for example in the paper Scaling of the atmosphere of self-avoiding walks', by Aleks Owczarek and Thomas Prellberg, dx.doi.org/10.1088/1751-8113/41/37/375004 N.B. the aforementioned paper is for self-avoiding walks. It sounds like you may be more interested in kinetic growth walks, which are discussed for example in dx.doi.org/10.1103/PhysRevA.34.3304 (I know little about this topic).
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| Sep 9, 2012 at 17:38 | vote | accept | CKura | ||
| Sep 8, 2012 at 17:55 | comment | added | CKura | I suspect I'll have to run a simulation... | |
| Sep 8, 2012 at 17:55 | comment | added | CKura | @Vincent Beffara, I could imagine that there is some c at the limit of large N, but intuitively I feel that, at short times (time ~ steps) the probability of self-trapping at the (N + 1)th step will grow with N. Also, a thank you for thinking about my question, and though your answer isn't quite what I was looking for, it was interesting nevertheless. | |
| Sep 8, 2012 at 15:11 | comment | added | Vincent Beffara | Then the survival probability will decay like $c^N$, $c\in(0,1)$, because at every step the walk will have a way to block itself in, say, 12 steps, and will do just that with positive probability. (Well, except if there is a very long corridor, but that is also exponentially unlikely to happen). I don't think the value of $c$ is known though. | |
| Sep 7, 2012 at 17:16 | comment | added | CKura | @Peter Shor, yes, your interpretation is correct. The walker will choose unvisited nearest-neighbor vertices with uniform probability. | |
| Sep 7, 2012 at 12:29 | comment | added | Peter Shor | Even with your clarification, you still haven't completely specified the probability model for your random self-avoiding walk. I assume that your model is that, at every time step, the walk is equally likely to go to any unvisited adjacent vertex. | |
| Sep 7, 2012 at 8:33 | history | edited | CKura | CC BY-SA 3.0 |
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| Sep 7, 2012 at 8:24 | history | edited | CKura | CC BY-SA 3.0 |
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| Sep 7, 2012 at 7:10 | answer | added | Vincent Beffara | timeline score: 7 | |
| Sep 7, 2012 at 5:15 | history | asked | CKura | CC BY-SA 3.0 |