Timeline for Transience of self avoiding random walks on $\mathbb{Z}^d$
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 20, 2012 at 12:40 | vote | accept | David White | ||
| Apr 20, 2012 at 12:18 | comment | added | Vincent Beffara | SLE = Schramm-Loewner Evolution. It is hard to describe in 600 characters, but the moral is that it is an explicit description of the scaling limit of loop-erased random walk. BTW, said LERW in 2D is "very transient" in the sense that its $n$th step is at distance of order $n^{4/5}$ of the origin rather than $n^{1/2}$ [for SAW, this is expected to scale like $n^{3/4}$]. | |
| Apr 19, 2012 at 23:34 | comment | added | David White | @Vincent: Thanks for all your helpful comments and for the Kesten reference. I think I know now what to say in my thesis about SAWs (and the answer is: not very much, which is great from the point of view of getting it done). I may return at some point to the question of finding the right measure to ask the recurrence question. Please let me know if you think of anything. One last question for you: what do you mean by SLE_2 in your penultimate comment? | |
| Apr 19, 2012 at 22:57 | comment | added | Vincent Beffara | The "right" way to ask about transience would have to involve scaling limits, I believe (look at the definition of recurrence of 2D Brownian motion, it a.s. touches any disk but still has measure 0 so every point is a.s. not on the path ...) | |
| Apr 19, 2012 at 22:56 | comment | added | Vincent Beffara | The set of trapped paths is countable, the set of untrapped ones is not countable. The question is what you want this measure to satisfy ... You could look at the loop-erased random walk (i.e. start from a simple random walk, and remove all the loops it builds): if the SRW is transient, then the LERW is well-defined, self-avoiding, and actually pretty well understood (d=2, by Lawler-Schramm-Werner, looks like SLE_2; d=3, cf Kozma; d>4, it should be diffusive). But somehow it is not the natural measure, because it has little to do with the uniform measure on finite self-avoiding paths. | |
| Apr 19, 2012 at 21:36 | comment | added | David White | @Vincent: What about finding a good measure to put on the set $X$ of all self-avoiding paths? Let $A$ be the subset of self-avoiding paths $(v_0,v_1,\dots,v_n)$ which get trapped at $v_n$. Represent these paths as points $(v_0,\dots,v_n,v_n,v_n,\dots) \in X$. I wonder if there's a natural probability measure on $X$ which gives $A$ nontrivial size. I'd like to say the process is transitive if $\mu(A)<1$ and "gets trapped" if $\mu(A)=1$. One problem I see is that $A$ is countable but $X$ could be uncountable. Do you know if anyone has pursued this? What's the "right" way to ask about transience? | |
| Apr 19, 2012 at 20:34 | comment | added | Ori Gurel-Gurevich | David, there is a pdf at the JMP site: jmp.aip.org/resource/1/jmapaq/v5/i8/p1128_s1 . Drop me a line if you don't have access to it. | |
| Apr 19, 2012 at 20:15 | comment | added | Vincent Beffara | The thing is, constructing a measure on the set of infinite self-avoiding paths is easy (just go east all the time, look at a directed paths,whatever). Constructing a natural measure on those is the issue. And as I said, if you take a "natural" measure on the set of all paths, typically the set of self-avoiding paths will be $0$. | |
| Apr 19, 2012 at 20:07 | comment | added | David White | Also, I found a PDF of the first of the two articles on the pattern theorem: scitation.aip.org/getpdf/servlet/… but I can't find a PDF of the second, which is called "On the number of self-avoiding walks II." Maybe someone can leave a comment if they know where one can be found. | |
| Apr 19, 2012 at 20:04 | comment | added | David White | I see what you mean about the conditioning; thanks for pointing it out. However, I'm confused by the first line in your second paragraph. Any infinite self-avoiding path goes to infinity, so I feel like what I need more than a measure on the set of infinite self-avoiding paths is a way to figure out their measure in the set of all paths (infinite or otherwise). If this measure is positive then I would say the process is transitive, since a non-vanishing proportion of walks do escape. Is your second paragraph saying it's possible to get a measure on the set of all paths? | |
| Apr 19, 2012 at 19:51 | history | answered | Vincent Beffara | CC BY-SA 3.0 |