Talk:Random walk

message: Please, would you be so kind to give an example of a random walk under this heading, for example, the geometric distribution. It would especially be helpful if you could show how the probability distribution P(N = n) = p(1-p)^{(n-1)} for n = 1, 2, 3, ... could be derived in the framework of a random walk as well as the central moments and, if possible, the maximum likelihood estimator for p.  — Preceding unsigned comment added by Ad van der Ven (talk • contribs) 15:27, 17 June 2011 (UTC)
thread_title: Probabilistic interpretation

message: The article says: "Will the drunkard ever get back to his home from the bar? It turns out that he will"
Wouldn't "Will the drunkard always get back to his home from the bar? It turns out that he will" be more accurate? For me "ever" implies any probabilty > 0 (which is trivially true) whereas "always" would more accurately describe "p = 1". 
--flatfish89 (talk) 17:46, 19 October 2008 (UTC)
replies: text: As I understand it, with infinite time (as implied by "ever"), any probability > 0 would mean that the drunkard will get home - saying that he always gets home is redundant. --Aseld talk 15:48, 7 February 2009 (UTC)
thread_title:  Higher Dimensions 

message: Well, I didn't really see the point here. A graph is a graphic representation of something and there is no "strict sense" in which it has to be one way or the other. The time axis is just as important as the space axis. Could you please clarify?
Gadykozma 05:17, 24 Jul 2004 (UTC)
thread_title:  Stricto Sensu 

message: I don't agree that cities are infinite.  Maybe the word can be changed to indicate that cities are vast, but not infinite.
"Imagine now a drunkard walking around in the city. The city is infinite and completely ordered, and at every corner he chooses one of the four possible routes (including the one he came from) with equal probability."
:it doesn't say cities are infinite, it says "imagine a city [which is] infinite". This is is mathematics. --Taejo | Talk 15:31, 11 November 2005 (UTC)
thread_title:  Cities are Infinite? 

message: I just added a sentence at the end of the first paragraph of the relation to brownian motion section.
Basically the reason is that this article says:
:"Brownian motion is the scaling limit of random walk in dimension 1."
Whereas the brownian motion article says:
:"The term Brownian motion (in honor of the botanist Robert Brown) refers to either
:1. The physical phenomenon that minute particles immersed in a fluid move about randomly; or
 
:2. The mathematical models used to describe those random movements."
It's only 2. that is the scaling limit for a random walk. 1. is just a particular type of random movement (follwing the langevin equation).
:Right; I've made this more precise by changing "Brownian motion" to "Wiener process" in that section. David-Sarah Hopwood (talk) 17:16, 12 February 2009 (UTC)
thread_title:  clarification and consistency of figure

message: Should Langton's Ant be mentioned (and linked to) as another example of a random walker, or is that diverting too far from the main point?
213.106.64.203 17:09, 20 January 2006 (UTC)
replies: text: There doesn't seem to be any randomness in Langton's ant.--GrafZahl 07:41, 21 January 2006 (UTC)
thread_title:  Langton's Ant 

message: This article tells the reader to imagine things and asks rhetorical questions. This is not written from an encyclopedic POV, and these parts should be replaced or deleted. Steevven1 (Talk) (Contribs) (Gallery) 13:40, 5 February 2007 (UTC)
replies: text: The language used here is fairly standard fare for a mathematics discussion.  The "Imagine..." is setting up an analogy to the problem, and the question "Will the drunkard ever get back to his home from the bar?" then asked is not rhetorical, it is also analogy to the equivalent hypothesis, "A random walk from point A will eventually reach point B".  While it may seem "non-encyclopedic" to someone who does not frequently read lay-discussions of mathematics, this is probably the clearest way to illustrate this concept. Dolohov 19:57, 12 March 2007 (UTC)
replies: text: Dolohov, please see Wikipedia:Manual of Style (mathematics) especially the section about writing style in mathematics articles. While it is pretty standard for the mathematics community, it is inappropriate and unnecessary in an encyclopedia. Use of the "royal we" should be avoided whenever possible. Cliff (talk) 06:02, 5 April 2011 (UTC)
replies: text: Er... do you realize you replied to a comment more than 4 years old? I think it may be time to archive some things on this page. ~Amatulić (talk) 06:26, 5 April 2011 (UTC)
replies: text: Of course I do. This is my attempt to revive discussion on this point. It makes no sense to create a new section to discuss a topic that still (after 4 years!) Has not been fixed.  It's important to see old discussion on a topic. Cliff (talk) 15:57, 10 April 2011 (UTC)
thread_title:  Non-encyclopedic POV 

message: I added a picture and sorted the sections with the more elementary on top. This makes the article a bit non-rigorous (with examples and pictures before the definition) but should make for a better read I hope. Oleg Alexandrov (talk) 03:19, 18 April 2007 (UTC)
thread_title: Reorganization

message: What is wrong with this explanation, assuming that "In a one dimensional random walk every point (including the origin) is crossed an infinite number of times in an infinite amount of time." is true?
In a one dimensional random walk every point (including the origin) is crossed an infinite number of times in an infinite amount of time.  If we add a second dimension the line representing the point in the original 1 dimension will be crossed an infinite number of times.  Since both dimensions are independent of each other there will be an infinite number of occurrences of both dimensions crossing the origin at the same time. This analogy can be extended to any number of dimensions where the third dimension will cross the plane, representing the original 2 dimensions, an infinite number of times.
From this I would conclude that a random walk will always return to the point of origin regardless of the number of dimensions if an infinite amount of time is allowed.  In other words there always exists the chance that a random walk will take the shortest possible path to it's origin at any point in time and space.  For example after n steps the random walker could be at coordinate (0,0,n) with probability p=\left({1\over 6}\right)^n thus there exists with equal probability that the random walker would take the shortest path to the origin from the farthest possible distance.  How can their exist a chance that it would never(p=0) return to the origin if their always exists(p>0) a chance that it could return to the origin in any finite number of steps from any conceivable (Manhatten) distance from the origin.  --ANONYMOUS COWARD0xC0DE 03:46, 2 June 2007 (UTC)
replies: text: What mathematicans are usually concerned with is if the random walk visits a given point infinitely many times, not just one time. In particular, a simple symmetric random walk on the d-dimensional lattice doesn't visit any point infinitely often for dimension greater than two. The section on higher dimension needs this distinction to be brought up and clarified. Filam3nt 22:46, 3 June 2007 (UTC)

text: Lets generalize to 'd' dimensions travelling 'n' units straight out and back to the origin p=\left({1\over 2d}\right)^{2n}.  This means their exists a chance for a random walk to return to the origin once in any number of dimensions.  Lets say k times: p=\left(\left({1\over 2d}\right)^{2n}\right)^k and \lim_{k\rightarrow\infty}{\left(\left({1\over 2d}\right)^{2n}\right)^k}=0, but this also applies to the 2-D case so their must be something I am missing.  In addition  never mentions a random walk returing to the origin an infinite number of times. --ANONYMOUS COWARD0xC0DE 03:46, 29 June 2007 (UTC)
replies: text: Nobody's claiming that the probability of returning to the origin is zero.  The claim is that in 3 or more dimensions, the probability is less than one.  The flaw in your "take each dimension separately" argument is that, although the path will almost certainly cross the x-y plane infinitely often, there's no reason to suppose that any of those crossings will coincide with crossing the y-z plane and the z-x plane.  65.57.245.11 05:26, 10 September 2007 (UTC)
replies: text: You're right, the crossing coincidence is not certain, just almost certain. However, I don't see why the chance of hitting an arbitrary bound with a 1D walk is anything but almost certain either, which is why I call into question the statement "A simple random walk on \mathbb Z will cross every point an infinite number of times." 80.203.160.34 (talk) 14:09, 9 October 2015 (UTC)

text: I had the same difficulty understanding this so I had to read a book describing the proof.  I ended up conceptualizing it as similar in nature to the issue of an escape velocity, although there is a force pulling the two objects together, they never meet.  In the same way although there is always a certain non-zero probability that the random walk will return to the origin the probability does not sum to one even after considering an infinite sum in the same way that an infinitely long integral of the force over time applied to an object at it's escape velocity will only, manage to cancel out the momentum, not reverse it (in order to meet).  So just think of the third dimension as the "probabilistic" escape velocity of a random walker.  I think such an analogy works quite well.  71.147.50.115 (talk) 21:15, 25 March 2010 (UTC)
replies: text: I just realized that the analogy works quite well in another regard.  In three dimensions the inverse square law holds for forces.  So the force at a distance 'd' is proportional to 1/d^2.  Since \int_{1}^{\infty} \frac{1}{x^2} \, dx=1 is finite it only takes a finite amount of kinetic energy to achieve an escape velocity.  Whereas if the forces applied only in two dimensions the force of gravity would be proportional to 1/d, and since \int_{1}^{\infty} \frac{1}{x} \, dx=\infty is infinite it would take an infinite amount of kinetic energy to achieve an escape velocity (which is impossible, therefore the probability of returning is always one in less than or equal to two dimensions). 71.147.50.115 (talk) 21:59, 25 March 2010 (UTC)
thread_title:  A random walk will always return to the origin regardless of the number of dimensions. 

message: Rather than drunkard's walk, I remember the term "drunk sailor's walk" - or is this only used in German?  —Preceding unsigned comment added by 84.136.218.223 (talk) 14:58, 27 September 2007 (UTC)
thread_title:  Drunk Sailor 

message: Would someone please clarify the sentence leading up to the  b/(a+b) formula. a and b are fixed positions. The current usage of "steps" could refer either to steps taken or a fixed number of steps from the origin. It would probably read better like this: "The expected number of steps until a one dimensional random walk goes up to position b or down to position -a is ab. The probability that the random walk will go up to position b before going down to position a is ..."  —Preceding unsigned comment added by Speculator mike (talk • contribs) 21:45, 28 September 2007 (UTC)
thread_title:  Some clarification on the a and b formula in the one dimensional case 

message: Does the "move distribution" for a generic random walk have to be independent of the current position? If so (or if not so) could this be specified somewhere in the introduction. If so then presumably the sample space has to follow certain symmetry conditions which depend on the "move distribution." For example you couldn't have a Gaussian random walk on the interval [0,1] because of the boundaries.  —Preceding unsigned comment added by 129.31.244.53 (talk) 20:01, 16 October 2010 (UTC)
thread_title: Clarification of the general definition 

message: I revised/expanded the intro somewhat and totally changed the definition.  Hopefully the intro covers the topic more broadly.  As for the definition, it formerly said: 
This is far too narrow since RW's can be correlated, biased and have any step-length distribution.  However, there is some inconsistency in the literature about definitions (for example: is a random walk a kind of Markov process?  or are some Markov processes a limited form of a random walk?) and I do not presume that the definition/notation I provide is canonical.  Best, Eliezg (talk) 19:57, 14 January 2008 (UTC)
thread_title:  Redid intro and defition 

message: I found this article over the internet: http://qjmath.oxfordjournals.org/cgi/reprint/4/1/120.pdf . However, some amount of dollars must be paid in order to read it. Could someone please read this article? I think it would be useful for the improvement of this page.  —Preceding unsigned comment added by Leoisiah (talk • contribs) 09:17, 13 February 2008 (UTC)
thread_title:  The general case random walk 

message: Does the "move distribution" for a generic random walk have to be independent of the current position? If so (or if not so) could this be specified somewhere in the introduction. If so then presumably the sample space has to follow certain symmetry conditions which depend on the "move distribution." For example you couldn't have a Gaussian random walk on the interval [0,1] because of the boundaries.
129.31.244.53 (talk) 20:02, 16 October 2010 (UTC)
thread_title: Clarification of the general definition 

message: I don't see anything about bounded random walks here. That seems a shame. A walk on a finite graph is finite, OK, but what about the continuous case? e.g. random walk in a space with one boundary (e.g. position can't go below zero). --mcld (talk) 15:32, 19 February 2011 (UTC)
thread_title:  Bounded random walk 

message: The section Random walk on graphs begins as follows:
"Assume now that our city is no longer a perfect square grid. When our drunkard reaches a certain junction he picks between the various available roads with equal probability. Thus, if the junction has seven exits the drunkard will go to each one with probability one seventh. This is a random walk on a graph. Will our drunkard reach his home? It turns out that under rather mild conditions, the answer is still yes. For example, if the lengths of all the blocks are between a and b (where a and b are any two finite positive numbers), then the drunkard will, almost surely, reach his home. Notice that we do not assume that the graph is planar, i.e. the city may contain tunnels and bridges."
Can someone please explain how this is consistent with the fact that the symmetric random walk in 3D (i.e., on the 1-skeleton of the cubical tiling) is known to be transient?
Certainly all "blocks" have equal lengths.  The statement "Notice that we do not assume that the graph is planar" seems to allow a grid like this 3D one.   What properties of the graph are assumed???
If this is not what is intended here, perhaps the above-quoted statement could  be modified to make clear just why such grids as the 3D one (and for that matter all the corresponding nD ones, n ≥ 3) do not qualify.  Thanks.Daqu (talk) 23:50, 5 March 2011 (UTC)
thread_title:  How does this passage square with the transience in 3d ??? 

message: I created an account so I could submit my Random Walk animated gifs but now it tells me I must be Autoconfirmed.  Can I just give these to someone to put on the page?  I think they're perfect ... small and actually a Random Walk, unlike the Brownian motion simulated by Random Walk video.  — Preceding unsigned comment added by DrDInfinity (talk • contribs) 00:59, 2 July 2011 (UTC)
thread_title:  I want to submit a couple of Random Walk .gif's 

message: I saw Firefox 13.0.1 cannot play the video, it previews, but when I click play, nothing happens.
Now3d (talk) 12:22, 6 July 2012 (UTC)
thread_title:  Video not working 

message: The current section about random walk on graphs is in my opinion terribly written, being vague and unstructured even though the contents are mostly fine. I have written a new version (you can see it at User:Jean Raimbault/sandbox/Random walk; I think the contents are OK for now but I'll probably edit it a bit for readability before moving it to mainspace) with which I would like to replace the current one. On the other hand the material is perhaps a bit technical for a "entry-level" page such as this one and it may be preferable to create a new page for this content, rewriting the material on the main page to be much more lightweight. I personally favour the latter option, in fact I would probably put a large part of the current content in separate pages (in particular the stuff on Pólya's walk feels too massive) and try to make this page more accessible (this seems like a rather long-term project so I cannot claim I'll be able to do a substantial part of this myself). Opinions and comments? jraimbau (talk) 16:46, 30 August 2016 (UTC)
thread_title:  Random walks on graphs