most recent 30 from http://mathoverflow.net 2013-05-19T11:35:06Z http://mathoverflow.net/feeds/question/4068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4068/how-many-dimensions-is-it-safe-to-get-drunk-in petef 2009-11-04T09:55:47Z 2009-11-04T17:12:42Z

<p>In Michael Lugo's blog post <a href="http://godplaysdice.blogspot.com/2007/09/variations-on-drunken-bird-theorem-and.html" rel="nofollow">Variations on the drunken-bird theorem, and real-world sightings</a> he wonders (without coming to a conclusion) what the maximum 'safe' number of dimensions to get drunk in might be. </p> <blockquote> <p>So where's the line between "always finds his way home" and "gets lost forever"? It seems to me that this should be at 2+&epsilon; dimensions, since that's what we need to make the sum convergent, but I don't know if anyone's studied random walks on fractal spaces.</p> </blockquote> <p>In the comments Matt Noonan recalls an unverified claim that the line is at 2.5 dimensions. </p> <p>Can anybody shed any further light on these musings?</p>

http://mathoverflow.net/questions/4068/how-many-dimensions-is-it-safe-to-get-drunk-in/4072#4072 Tim vB 2009-11-04T12:02:55Z 2009-11-04T12:02:55Z

<p>Have a look at</p> <p><a href="http://mathworld.wolfram.com/PolyasRandomWalkConstants.html" rel="nofollow">http://mathworld.wolfram.com/PolyasRandomWalkConstants.html</a></p> <p>The answer turns out that at as soon as dim > 2 the probability of recurrence is lower than 100%.</p>

http://mathoverflow.net/questions/4068/how-many-dimensions-is-it-safe-to-get-drunk-in/4077#4077 Michael Lugo 2009-11-04T13:22:57Z 2009-11-04T13:22:57Z

<p>In that post, my intuition was simply that the probability of a random walk in d dimensions being at the origin at time t scales like t^-d/2, and &sum;_{t &gt; 0} t^-d/2 converges if and only if d &gt; 2. Obviously this can only be turned into a proof if <i>d</i> is an integer.</p>

http://mathoverflow.net/questions/4068/how-many-dimensions-is-it-safe-to-get-drunk-in/4084#4084 Gerald Edgar 2009-11-04T14:10:07Z 2009-11-04T14:10:07Z

<p>There are some "fractal" meanings to a question like this.</p> <p>See the delightful little Carus Monograph <em>Random Walks and Electric Networks</em> by Doyle and Snell. At the end of Chapter 6 they discuss a certain tree that has dimension log 6/log 2 = 2.5849... It is used to show that the 3-dimensional lattice is transient.</p>

http://mathoverflow.net/questions/4068/how-many-dimensions-is-it-safe-to-get-drunk-in/4118#4118 Ori Gurel-Gurevich 2009-11-04T17:12:42Z 2009-11-04T17:12:42Z

<p>Indeed, the answer is 2+&#1013; in most senses. For example, consider a <b>wedge</b> in Z³ which is all the points (x,y,z) such that |z|&lt;f(x) for some specific increasing function f:N->N .</p> <p>These wedges are transient already for pretty slowly growing functions. If you take f(x)=x^&#1013; which correspond, in the sense of volume growth, to 2+&#1013; dimensions, then it is transient for any &#1013;>0.</p> <p>One can get even more refined results by taking f(x)=log(x)^a, in which case the wedge is transient if and only if a>1.</p> <p>You can find more information in the <a href="http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html" rel="nofollow">book of Lyons with Peres</a>.</p>