From: (David Bernier).XOR.(Michel Bernier) Subject: equivalent resistance of 3-D grid from node to oo Date: Thu, 11 Mar 1999 17:35:15 GMT Newsgroups: sci.math,sci.physics Keywords: probability of return in random walk on Z^n In article <7c7nll$9mg$1@nnrp1.dejanews.com>, (David Bernier).XOR.(Michel Bernier) wrote: [revised and corrected posting below...] In article <36CD2251.A7B25E8C@pisquaredoversix.force9.co.uk>, Clive Tooth wrote: [...] > btw... I would be grateful to learn the resistance of a 3-dimensional > lattice of one ohm resistors between some node and infinity. I am kinda > hoping that it may be some simple multiple of pi^2/6 :) This problem is related to random walks on the group ZxZxZ . (See the reference given by Clive Tooth: http://math.ucsd.edu/~doyle/docs/net/net/net.html ) Unfortunately, problems such as "what is the probability of ever returning to the origin in the usual random walk on the 3-D grid" are very difficult. Fortunately, Ian Stewart, in his January 1999 "Mathematical Recreations" column (Sci. Amer.), in the Feedback Section, gives an expression for this probability (of eventually returning to the origin) that is attributed to the 1939 English mathematician George N. Watson. Unfortunately, the expression given in SA appears to be not quite correct, since it's numerical value is nowhere near the accepted value of ~= 0.35 . Fortunately, it seems like a typo crept in and the said probability, say q, has the value (?) _ _ _ _ _ _ 1-1/{3*(18 +12\/2 -10\/3 -7\/6)*[K( 2\/3 +\/6 -2\/2 -3)]^2} Then an argument with infinite sums shows that the expected number of visits at the origin in the usual walk on the 3-D grid is 1/(1-q), which amounts to the expression in curly brackets above. In this context, K(k) = (2/Pi) * [\int_{0,Pi/2} {[1-(k*sin(x))^2]^(-1/2) dx}] (known as a "complete elliptic integral of the first kind"). Fortunately, 1/(1-q) has a value of about 1.516, which agrees with some Monte Carlo simulations I did recently. So my 3 cents ( Canadian...) contribution is to say that the required equivalent resistance of the 3-D grid of 1 ohm resistors is about 1.516386059152. David Bernier www.mapblast.com www.nytimes.com www.blackvault.com (;-) www.terraserver.microsoft.com www.gsoc.dlr.de/satvis dictionaries.travlang.com www.bldrdoc.gov/timefreq/javaclck.htm -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: proba of return in R^3 Date: 19 May 1999 00:02:16 GMT Newsgroups: sci.math To: bfcedm02@club-internet.fr (Brieuc) In article <3741de84.6094003@news.club-internet.fr>, bfcedm02@club-internet.fr (Brieuc) writes: > For a random-walk, or a brownian-motion > in the real-line or in the plane, > the probability of return to the starting > point is 1. For random walk on a 2-dimensional lattice the probability of return is 1. For Brownian motion in the plane it is 0 (i.e. you never return exactly to the starting point, although with probability 1 you come arbitrarily close). > I can't retrieve where I have read that > in the 3D-space, this probability is a > number inside (0,1). (like 2.9.. or 3.7..) > > - What is the right 3D-proba-number ? For random walk on a 3-dimensional square lattice, the probability of return is approximately 0.3405373296: see http://www.mathsoft.com/asolve/constant/polya/polya.html. For Brownian motion, again, it is 0. > - Is there a proper-definition of what > should be understood by "proba-of-return" ? > May-be its a density-of-return, in the > continuous case ? Probability of return is exactly what you would expect: the probability that the process, starting at a given point, eventually returns to that point. In continuous cases you can ask something like: given that you start at 0, what is the probability of ever hitting a sphere of radius r whose centre is a distance x from 0? Of course, this could depend on r and x. But for 1- or 2-dimensional Brownian motion it is 1. > - Is the calculus of this proba in the 3D-case > easy-enough so you can give me a hint ? Finding the exact probability is not easy, proving it is less than 1 is not so hard. A heuristic argument is rather easy. The point is that after N steps, each coordinate has probability approximately proportional to N^(-1/2) of being 0, so the probability that all three are 0 is approximately a constant times N^(-3/2). The expected number of times to hit 0 is the sum of these probabilities for N = 1 to infinity, and this converges. But if the return probability was 1 the expected number of hits would be infinite. > - What would happen in 4D-space ? > (euclidean-isotropic-one, not relativity-one). Similar to 3D. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: "G. A. Edgar" Subject: Re: proba of return in R^3 Date: Wed, 19 May 1999 10:49:59 -0400 Newsgroups: sci.math In article <3741de84.6094003@news.club-internet.fr>, Brieuc wrote: > - What would be a fractional-dim-space, if any, > with max-dim between 2 and 3, where a supposed > definable motion would be coming-back-home > with proba 1 ? For some thoughts on this, and general information on this type of problem, see the book: Doyle & Snell, RANDOM WALKS AND ELECTRIC NETWORKS. A charming little book, suitable for undergraduates. -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax)