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Chip Eastham
science forum Guru

Joined: 01 May 2005
Posts: 412

Posted: Tue May 30, 2006 3:29 pm    Post subject: Re: 3 Dimensional Random Walk

Jim Dars wrote:
 Quote: "Barry Schwarz" wrote in message news:ssoh72tir6bict6kvvqmhgqgpldp7mqo6r@4ax.com... On Sat, 27 May 2006 13:32:50 -0700, "Jim Dars"

I can recommend this (free) monograph by Doyle and Snell on the
subject:

http://front.math.ucdavis.edu/math.PR/0001057

In two dimensions, a random walk on the "graph" connecting
the usual integer lattice points visits every point in the plane
with probability 1. So in that setting it doesn't matter whether
we ask about "returning" to the origin or going from one point
to another, the probability is the same. This can be deduced
from the divergence of SUM 1/n.

In three dimensions, as Jim quotes Feller, the chance of
returning to an original starting point is positive and strictly
less than 1 (about .35). This is related to the convergence
of SUM 1/(n^2), although the connection is not elegant when
made rigorous. The search for a simple connection motivates
much of the monograph above.

In short we can define a function that gives the probability
that a random walk starting at the origin will visit (at some
subsequent step) any specific point in the integer lattice.
Evaluation of this function at (1000,1000,1000) will give by
symmetry the answer to Jim's question, and in general
this function is monotone decreasing as we go farther
from the origin.

regards, chip
Jim Dars
science forum beginner

Joined: 03 Jul 2005
Posts: 41

Posted: Wed May 31, 2006 9:03 pm    Post subject: Re: 3 Dimensional Random Walk

"Chip Eastham" <hardmath@gmail.com> wrote in message
 Quote: Jim Dars wrote: "Barry Schwarz" wrote in message news:ssoh72tir6bict6kvvqmhgqgpldp7mqo6r@4ax.com... On Sat, 27 May 2006 13:32:50 -0700, "Jim Dars"

Thank you. I suspected as much in that if you start at the origin you have
a 1/6 probability of reaching the origin on the second move. It's
interesting to learn that, as you move the starting point away from the
origin, the probability of reaching the origin diminishes. A little
reflection does indicate this as a logical conclusion, but seemingly logical
conclusions often blow up in your face.

Best wishes, Jim
Mike Amling
science forum Guru

Joined: 05 May 2005
Posts: 525

Posted: Thu Jun 08, 2006 10:50 pm    Post subject: Re: 3 Dimensional Random Walk

Chip Eastham <hardmath@gmail.com> wrote:
 Quote: In short we can define a function that gives the probability that a random walk starting at the origin will visit (at some subsequent step) any specific point in the integer lattice. Evaluation of this function at (1000,1000,1000) will give by symmetry the answer to Jim's question, and in general this function is monotone decreasing as we go farther from the origin.

THe probability that a random walk starting at the origin will
eventually return to the origin should be the same as the probability
that a random walk starting at (1,0,0), (0,1,0), (0,0,1), (-1,0,0),
(0,-1,0), or (0,0,-1) will eventually reach the origin.

--
pciszek at panix dot com | indistinguishable from malice."

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