No. of 2-D random walks of fixed length

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#26

From: David L Dowe
Date: Fri, 13 Oct 1995 00:00

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   The 1-dimensional random walk, be it along a continuum or at the
discrete, integer-valued points, is guaranteed (with probability 1)
to visit each point infinitely often.

   The number of ways that the 1-D random walk can have length (2m + 1)
is of the order of the Catalan number, (2m)!/(m! (m+1)!) .

I use an odd (non-even) length, 2m+1, above since I wish to stop the
walk above on the first occasion it steps immediately left after a
return to the origin.


Let us now consider a 2-D walk, which we shall say terminates (e.g.)
upon first return to the origin.

What is the number of ways that such a walk can have length l ?


Thanks.   - David Dowe.

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From: dld@cs.monash.edu.au (David L Dowe)
Subject: No. of 2-D random walks of fixed length
Date: 1995/10/13
Message-ID: <dld.813562226@bruce.cs.monash.edu.au>
X-Deja-AN: 117518693
sender: news@bruce.cs.monash.edu.au (USENET News System)
organization: Computer Science, Monash University, Australia
keywords: random walk, random, walk, count
newsgroups: sci.stat.math,sci.math,aus.mathematics
summary: Query re number of No. of 2-D random walks of fixed length