Suppose I take a random walk on a 2 dimensional square lattice, but this lattice plane has a finite size, e.g. Dx*Dy. I can not cross the boundary, my step length is the lattice cell size, I either go straight or make turns with right angle. Is there any work on this type of random walk?(adsbygoogle = window.adsbygoogle || []).push({});

If the total walking distance is fixed, e.g. 2N, in this confined region, how may ways are there that at the 2Nth step I am at the starting position? Or equivalently, how many configurations of loops of length 2N are there in this confined square lattice? How about if this loop is a self avoiding loop?

Anybody knows related work on this? Since my major is physics, I am not sure if some mathematicians have done this. Thanks in advance.

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# Random Walk in confined region and loop configurations

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