Random Walk in confined region and loop configurations

[allanqunzi]

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?

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.

 

[tiny-tim]

welcome to pf!

hi allanqunzi! welcome to pf!

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?

if it was a 1D lattice, the answer would be easy …

just write 1 for left and 0 for right, and count the number of ways of writing N 1s and N 0s ​

you should be able to find a similar method for 2D

(alternatively, googling "random walk" and "lattice" should give you plenty of hints)

 

[techmologist]

Definitely try it for the 1-D case first if you haven't already. Even that requires some thought. If there were no boundaries, both the 1-D and 2-D cases would be easy. But you have to subtract those paths that cross a boundary. To find those, use the method of images (or principle of reflection). If you have two points, A and B, on the same side of a line, the number of paths between those two points that cross or touch the line is the same as the total number of paths between A* and B, where A* is the reflection of A through that line.