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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.