Enumeration of closed random walks in the square lattice according to their areas

Details Enumeration of closed random walks in the square lattice according to their areas Enumeration of closed random walks in the square lattice according to their areas

2010-12-16

arxiv.org/abs/1012.3720v1

Mathematics

Combinatorics

12 pages, 1 Figure

Scientific paper

We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple argument. It is concluded that the number of walks of length $n$ and area $s$ equals to the coefficient of $z^s$ in the expression $(x+x^{-1}+y+y^{-1})^n$, where the calculations are performed in a special group ring $R[x,y,z]$. A polynomial time algorithm for calculating these values, is then concluded. Finally, the provided algorithm and the results of implementation are compared with previous works.

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