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

TL;DR

This paper investigates the distribution of areas of closed walks on a square lattice, introducing a generalized concept of walk area, and provides an efficient polynomial-time algorithm for enumeration, comparing results with prior research.

Contribution

It introduces a new generalized concept of walk area and develops a polynomial-time algorithm for enumerating closed walks by their area in the square lattice.

Findings

Number of walks of length n and area s equals the coefficient of z^s in (x+x^{-1}+y+y^{-1})^n

Developed a polynomial-time algorithm for calculating walk areas

Compared new algorithm and results with previous studies

Abstract

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.