An appropriate candidate for exact distribution of closed random walks using quantum groups

TL;DR

This paper explores the relationship between quantum groups and random walks on a 2D lattice, proposing an exact area distribution for closed walks using quantum group structures and validating with enumeration.

Contribution

It introduces a novel approach linking quantum groups to random walk distributions, providing an exact formula for the area distribution of closed walks.

Findings

Derived an exact area distribution for closed random walks

Established a connection between quantum groups and lattice walks

Validated results with exact enumeration

Abstract

We show that the structure of the quantum group $su_{q}(2)$ is intimately related to the random walks on a two dimentional lattice. Using this connection we obtain an appropriate candidate for the exact area distribution of closed random walks of length $N$ on a two dimensional square lattice. We compare our results with exact enumeration.