# Exclusion statistics and lattice random walks

## TL;DR

This paper reveals a novel link between exclusion statistics with arbitrary integer parameters and planar lattice random walks, connecting generating functions of walk areas to grand partition functions of exclusion particles.

## Contribution

It introduces a new correspondence between exclusion statistics and lattice random walks, including explicit models for g=2 and g=3, and derives cluster coefficients for arbitrary g.

## Key findings

- Square lattice walks relate to g=2 exclusion statistics.
- Constructed a g=3 chiral walk on a triangular lattice.
- Derived microscopic cluster coefficients for any g.

## Abstract

We establish a connection between exclusion statistics with arbitrary integer exclusion parameter $g$ and a class of random walks on planar lattices. This connection maps the generating function for the number of closed walks of given length enclosing a given algebraic area on the lattice to the grand partition function of particles obeying exclusion statistics $g$ in a particular single-particle spectrum, determined by the properties of the random walk. Square lattice random walks, described in terms of the Hofstadter Hamiltonian, correspond to $g=2$. In the $g=3$ case we explicitly construct a corresponding chiral random walk model on a triangular lattice, and we point to potential random walk models for higher $g$. In this context, we also derive the form of the microscopic cluster coefficients for arbitrary exclusion statistics.