Exclusion statistics and lattice random walks

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.