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, relating the generating function for the algebraic area of closed walks on the lattice to the grand partition function of particles obeying exclusion statistics g. Square lattice random walks, described in terms of the Hofstadter Hamiltonian, correspond to g=2. In the g=3 case we construct a corresponding chiral random walk 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 and one-body spectrum.