Algebraic area enumeration of random walks on the honeycomb lattice

TL;DR

This paper explores the enumeration of closed walks on the honeycomb lattice by mapping the problem to a Hofstadter-like Hamiltonian, revealing connections to exclusion statistics and Fibonacci sequences.

Contribution

It introduces a novel algebraic approach to counting lattice walks, linking combinatorics with quantum statistical models and uncovering unexpected Fibonacci patterns.

Findings

Mapping of walk enumeration to a Hofstadter-like Hamiltonian.

Identification of exclusion statistics of order g=2 in the generating function.

Discovery of Fibonacci sequences in combinatorial weights.

Abstract

We study the enumeration of closed walks of given length and algebraic area on the honeycomb lattice. Using an irreducible operator realization of honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian and show that the generating function of closed walks maps to the grand partition function of a system of particles with exclusion statistics of order $g=2$ and an appropriate spectrum, along the lines of a connection previously established by two of the authors. Reinterpreting the results in terms of the standard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) and $g=2$ exclusion whose physical meaning and properties require further elucidation. In this context we also obtain some unexpected Fibonacci sequences within the weights of the combinatorial factors appearing in the counting of walks.