Algebraic area enumeration of random walks on the honeycomb lattice

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.

26 pages, 2 figures

References in corpus (5)

• Room-Temperature Quantum Hall Effect in Graphene
• Interactions and phase transitions on graphene's honeycomb lattice
• The Colored Hofstadter Butterfly for the Honeycomb Lattice
• Quantum Hall effect in carbon nanotubes and curved graphene strips
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Cited by in corpus (5)

• Combinatorics of generalized Dyck and Motzkin paths
• Algebraic area enumeration for open lattice walks
• On the algebraic area of cubic lattice walks
• Walking on Archimedean Lattices: Insights from Bloch Band Theory
• Lattice random walks and quantum A-period conjecture