Algebraic area enumeration of random walks on the honeycomb lattice
arXiv:2107.10851 · doi:10.1103/PhysRevE.105.014112
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 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 (fermion) and 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.
arXiv comment26 pages, 2 figures
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