Hamiltonian and exclusion statistics approach to discrete forward-moving paths.

@article{Ouvry2021HamiltonianAE,
  title={Hamiltonian and exclusion statistics approach to discrete forward-moving paths.},
  author={St'ephane Ouvry and Alexios P. Polychronakos},
  journal={Physical review. E},
  year={2021},
  volume={104 1-1},
  pages={
          014143
        }
}
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in… 

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  • S. OuvryShuang Wu
  • Mathematics, Computer Science
    Journal of Physics A: Mathematical and Theoretical
  • 2019
TLDR
A formula for the enumeration of closed lattice random walks of length n enclosing a given algebraic area is proposed, contained in the Kreft coefficients which encode the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field.
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