Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers

@article{Ouvry2020LatticeWA,
  title={Lattice walk area combinatorics, some remarkable trigonometric sums and Ap{\'e}ry-like numbers},
  author={St'ephane Ouvry and Alexios P. Polychronakos},
  journal={Nuclear Physics},
  year={2020},
  volume={960},
  pages={115174}
}
View PDF on arXiv
8 Citations
Background Citations
2
Methods Citations
1

Figures from this paper

8 Citations

Algebraic area enumeration of random walks on the honeycomb lattice.
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…
A shifted binomial theorem and trigonometric series
We introduce a shifted version of the binomial theorem, and use it to study some remarkable trigonometric integrals and their explicit rewriting in terms of binomial multiple sums. Motivated by the…
Algebraic area enumeration for lattice paths
We give a summary of recent progress on the algebraic area enumeration of closed paths on planar lattices. Several connections are made with quantum mechanics and statistical mechanics. Explicit…
Hamiltonian and exclusion statistics approach to discrete forward-moving paths.
TLDR
This work expresses the length and area generating function for paths with arbitrary starting and ending points in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and presents an alternative, simpler form for its logarithm that makes its polynomial structure explicit.
Exclusion statistics for particles with a discrete spectrum
Length and area generating functions for height-restricted Motzkin meanders.
We derive the length and area generating function of planar height-restricted forward-moving discrete paths of increments ±1 or 0 with arbitrary starting and ending points, the so-called Motzkin…
Algebraic area enumeration for open lattice walks
We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area…
2 Square lattice walks : the Hofstadter model

    References

    SHOWING 1-8 OF 8 REFERENCES
    The algebraic area of closed lattice random walks
    • S. Ouvry, Shuang 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.
    Bijective counting of Kreweras walks and loopless triangulations
    Exclusion statistics and lattice random walks
    Non-relativistic bosonization and fractional statistics☆
    Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields
    An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing…
    Equation of state of an anyon gas in a strong magnetic field.
    The statistical mechanics of an anyon gas in a magnetic field is addressed. A harmonic regulator is used to define a proper thermodynamic limit. When the magnetic field is sufficiently strong, only…
    "Fractional statistics" in arbitrary dimensions: A generalization of the Pauli principle.
    • Haldane
    • Physics
      Physical review letters
    • 1991
    TLDR
    Fractional statistics is reformulated as a generalization of the Pauli exclusion principle, and a definition independent of the dimension of space is obtained, which is used to classify spinons in gapless spin-1/2 antiferromagnetic chains as semions.
    Statistical distribution for generalized ideal gas of fractional-statistics particles.
    • Wu
    • Physics
      Physical review letters
    • 1994
    TLDR
    The occupation-number distribution in a generalized ideal gas of particles obeying fractional statistics, including mutual statistics, is derived by adopting a state-counting definition and applications to the thermodynamic properties of quasiparticle excitations in the Laughlin quantum Hall fluid are discussed.