# 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…

## 4 Citations

Length and area generating functions for height-restricted Motzkin meanders.

- MathematicsPhysical review. E
- 2022

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

- Mathematics
- 2022

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…

## References

SHOWING 1-10 OF 25 REFERENCES

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

- Mathematics
- 2020

Skew Schur Function Representation of Directed Paths in a Slit

- Mathematics
- 2019

In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Melou, who expressed generating functions of…

Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields

- Physics
- 1976

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…

Discrete excursions

- Mathematics
- 2007

for instan e, when S = { a, − b } with a and b oprime, D ( t,z ) is irredu ible, and is thus the minimal polynomial of the ex ursion generating fun tion E ( t ) . The proofs of these results involve…

The algebraic area of closed lattice random walks

- Mathematics, Computer ScienceJournal of Physics A: Mathematical and Theoretical
- 2019

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.