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Hamiltonian and exclusion statistics approach to discrete forward-moving paths
Stéphane Ouvry1, Alexios P Polychronakos2
1LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France.
Physical Review. E
|August 20, 2021
Summary
This study introduces a Hamiltonian description for height-restricted Dyck paths, yielding a novel method to calculate path length and area generating functions using determinants and quantum exclusion statistics.
Area of Science:
- Combinatorics and Statistical Physics
- Mathematical Physics
Background:
- Dyck paths are fundamental in combinatorics and probability.
- Generating functions are key tools for enumerating combinatorial objects.
Purpose of the Study:
- To develop a Hamiltonian (transition matrix) description for height-restricted Dyck paths.
- To evaluate the length and area generating function for these paths with arbitrary start and end points.
Main Methods:
- Utilizing a Hamiltonian description where generating functions are matrix elements of the propagator.
- Expressing the generating function as a rational combination of determinants.
- Connecting random walks to quantum exclusion statistics.
- Formulating the generating function using grand partition functions for exclusion particles.
Main Results:
- The length and area generating function is expressed as a rational combination of determinants.
- An alternative, simpler form for the logarithm of the generating function is presented, explicitly revealing its polynomial structure.
Conclusions:
- The Hamiltonian approach provides an effective framework for analyzing height-restricted Dyck paths.
- The established connection to quantum exclusion statistics offers new insights into path enumeration problems.
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