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

Stéphane Ouvry1, Alexios P Polychronakos2

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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.

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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.