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Walks in the quadrant with interacting boundaries : genus zero case

Pierre Bonnet

Abstract

The study of lattice walks restricted to the first quadrant has shed a lot of interest in the past twenty years. In particular, there has been an important effort to classify models of weighted walks with small steps with respect to the algebraic-differential nature of their generating function. The techniques that were developed in the course of this work are now applied to different extensions of those walks. One of these extensions, called walks with interacting boundaries, consists in accounting for the number of contacts of the walk with the axes, with motivation coming from statistical physics. These contacts are encoded as two additional parameters for the generating function, the Boltzmann weights. For one notable family of models, called genus zero models, we establish in this paper the complete classification of their generating function, for all real values of the parameters. We do this by adapting to this more general case a method due to Dreyfus, Hardouin, Roques and Singer, used in the former classification, and which consists in studying the rational solutions to a $q$-difference equation. In almost all cases, we show that the generating function is hypertranscendental, regardless of the values of the weights. In the remaining cases, we prove that specific algebraic relations between the Boltzmann weights make the generating function $\mathbb{N}$-algebraic or $\mathbb{N}$-rational, contrasting with the interaction-less case.

Walks in the quadrant with interacting boundaries : genus zero case

Abstract

The study of lattice walks restricted to the first quadrant has shed a lot of interest in the past twenty years. In particular, there has been an important effort to classify models of weighted walks with small steps with respect to the algebraic-differential nature of their generating function. The techniques that were developed in the course of this work are now applied to different extensions of those walks. One of these extensions, called walks with interacting boundaries, consists in accounting for the number of contacts of the walk with the axes, with motivation coming from statistical physics. These contacts are encoded as two additional parameters for the generating function, the Boltzmann weights. For one notable family of models, called genus zero models, we establish in this paper the complete classification of their generating function, for all real values of the parameters. We do this by adapting to this more general case a method due to Dreyfus, Hardouin, Roques and Singer, used in the former classification, and which consists in studying the rational solutions to a -difference equation. In almost all cases, we show that the generating function is hypertranscendental, regardless of the values of the weights. In the remaining cases, we prove that specific algebraic relations between the Boltzmann weights make the generating function -algebraic or -rational, contrasting with the interaction-less case.
Paper Structure (23 sections, 35 theorems, 77 equations, 5 figures, 6 tables)

This paper contains 23 sections, 35 theorems, 77 equations, 5 figures, 6 tables.

Table of Contents

  1. Quadrant walks and $q$-difference equations
  2. Quadrant walks with interacting boundaries
  3. Genus zero models
  4. The q-difference equations
  5. The D-algebraicity of $Q(x,y)$, $F(s)$, $G(s)$
  6. Classification strategy
  7. Showing non D-algebraicity
  8. Retrieving D-algebraic solutions
  9. Decoupling equations
  10. Homogeneous equation
  11. Inhomogeneous equation
  12. σ-distance
  13. Computing the $\sigma$-distance
  14. Classification
  15. Some decouplings and homogeneous solutions
  16. ...and 8 more sections

Key Result

Theorem 1

For any weighted genus $0$ model, the generating function $Q(x,y)$ of weighted walks in the quadrant with interacting boundaries has the following nature in the variables $x$ and $y$:

Figures (5)

  • Figure 1: A walk in the quadrant
  • Figure 1.1: A walk in the quadrant with set of steps $\mathcal{S} = \{(-1,1), (1,-1), (1,1)\}$, using 13 steps, 2 contacts with the $x$-axis and 1 contact with the $y$-axis.
  • Figure 1.2: The five models of genus $0$
  • Figure 6.1: Phase diagrams for model $\mathcal{S}_2$ with $d_{-1,1} = d_{1,-1} = d_{1,0} = d_{0,1} = 1$.
  • Figure : The five considered sets of steps

Theorems & Definitions (71)

  • Theorem : Theorem \ref{['thm:thm_clas']}, Section \ref{['sec:full-classification']}
  • Proposition 1.2: Section 4.1 of DreyfusHardouinRoquesSingerGenuszero2
  • Proposition 1.3: Section 1.3 of DreyfusHardouinRoquesSingerGenuszero
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • ...and 61 more