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Length derivative of the generating series of walks confined in the quarter plane

Thomas Dreyfus, Charlotte Hardouin

Abstract

In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of walks confined in the first quadrant of the plane with a fixed set of admissible steps, called the model of the walk. While the variables $x$ and $y$ are associated to the ending point of the path, the variable $t$ encodes its length. In this paper, we prove that in the unweighted case, $Q(x,y,t)$ satisfies an algebraic differential relation with respect to $t$ if and only if it satisfies an algebraic differential relation with respect $x$ (resp. $y$). Combined with other papers, we are able to characterize the $t$-differential transcendence of the $79$ models of walks listed by Bousquet-Mélou and Mishna.

Length derivative of the generating series of walks confined in the quarter plane

Abstract

In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series that count the number of walks confined in the first quadrant of the plane with a fixed set of admissible steps, called the model of the walk. While the variables and are associated to the ending point of the path, the variable encodes its length. In this paper, we prove that in the unweighted case, satisfies an algebraic differential relation with respect to if and only if it satisfies an algebraic differential relation with respect (resp. ). Combined with other papers, we are able to characterize the -differential transcendence of the models of walks listed by Bousquet-Mélou and Mishna.

Paper Structure

This paper contains 26 sections, 55 theorems, 150 equations, 3 figures.

Table of Contents

  1. The walks in the quadrant
  2. The walks
  3. The kernel curve
  4. The automorphism of the walk
  5. Generating functions for walks, genus zero case
  6. Uniformization of the kernel curve
  7. Meromorphic continuation of the generating functions
  8. Differential transcendence in the genus zero case
  9. Generating functions of walks, genus one case
  10. Uniformization of the kernel curve
  11. The group of the walk
  12. Meromorphic continuation
  13. Differential transcendence
  14. Nonarchimedean estimates
  15. Discriminants of the kernel equation
  16. ...and 11 more sections

Key Result

Theorem 1

For any of the $79$ models of Figure figcas, the complete generating function is $\frac{d}{dt}$-differentially algebraic over $\mathbb{Q}$ if and only if it is $\frac{d}{dx}$-differentially algebraic over $\mathbb{Q}$.

Figures (3)

  • Figure 1: Classification of the $79$ models with respect to the $x$ and $y$-variables.
  • Figure 2: The maps $\iota_{1},\iota_{2}$ restricted to the kernel curve $E$
  • Figure 3: The plain circles correspond to $|s|=|q|^{\pm 1/2}$. The dashed circles correspond to $|\overline{x}(s)|= 1$.

Theorems & Definitions (112)

  • Theorem 1: Theorem \ref{['thm:genre0']} and Corollary \ref{['cor:diffalgxequivalentdiffalgtgenusone']} below
  • Theorem 2: Theorems \ref{['thm:genre0']} and \ref{['theo2']} below
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Proposition 1.6: Lemma 2.3.2 in FIM
  • Definition 1.7: § 2.3.5 in DuistQRT
  • Proposition 1.8: Proposition 2.4.3 in DuistQRT and Proposition 2.1 in DreyfusHardouinRoquesSingerGenuszero2
  • ...and 102 more