Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counting quadrant walks via Tutte's invariant method

Olivier Bernardi, Mireille Bousquet-Mélou, Kilian Raschel

TL;DR

The paper develops a two-pronged framework for quadrant lattice walks: a Tutte-inspired algebraic approach using rational invariants and decoupling functions to prove algebraicity for a class of models (notably including Gessel’s model), and a complex-analytic framework with weak invariants that yields integral-free, D-algebraic expressions for non-D-finite cases. It establishes that a finite group of the model guarantees a rational invariant and, when the orbit-sum vanishes, a decoupling function, enabling uniform algebraicity proofs across the algebraic models; for infinite groups, it introduces a weak invariant w(y) and shows DA properties for decoupled models, with explicit differential equations in y and t. The work also develops an invariant-analytic bridge, connecting algebraic/invariant methods to analytic tools, and extends the analysis to starting points other than (0,0). Overall, the paper provides a near-complete classification of small-step quadrant models by their invariant/decoupling structure and demonstrates powerful, uniform techniques to derive algebraicity, D-algebraicity, or non-D-finiteness. This advances both the combinatorial enumeration of quadrant walks and the broader theory of functional equations with multiple catalytic variables.

Abstract

In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 20 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in $\{-1, 0,1\}^2$. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically. Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).

Counting quadrant walks via Tutte's invariant method

TL;DR

The paper develops a two-pronged framework for quadrant lattice walks: a Tutte-inspired algebraic approach using rational invariants and decoupling functions to prove algebraicity for a class of models (notably including Gessel’s model), and a complex-analytic framework with weak invariants that yields integral-free, D-algebraic expressions for non-D-finite cases. It establishes that a finite group of the model guarantees a rational invariant and, when the orbit-sum vanishes, a decoupling function, enabling uniform algebraicity proofs across the algebraic models; for infinite groups, it introduces a weak invariant w(y) and shows DA properties for decoupled models, with explicit differential equations in y and t. The work also develops an invariant-analytic bridge, connecting algebraic/invariant methods to analytic tools, and extends the analysis to starting points other than (0,0). Overall, the paper provides a near-complete classification of small-step quadrant models by their invariant/decoupling structure and demonstrates powerful, uniform techniques to derive algebraicity, D-algebraicity, or non-D-finiteness. This advances both the combinatorial enumeration of quadrant walks and the broader theory of functional equations with multiple catalytic variables.

Abstract

In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 20 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in . We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically. Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).

Paper Structure

This paper contains 47 sections, 38 theorems, 242 equations, 9 figures, 11 tables.

Table of Contents

  1. Introduction
  2. First steps to quadrant walks
  3. A new solution of Gessel's model
  4. Extensions and obstructions: uniform algebraicity proofs
  5. Invariants
  6. Decoupling functions
  7. The finite group case
  8. The infinite group case.
  9. Case 1: $\boldsymbol{a(x)=t(1+x^2)}$.
  10. Case 2: $\boldsymbol{a(x)=t(1+x+x^2)}$.
  11. Case 3: $\boldsymbol{a(x)=t(1+x)}$.
  12. Case 4: $\boldsymbol{a(x)=tx}$.
  13. Case 5: the last two models.
  14. The invariant lemma
  15. An alternative to the invariant lemma
  16. ...and 32 more sections

Key Result

Lemma 3.1

Let $X=t+t^2(u+\bar{u})$, where $u$ is a new variable and $\bar{u}$ stands for $1/u$. We slightly abuse notation by denoting $Y_0$ and $Y_1$ the roots of $K(X,y)$. Then $Y_0$ and $Y_1$ are Laurent series in $t$ with coefficients in $\mathbb{Q}(u)$, satisfying The series $Y_0$ and $Y_1$ simply differ by the transformation $u\mapsto \bar{u}$. For $i\in\{0,1\}$, the series $Q(X,Y_i)$ and $Q(0,Y_i)$

Figures (9)

  • Figure 1: Three models of walks in the quadrant. The generating function $Q(x,y;t)$ is algebraic for the first one, but only D-algebraic for the second. For the third one, it is not even D-algebraic.
  • Figure 3: The nine D-algebraic models having an infinite group.
  • Figure 4: The support of the series $A(y)$, shown with dots, and the definition of $m$.
  • Figure 5: Plot of $d(x)$ for $x$ real: the main two possibilities, depending on the sign of $x_4$. Note that $x_1$ may be non-negative, and $x_4$ may be $+\infty$.
  • Figure 6: The curves $\mathcal{L}$ for model ${\#3}$ of Table \ref{['tab:decoupling_functions-infinite']} (for $t=0.03$, $0.1$, $0.2$, $0.25=1/\vert\mathcal{S}\vert$ and $0.263185\ldots$ as one moves closer to the origin) and for the reverse Kreweras model (second model in Table \ref{['tab:decoupling_functions-finite']}; from right to left, $t=0.2$, $0.25$, $0.28$, $0.3$ and $1/3=1/\vert\mathcal{S}\vert$). The dashed curve is obtained for a value $t_c\geqslant 1/\vert\mathcal{S}\vert$, where $\mathcal{L}$ stops being smooth, but here we only consider values of $t$ less than $1/\vert\mathcal{S}\vert$.
  • ...and 4 more figures

Theorems & Definitions (77)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Lemma 4.4
  • ...and 67 more