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Counting quadrant walks via Tutte's invariant method (extended abstract)

O Bernardi, M Bousquet-Mélou, Kilian Raschel

TL;DR

The paper tackles the enumeration of quadrant lattice walks by adapting Tutte's invariant method to derive generating-function properties. It combines a finite-group invariant theory with a decoupling function to obtain algebraicity for all algebraic quadrant models (including Gessel's model) and introduces an analytic invariant framework based on weak invariants and a conformal gluing function to handle infinite-group cases, yielding integral-free expressions and differential-algebraicity results. A new Tutte-style proof for Gessel's model illustrates the effectiveness of invariants, while the analytic approach produces DA results for nine infinite-group decoupled models and clarifies the structure of the corresponding generating functions. The work outlines a unified program linking invariants, decoupling, and analytic methods to classify and bound the nature of quadrant-walk generating functions, with open questions about the reach of decoupling functions across models and the boundary between DA/D-finite/algebraic cases.

Abstract

In the 1970s, 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 decade 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. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). 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 comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of 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 of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

Counting quadrant walks via Tutte's invariant method (extended abstract)

TL;DR

The paper tackles the enumeration of quadrant lattice walks by adapting Tutte's invariant method to derive generating-function properties. It combines a finite-group invariant theory with a decoupling function to obtain algebraicity for all algebraic quadrant models (including Gessel's model) and introduces an analytic invariant framework based on weak invariants and a conformal gluing function to handle infinite-group cases, yielding integral-free expressions and differential-algebraicity results. A new Tutte-style proof for Gessel's model illustrates the effectiveness of invariants, while the analytic approach produces DA results for nine infinite-group decoupled models and clarifies the structure of the corresponding generating functions. The work outlines a unified program linking invariants, decoupling, and analytic methods to classify and bound the nature of quadrant-walk generating functions, with open questions about the reach of decoupling functions across models and the boundary between DA/D-finite/algebraic cases.

Abstract

In the 1970s, 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 decade 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. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). 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 comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of 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 of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

Paper Structure

This paper contains 9 sections, 10 theorems, 32 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. A new solution of Gessel's model
  3. Extensions and obstructions
  4. An analytic invariant method
  5. Weak invariants
  6. The analytic invariant lemma --- Application to quadrant walks
  7. Differential algebraicity
  8. Further results and final comments
  9. Rational invariants for finite groups

Key Result

Lemma 1

Let $F(y)$ be a Laurent series in $t$ with coefficients in $\mathbb{Q}[y]$, of the form for some $n_0\ge 0$. Then for $x=t+t^2(u+1/u)$, the series $F(Y_0)$ and $F(Y_1)$ are well defined Laurent series in $t$, with coefficients in $\mathbb{Q}(u)$. If they coincide, then $F(y)$ is in fact independent of $y$.

Figures (1)

  • Figure 1: The curves $\mathcal{L}$ for model ${\#3}$ of Table \ref{['tab:decoupling_functions-infinite']} (for $t=0.03$, $0.1$, $0.2$ and $0.25$, as one moves closer to the origin) and for the reverse Kreweras model (second model in Table \ref{['tab:decoupling_functions-finite']}; $t=0.2$, $0.25$, $0.28$ and $0.33$, from right to left).

Theorems & Definitions (14)

  • Lemma 1
  • Definition 2
  • Lemma 3
  • Proposition 4
  • proof
  • Definition 5
  • Proposition 6
  • Lemma 7: FIM-99
  • Definition 8
  • Lemma 9
  • ...and 4 more