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Square lattice walks avoiding a quadrant

Mireille Bousquet-Mélou

TL;DR

This work advances the enumeration of two-dimensional lattice walks confined to a non-convex three-quadrant cone by square- and diagonal-lattice models, showing that the generating function $C(x,y)$ splits into a simple D-finite part plus an algebraic correction. The authors develop and apply an orbit-sum/kernel framework, introducing a quadrant-like reduction and a generalized quadratic method to prove algebraicity of key components, and they obtain rational parametrizations and degree bounds for the resulting series. They connect their results to Gessel's walks via reflection principles, yielding new algebraic and hypergeometric expressions for endpoint counts and asymptotics. Overall, the paper extends the quadrant-walk methodology to non-convex cones, providing a structured approach and concrete parametrizations that could guide broader studies of walks avoiding a quadrant and related models.

Abstract

In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear transformation). But what about walks in non-convex cones? We investigate the two most natural cases: first, square lattice walks avoiding the negative quadrant Q 1 = {(i, j) : i \textless{} 0 and j \textless{} 0}, and then, square lattice walks avoiding the West quadrant Q 2 = {(i, j) : i \textless{} j and i \textless{} --j}. In both cases, the generating function that counts walks starting from the origin is found to differ from a simple D-finite series by an algebraic one. We also obtain closed form expressions for the number of n-step walks ending at certain prescribed endpoints, as a sum of three hypergeometric terms. One of these terms already appears in the enumeration of square lattice walks confined to the cone {(i, j) : i +j $\ge$ 0 and j $\ge$ 0}, known as Gessel's walks. In fact, the enumeration of Gessel's walks follows, by the reflection principle, from the enumeration of walks starting from (--1, 0) and avoiding Q 1. Their generating function turns out to be purely algebraic (as the generating function of Gessel's walks). Another approach to Gessel's walks consists in counting walks that start from (--1, 1) and avoid the West quadrant Q 2. The associated generating function is D-finite but transcendental.

Square lattice walks avoiding a quadrant

TL;DR

This work advances the enumeration of two-dimensional lattice walks confined to a non-convex three-quadrant cone by square- and diagonal-lattice models, showing that the generating function splits into a simple D-finite part plus an algebraic correction. The authors develop and apply an orbit-sum/kernel framework, introducing a quadrant-like reduction and a generalized quadratic method to prove algebraicity of key components, and they obtain rational parametrizations and degree bounds for the resulting series. They connect their results to Gessel's walks via reflection principles, yielding new algebraic and hypergeometric expressions for endpoint counts and asymptotics. Overall, the paper extends the quadrant-walk methodology to non-convex cones, providing a structured approach and concrete parametrizations that could guide broader studies of walks avoiding a quadrant and related models.

Abstract

In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear transformation). But what about walks in non-convex cones? We investigate the two most natural cases: first, square lattice walks avoiding the negative quadrant Q 1 = {(i, j) : i \textless{} 0 and j \textless{} 0}, and then, square lattice walks avoiding the West quadrant Q 2 = {(i, j) : i \textless{} j and i \textless{} --j}. In both cases, the generating function that counts walks starting from the origin is found to differ from a simple D-finite series by an algebraic one. We also obtain closed form expressions for the number of n-step walks ending at certain prescribed endpoints, as a sum of three hypergeometric terms. One of these terms already appears in the enumeration of square lattice walks confined to the cone {(i, j) : i +j 0 and j 0}, known as Gessel's walks. In fact, the enumeration of Gessel's walks follows, by the reflection principle, from the enumeration of walks starting from (--1, 0) and avoiding Q 1. Their generating function turns out to be purely algebraic (as the generating function of Gessel's walks). Another approach to Gessel's walks consists in counting walks that start from (--1, 1) and avoid the West quadrant Q 2. The associated generating function is D-finite but transcendental.

Paper Structure

This paper contains 39 sections, 9 theorems, 216 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The square lattice
  3. Warming up: walks confined to the positive quadrant
  4. Reduction to an equation with orbit sum zero
  5. Reduction to a quadrant-like problem
  6. Cancelling the kernel: an equation between $\boldsymbol{M(x,0)}$, $\boldsymbol{M(0,x)}$ and $\boldsymbol{M(0,\bar{x})}$
  7. An equation between $\boldsymbol{M(0,x)}$ and $\boldsymbol{M(0,\bar{x})}$
  8. An equation for $\boldsymbol{M(0,x)}$ only
  9. The generalized quadratic method: algebraicity of $\boldsymbol{A(x,y)}$
  10. General principle.
  11. Identifying the series $\boldsymbol{X_i}$.
  12. Elimination.
  13. Rational parametrizations and degrees
  14. Walks ending at a prescribed position
  15. The diagonal square lattice
  16. ...and 24 more sections

Key Result

Theorem 1

The generating function of square lattice walks starting at $(0,0)$, confined to $\mathcal{C}$ and ending in the first quadrant (resp. at a negative abscissa) is where $M(x,y)$ and $P(x,y)$ are algebraic of degree $72$ over $\mathbb{Q}(x,y,t)$. More precisely, $P$ can be expressed in terms of $M$ by: $M$ satisfies the functional equation and the specializations $M(x,0)$ and $M(0,y)$ have respec

Figures (5)

  • Figure 1: Square lattice walks staying in a $135\degree$ wedge are equivalent to quadrant walks with steps $\rightarrow, \nearrow, \leftarrow, \swarrow$.
  • Figure 2: Walks confined to the non-convex cone $\mathcal{C}$: on the square lattice (left), and on the diagonal square lattice (right).
  • Figure 3: The reflection principle: a walk from $(-1,0)$ to $(i,j)$ that crosses the line $y=x+1$ can be transformed bijectively into a walk ending at $(j,i)$.
  • Figure 4: Second application of the reflection principle.
  • Figure 5: The four algebraic quadrant models.

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 2: Walks ending at a prescribed position
  • Proposition 3
  • Theorem 4
  • Corollary 5: Walks ending at a prescribed position
  • Theorem 6
  • Corollary 7: Walks ending at a prescribed position
  • Theorem 8
  • Corollary 9: Walks ending at a prescribed position