Walks avoiding a quadrant and the reflection principle

TL;DR

This work advances the enumeration of plane lattice walks avoiding the negative quadrant by solving the king model in the three-quadrant setting, establishing that the generating function $C(x,y;t)$ splits into a simple D-finite part plus an algebraic part of high degree (216). The authors develop and apply a kernel-based functional equation, orbit-sum techniques, and a one-catalytic-variable reduction to obtain a full algebraic description of the king model, including univariate and bivariate auxiliary series and explicit degree bounds. They also connect these results to Weyl-models, conjecturing and proving algebraicity for several cases via vanishing orbit sums, and provide combinatorial proofs of reflection-type identities for square lattice variants. The results yield precise asymptotics for the number of king walks in the three-quadrant cone and illustrate a broad algebraicity phenomenon extending the reflection principle to multiple cone configurations with finite groups. Overall, the paper deepens the understanding of how kernel methods and group actions structure the enumeration of constrained lattice walks and highlights a pathway for tackling remaining Weyl-model cases.

Abstract

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We expect a similar algebraicity phenomenon to hold for the seven Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.

Figures (6)

• Figure 1: A king walk in the three-quadrant plane $\mathcal{C}$. The associated generating function is D-finite and transcendental (i.e., non-algebraic).
• Figure 2: Structure of the various fields involved in the solution of king walks in $\mathcal{C}$. We have indicated the degrees, and where the main series lie.
• Figure 3: By the reflection principle, walks in the three-quarter plane $\mathcal{C}$ from $(0,0)$ to $(i,j)$ with $i,j \geq 0$ are in bijection with the union of three sets of walks: walks in $\mathcal{C}$ ending at $(-i-2,j)$, walks in $\mathcal{C}$ ending at $(i,-j-2)$, and walks staying completely in the first quadrant $\mathcal{Q}$, ending at $(i,j)$. For more such identities see Proposition \ref{['prop:bijection']}.
• Figure 4: The $2d$ domains $\mathcal{Q}_g$ for $g \in G$, where $G$ has order $2d=4,6, 8$. They are separated by the $d$ walls $W_g$, for $g\in G$ such that $\varepsilon_g=-1$.
• Figure 5: The involution of Proposition \ref{['prop:bij3v-general']} for walks starting at $(0,0)$ and a group of order $6$. The above eight walks capture all possible values of the pair $(g,h)$, where $\mathcal{Q}_g$ is the domain in which the walk ends and $W_h$ the last visited wall. The figure also shows the action of $h$ on the steps.

Theorems & Definitions (36)

• Remark
• Remark
• Proposition 3.1
• proof
• Remark
• Conjecture 3.2
• Theorem 4.1: The GF of king walks
• Proposition 4.2: Walks ending at a prescribed position
• Corollary 4.3
• Lemma 5.1