The Weighted Walks in Quadrant with Finite Groups: an Algebro-Geometric Approach

We classify weighted small-step lattice walks in the quadrant whose associated birational group is finite. Using an algebro-geometric description of the kernel curves and Cayley-type finite-order conditions, we relate the group $G_W$ of a walk to the family of groups $Γ_t$ acting on its kernel curves. Together with the uniform upper bound on the order of $G_W$, this allows us to analyse all possible finite orders. We obtain explicit necessary and sufficient conditions for the group to have order 4, 6, 8, or 10, and prove that no weighted walk in the quadrant has a group of order 12. This yields a complete classification of weighted quadrant walks with finite groups.