Integrability of Combinatorial Riemann Boundary Value Problem and Lattice Walks Avoiding a Quadrant

Abstract

We introduce a general framework of matrix-form combinatorial Riemann boundary value problem (cRBVP) to characterize the integrability of functional equations arising in lattice walk enumerations. A matrix cRBVP is defined as integrable if it can be reduced to enough polynomial equations with one catalytic variable. Our central results establish that the integrability depends on the eigenspace of some matrix associated to the problem. For lattice walks in three quadrants, we demonstrate how the obstinate kernel method transforms a discrete difference equation into a $3\times 3$ matrix cRBVP. The special double-roots eigenvalue $1/4$ yields two independent polynomial equations in the problem. The other single-root eigenvalue yields a linear equation. We obtain three independent equations from a $3\times 3$ system. Crucially, our framework generalizes three-quadrant walks with Weyl symmetry to models satisfying only orbit-sum conditions. It explains many criteria about the orbit-sum proposed by various researchers and it also explains the counter-example of lattice walks starting outside the quadrant.