标题： 组合Riemann边值问题的可积性与避免一个象限的格子行走 显示英文标题

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

摘要： 我们引入了一种矩阵形式的组合黎曼边值问题（cRBVP）的一般框架，用于刻画出现在格路计数中的函数方程的可积性。 如果一个矩阵cRBVP可以约化为足够多的只有一个催化变量的多项式方程，则称其为可积的。 我们的中心结果表明，可积性取决于与该问题相关的某个矩阵的特征空间。 对于三维象限中的格路，我们展示了顽固核方法如何将离散差分方程转化为一个$3\times 3$矩阵cRBVP。 特殊的二重根特征值$1/4$在问题中产生两个独立的多项式方程。 另一个单根特征值产生一个线性方程。 我们从$3\times 3$系统中得到三个独立方程。 至关重要的是，我们的框架将具有外尔对称性的三维象限行走推广到仅满足轨道和条件的模型。 它解释了许多研究人员提出的关于轨道和的许多标准，并且也解释了起始于象限外部的格路的反例。 显示英文摘要

摘要： 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.