In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard decomposition, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision.

在本文中，我们探索了局限于正八分音符的三维偏移的渐近枚举。如[29]所示，指数增长和临界指数都分别接受通用公式，分别涉及步骤集的清单和某个球形三角形的主要狄利克雷特征值的本身，其特征在于模型的步骤。我们关注临界指数，我们的主要目标是将步骤集的组合属性（所谓的步态组结构，存在的Hadamard分解，存在的生成函数满足的微分方程）与几何相关联。或相关球形三角形的分析属性（显着角度，平铺属性，主特征值存在一个特殊的封闭形式公式）。通常，球形三角形上的Dirichlet问题的特征值不是以闭合形式已知的，因此，我们还开发了一种有限元方法来计算近似值，通常具有十位数的精度。