  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
factorization, 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.
